Modeling and Forecasting the Spread of COVID-19 with Stochastic and Deterministic Approaches: Africa and Europe

Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 https://doi.org/10.1186/s13662-021-03213-2

R E S E A R C H Open Access Modeling and forecasting the spread of COVID-19 with stochastic and deterministic approaches: Africa and Europe

Abdon Atangana1,2* and Seda ˙Igret˘ Araz3

*Correspondence: [email protected] Abstract 1Institute for Groundwater Studies, Faculty of Natural and Agricultural Using the existing collected data from European and African countries, we present a Sciences, University of the Free statistical analysis of forecast of the future number of daily deaths and infections up to State, Bloemfontein, South Africa 10 September 2020. We presented numerous statistical analyses of collected data 2Department of Medical Research, China Medical University Hospital, from both continents using numerous existing statistical theories. Our predictions China Medical University, Taichung, show the possibility of the second wave of spread in Europe in the worse scenario Taiwan and an exponential growth in the number of infections in Africa. The projection of Full list of author information is available at the end of the article statistical analysis leads us to introducing an extended version of the well-blancmange function to further capture the spread with fractal properties. A mathematical model depicting the spread with nine sub-classes is considered, first converted to a stochastic system, where the existence and uniqueness are presented. Then the model is extended to the concept of nonlocal operators; due to nonlinearity, a modified numerical scheme is suggested and used to present numerical simulations. The suggested mathematical model is able to predict two to three waves of the spread in the near future. Keywords: Statistical analysis; Extended blancmange function; Stochastic model; COVID-19 spread with waves; Modified numerical scheme

1 Introduction Interdisciplinary research is the way forward for mankind to be in control of its environ- ment. Of course they will not be able to have total control since the nature within which they live is full of uncertainties, many complex phenomena that have not been yet under- stood with the current collections of knowledge and technology. For example, we cannot explicitly and confidently explain what is happening at the Bermuda Triangle, although many studies have been done around this place, some believe it is a devil’s triangle. There are many other natural occurrences that could not be explained so far with our knowledge. But it has been proven that putting together several concepts from different academic fields could provide better results. COVID-19 is an invisible enemy that left humans with no choice than to put all their efforts from all backgrounds with the aim to protect the survival of their kind. Many souls have been taken, many humans have been infected and some recovered, but still the spread has not yet reached its peak in many countries. While

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in some countries the curve of daily new infected has nearly reached zero, in others the spread is increasing exponentially. For some statistical analysis, we investigated daily cases of infections and deaths due to the COVID-19 spread that occurred in 54 countries in the European continent and 47 countries in the African continent from the beginning of the outbreak to 15 June 2020. To do this, we used the available data on the website of the World Health Organization (WHO) [1, 2]. Although mathematicians cannot provide vaccine or cure the disease in an infected person, they can use their mathematical tools to foresee what could possibly happen in the near future with some limitations [3–14]. With the new trend of spread, it is possible that the world will face a second wave of COVID-19 spread, this will be the aim of our work. The paper is organized as follows. In Sect. 2, we present the definitions of differential and integral operators where singular and nonsingular kernels are used. In Sect. 3,thepa- rameter estimations are presented for the infected and deaths in Africa and Europe using the Box–Jenkins model. In Sect. 4, the simulations for smoothing method for the infected and deaths in Africa and Europe are presented. In Sect. 5, the predictions about the cases of infections and deaths in Africa and Europe are provided. In Sect. 6,wegiveananaly- sis of COVID-19 spread based on fractal interpolation and fractal dimension. In Sect. 7, existence and uniqueness for a mathematical model with stochastic component are investigated. Also the numerical simulations for such a model are depicted. In Sect. 8,we present a modified scheme based on the Newton polynomial. In Sect. 9, we provide numerical solutions for the suggested COVID-19 model with different differential operators.

2 Differential and integral operators In this section, we present some definitions of differential and integral operators with singular and nonsingular kernels. The fractional derivatives with power-law, exponential decay, and Mittag-Leffler kernel are given as follows:

Deﬁnition 1 t C α 1 d –α 0 Dt f (t)= f (τ)(t – τ) dτ, (1 – α) 0 dτ t CF α M(α) d α 0 Dt f (t)= f (τ) exp – (t – τ) dτ,(1) 1–α 0 dτ 1–α t ABC α AB(α) d α α 0 Dt f (t)= f (τ)Eα – (t – τ) dτ. 1–α 0 dτ 1–α

The fractional integrals with power-law, exponential decay, and Mittag-Leﬄer kernel are given as follows: t C α 1 α–1 0 Jt f (t)= (t – τ) f (τ) dτ, (α) 0 t CF α,β 1–α α 0 Jt f (t)= f (t)+ f (τ) dτ,(2) M(α) M(α) 0 t AB α,β 1–α α α–1 0 Jt f (t)= f (t)+ (t – τ) f (τ) dτ. AB(α) AB(α)(α) 0 Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 3 of 107

The fractal-fractional derivatives with power-law kernel, exponential decay, and Mittag- Leﬄer kernel are given as follows: 1 d t FFP α,β τ τ –α τ 0 Dt f (t)= β f ( )(t – ) d , (1 – α) dt 0 M(α) d t α FFE α,β τ τ τ 0 Dt f (t)= β f ( ) exp – (t – ) d ,(3) 1–α dt 0 1–α AB(α) d t α FFM α,β τ τ α τ 0 Dt f (t)= β f ( )Eα – (t – ) d , 1–α dt 0 1–α

where

df (t) f (t)–f (t ) = lim 1 (2 – β). (4) β → 2–β dt t t1 2–β t – t1

The fractal-fractional integrals with power-law, exponential decay, and Mittag-Leﬄer kernel are as follows: t FFP α,β 1 α–1 1–β 0 Jt f (t)= (t – τ) τ f (τ) dτ, (α) 0 t FFE α,β 1–α 1–β α 1–β 0 Jt f (t)= t f (t)+ τ f (τ) dτ,(5) M(α) M(α) 0 t FFM α,β 1–α 1–β α α–1 1–β 0 Jt f (t)= t f (t)+ (t – τ) τ f (τ) dτ. AB(α) AB(α)(α) 0

3 Box–Jenkin’s model development Autoregressive integrated moving average (ARIMA) approach suggested by Box and Jenk- ins is one of the most powerful techniques used in time series analysis. The ARIMA model is composed of three parts. First, the autoregressive part is a linear regression which has a relation between past values and future values of data series; second, the integrated part expresses how many times the data series has to be diﬀerenced to obtain a stationary series; and the last one is the moving average part which has a relation between past forecast errors and future values of data series [14]. These processes can be presented by the models AR(p), MA(q), ARMA(p, q), and ARIMA(p, d, q). We should decide which model we will choose for our data series. To do this, partial autocorrelation (PACF) and the autocorrelation (ACF) are helpful to obtain parameters for the AR model and the MA model, respectively. Figures 1 and 2 depict graphs of autocorrelation functions for the infected and deaths in Africa and Europe.

Nowweintroducethesemodels.LetYt be the value of the time series at time t.Time series as a p-order autoregressive process is as follows:

Yt = δ + ϕ1Yt–1 + ϕ2Yt–2 + ···+ ϕpYt–p + εt,(6)

which is shown as AR(p). Here, δ and εt describe constant and error terms, respectively. Time series as a qth degree of moving average process is given by

Yt = μ + εt + θ1εt–1 + θ2εt–2 + ···+ θqεt–q,(7) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 4 of 107

Figure 1 Autocorrelation function for the infected and deaths in Africa

Figure 2 Autocorrelation function for the infected and deaths in Europe

which is shown as MA(q). The ARMA(p, q) expression is obtained as a combination of AR(p)andMA(q)equations:

Yt = δ + ϕ1Yt–1 + ϕ2Yt–2 + ···+ ϕpYt–p + εt + θ1εt–1 + θ2εt–2 + ···+ θqεt–q.(8)

When the time series is not stationary, we take the diﬀerence d times to make it stationary. The ARIMA(p, q)modelisgivenby 2 p d 1–ϕ1l – ϕ2l – ···– ϕpl Yt = δ + εt + θ1εt–1 + θ2εt–2 + ···+ θqεt–q.(9)

In the ARIMA technique, the model performance can be measured by using some criteria, for instance, Akaike information criteria(AIC), Bayesian information criteria(BIC). Here, we beneﬁt from the Akaike information criteria given as follows:

AIC =–2log(l)+2k, (10)

BIC =–2log(l)+k ln n,

where l states likelihood of the data, n is the number of data points, and k also deﬁnes the intercept of the ARIMA model. The numerical simulation are depicted in Figs. 3, 4, 5 and 6. Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 5 of 107

According to data series for the infected in Africa, we use the ARIMA(2,1,0) model which is given by

2 1–ϕ1l – ϕ2l (1 – l)Yt = c + εt. (11)

Here,

AIC = 1670.1734, (12)

BIC = 1680.9388.

In Table 1, we give parameter estimation for infections in Africa. According to data series for deaths in Africa, we use the AR(1) model which is given by

(1 – ϕ1l)Yt = c + εt. (13)

Here,

AIC = 1056.6482, (14)

BIC = 1064.7768.

In Table 2, we give parameter estimation for deaths in Africa.

Table 1 Model estimation for infections in Africa Parameter Value Standard error TStatistic Constant 89.2032 56.6511 1.5746 AR{1} –0.44796 0.099221 –4.5147 AR{2} –0.17789 0.068294 –2.6047 Variance 168,446.2911 12,738.3089 13.2236

Figure 3 ARIMA model for the infected in Africa Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 6 of 107

Table 2 Model estimation for deaths in Africa Parameter Value Standard error TStatistic Constant 12.581 6.0023 2.096 AR{1} 0.75094 0.082701 9.0802 Variance 694.3043 92.168 7.533

Figure 4 AR model for deaths in Africa

Table 3 Model estimation for the infected in Europe

Parameter Value Standard error TStatistic Constant 83.7826 118.7108 0.70577 AR{1} 0.3216 0.59303 0.5423 AR{2} 0.035772 0.16277 0.21977 MA{1} –0.53222 0.58359 –0.91197 Variance 7,214,609.9182 569,786.6944 12.6619

According to data series for the infected in Europe, we use the ARIMA(2,1,1)model which is given by

2 1–ϕ1l – ϕ2l (1 – l)Yt = c +(1+θ1l)εt. (15)

Here,

AIC = 2690.5358, (16)

BIC = 2705.2796.

In Table 3, we give parameter estimation for the infected in Europe. According to data series for deaths in Europe, we use the AR(1) model which is given by

(1 – ϕ1l)Yt = c + εt. (17) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 7 of 107

Figure 5 ARIMA model for the infected in Europe

Table 4 Model estimation for deaths in Europe

Parameter Value Standard error TStatistic Constant 151.4852 163.967 0.92388 AR{1} 0.8865 0.041096 21.5714 Variance 460,062.22 27,485.093 16.7386

Figure 6 AR model for deaths in Europe

Here,

AIC = 1670.1734, (18)

BIC = 1680.9388.

In Table 4, we give parameter estimation for deaths in Europe. Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 8 of 107

4 Brown’s exponential smoothing method Brown’s linear exponential smoothing is one type of double exponential smoothing based on two diﬀerent smoothed series. The formula is composed of an extrapolation of a line through the two centers. The Brown exponential smoothing method is helpful to model the time series having trend but no seasonality. For non-adaptive Brown exponential smoothing, the procedure can be described as follows. Firstly, we start with the following initialization:

1) S0 = u0,

2) T0 = u0,

3) a0 =2S0 – T0,

4) F1 = a0 + b0. Then we have the following calculations:

1) St = αut +(1–α)St–1,

2) Tt = αSt +(1–α)Tt–1,

3) at =2St + Tt63,

4) α(St – Tt)=(1–α)bt,

5) Ft+1 = at + bt,

where 0 < α < 1 is the smoothing factor. St and Tt are the simply smoothed value and

doubly smoothed value for the (t + 1)th time period, respectively. Also at and bt describe the intercept and the slope, respectively. In Figs. 7, 8, 9,and10, we present the simulation for smoothing method for the infected and deaths in Africa and Europe where the smoothing factor was chosen as α =0.99.

5 Future prediction of daily new numbers of the infected and deaths: Africa and Europe With the collected data using some statistical formula, it is possible to predict what will possibly happen in the near future. Having in mind what could possibly happen, several measures could be taken to avoid the worst case scenario. In this section, with the data collected for 101 countries from Africa (47) and Europe (54), we aim at presenting possible

Figure 7 Exponential smoothing for the infected in Africa Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 9 of 107

Figure 8 Exponential smoothing for deaths in Africa

Figure 9 Exponential smoothing for the infected in Europe

Figure 10 Exponential smoothing for deaths in Europe

scenarios or events that could be observed in the near future, the daily numbers of deaths and infections. Numerical simulation are presented in Figs. 11, 12, 13 and 14. In Figs. 15, 16, 17,and18, we present ﬁtting with smoothing spline for the infected and deaths in Africa and Europe. Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 10 of 107

Figure 11 Prediction for the infected in Africa using Forecast Sheet

Figure 12 Prediction for deaths in Africa using Forecast Sheet

6 An analysis of COVID-19 spread based on fractal interpolation and fractal dimension In this section, we present some information about fractal dimension, interpolation, and blancmange curve.

6.1 Fractal dimension Fractal dimensions enable us to compare fractals. Fractal dimensions are important be- cause they can be deﬁned in connection with real-world data, and they can be measured approximately by means of experiments. These numbers allow us to compare sets in the real world with the laboratory fractals. Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 11 of 107

Figure 13 Prediction for the infected in Europe using Forecast Sheet

Figure 14 Prediction for deaths in Europe using Forecast Sheet

Theorem (The box counting theorem) Let Nn(A) be the number of boxes of side length (1/2n). Then the fractal dimension D of A is given as [15] ln[N (A)] D = lim n . (19) n→∞ ln(2n)

6.2 Fractal interpolation Euclidean geometry and calculus enable us to model using some lines and curves, the shapes that we encounter in the nature [15, 16]. In this section, we present an interpolation function which interpolates the data. Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 12 of 107

Figure 15 Fitting for the infected in Africa

Figure 16 Fitting for deaths in Africa

Figure 17 Fitting for the infected in Europe

Figure 18 Fitting for deaths in Europe Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 13 of 107

Deﬁnition 2 An interpolation function f :[x0, xN ] → R correspondingtothesetofdata 2 {(xi, Fi) ∈ R : i =0,1,2,...,N} [15]

f (xi)=Fi for i =1,2,...,N, (20)

where x0 < x1 < x2 ···< xN .

Let f :[x0, xN ] → R denote the unique continuous function which is called a piece- wise linear interpolation function. Also this function is linear on each of the subintervals

[xi–1, xi], and it is represented by

(x – xi–1) f (x)=Fi–1 + (Fi – Fi–1)forx ∈ [xi–1, xi], i =1,2,...,N. (21) (xi – xi–1)

We have the following transformation, which is iterated: x tn 0 x vn fn = + . (22) y un yn y wn

When solving this system for tn, un, vn,andwn in terms of the data and yn,weobtainthe following:

xn – xn–1 tn = , xN – x0

Fn – Fn–1 Fn – F0 un = – yn , (23) xN – x0 xN – x0 xN xn–1 – x0xn vn = , xN – x0

xN Fn–1 – x0Fn xN F0 – x0Fn wn = – yn , xN – x0 xN – x0

where 0 ≤ yn < 1 is called the scaling factor [15].

6.3 Blancmange curve The blancmange function can be given as an example of fractal interpolation function, andthisfunctionisdeﬁnedby

∞ S(2nx) , x ∈ [0, 1], (24) 2n n=0

where S(x)=minm∈Z |x – m|, x ∈ R. However, many problems cannot be depicted when c =2[16]. Then we discuss the limitations of this blancmange; for example, t can only go from 0 to 1, the periodic parameter is 2. Therefore, we change 2 to c,wherec is a real number from 1 to a. Therefore, in this section, we extend the blancmange function to a large interval also with any given periodic parameter. So, we have the following formula:

∞ S(cnx) , x ∈ [0, a], (25) cn n=0 Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 14 of 107

Figure 19 Blancmange function c =2

Figure 20 Blancmange function c =3.7

Figure 21 Blancmange function c =1.3

where c is the real number. We now present the extended blancmange function for diﬀer- ent periodic parameters and diﬀerent w. The simulation are presented in Figs. 19, 20, 21,and18.

7 Mathematical model for COVID-19 outbreak We consider the following mathematical model of COVID-19 spread:

· S = – δ(t) αI + w(βID + γ IA + δ1IR + δ2IT )+γ1 + μ1 S, Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 15 of 107

· I = δ(t) αI + w(βID + γ IA + δ1IR + δ2IT ) S –(ε + ξ + λ + μ1)I, · IA = ξI –(θ + μ + χ + μ1)IA, · ID = εI –(η + ϕ + μ1)ID, · IR = ηID + θIA –(v + ξ + μ1)IR, (26) · IT = μIA + vIR –(σ + τ + μ1)IT , · R = λI + ϕID + χIA + ξIR + σ IT –( + μ1)R, · D = τIT , · V = γ1S + R – μ1V.

The above model was suggested by Atangana and Seda, the model has a deterministic character. In this section, we convert the model to a stochastic one by introducing the ef- fect of environmental white noise. To achieve this, we reformulate the model by adding the nonlinear perturbation into each equation of the system. The perturbation may de-

pend on square of the classes S, I, IA, ID, IR, IT , R, D,andV respectively. Here, we perturb only the rate of each class. However, for the vaccine class, it will be perturbed by a natural death rate.

· For the class S(t): –γ1 → –γ1 +(11S + 12)B1(t), · For the class I(t): –λ → –λ +(21I + 22)B2(t), · For the class IA(t): –θ → –θ +(31IA + 32)B3(t), · For the class ID(t): –η → –η +(41ID + 42)B4(t), · For the class IR(t): –v → –v +(51IR + 52)B5(t), · For the class IT (t): –σ → –σ +(61IT + 62)B6(t), · For the class R(t): – → – +(71R + 72)B7(t), For the class D(t): τ → τ no change, · For the class V(t): –μ1 → –μ1 +(81V + 82)B8(t).

Therefore, the associated stochastic model is given as follows: dS = – δ(t) αI + w(βID + γ IA + δ1IR + δ2IT )+γ1 + μ1 S dt

+(11S + 12)SdB1(t), dI = δ(t) αI + w(βID + γ IA + δ1IR + δ2IT ) S –(ε + ξ + λ + μ1)I dt

+(21I + 22)IdB2(t), dIA = ξI –(θ + μ + χ + μ1)IA dt +(31IA + 32)IA dB3(t), Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 16 of 107

dID = εI –(η + ϕ + μ1)ID dt +(41ID + 42)ID dB4(t), (27) dIR = ηID + θIA –(v + ξ + μ1)IR dt +(51IR + 52)ID dB5(t), dIT = μIA + vIR –(σ + τ + μ1)IT dt +(61IT + 62)IT dB6(t), dR = λI + ϕID + χIA + ξIR + σ IT –( + μ1)R dt +(71R + 72)RdB7(t),

dV =[γ1S + R – μ1V] dt ++(71V + 72)VdB8(t).

In this conversion, the function Bi(t) represents the standard Brownian motions valid

within the set of probability (, A, {At}t≥0, P), where {At}t≥0 is ﬁltration valid under the

condition described in [17]. Here, i,j∈[1,2,3,4,5,6,7,8] are positive and are the intensities of the environmental random disturbance.

7.1 Existence and uniqueness In this subsection, we present the existence and uniqueness of the system solutions of the stochastic model. To achieve the existence and uniqueness, we convert the system into Volterra type. But ﬁrst we do the following for simplicity:

dS = F1(t, S, I, IA, ID, IR, IT , R, V) dt + G1(t, S) dB1(t),

dI = F2(t, S, I, IA, ID, IR, IT , R, V) dt + G2(t, I) dB2(t),

dIA = F3(t, I, IA) dt + G3(t, IA) dB3(t),

dID = F4(t, I, ID,)dt + G4(t, ID) dB4(t), (28)

dIR = F5(t, IA, ID, IR) dt + G5(t, IR) dB5(t),

dIT = F6(t, IA, IR, IT ) dt + G6(t, IT ) dB6(t),

dR = F7(t, I, IA, ID, IR, IT , R) dt + G7(t, R) dB7(t),

dV = F8(t, S, R, V) dt + G8(t, V) dB8(t).

Therefore, converting to Volterra, we get

t t S(t)=S(0) + F1(τ, S, I, IA, ID, IR, IT , R, V) dτ + G1(τ, S) dB1(τ), 0 0 t t I(t)=I(0) + F2(τ, S, I, IA, ID, IR, IT , R, V) dτ + G2(τ, I) dB2(τ), 0 0 t t IA(t)=IA(0) + F3(τ, I, IA) dτ + G3(τ, IA) dB3(τ), 0 0 t t ID(t)=ID(0) + F4(τ, I, ID) dτ + G4(τ, ID) dB4(τ), 0 0 t t IR(t)=IR(0) + F5(τ, IA, ID, IR) dτ + G5(τ, IR) dB5(τ), (29) 0 0 t t IT (t)=IT (0) + F6(τ, IA, IR, IT ) dτ + G6(τ, IT ) dB6(τ), 0 0 Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 17 of 107

t t R(t)=R(0) + F7(τ, I, IA, ID, IR, IT , R) dτ + G7(τ, R) dB7(τ), 0 0 t t V(t)=V(0) + F8(τ, S, R, V) dτ + G8(τ, S) dB8(τ). 0 0

We present the existence and uniqueness of the stochastic system of COVID-19 model. This will be achieved via the following theorem.

Theorem Assume that there exist positive constants Ki, K i such that (i) 2 2 Fi(x, t)–Fi(xi, t) < Ki|x – xi| , (30) 2 2 Gi(x, t)–Gi(xi, t) < K i|x – xi|

(ii) ∀(x, t) ∈ R8 × [0, T] 2 2 2 Fi(x, t) , Gi(x, t) < K 1+|x| . (31)

Then there exists a unique solution X(t) ∈ R8 for our model and it belongs to M2([0, T], R8).

Theproofcanbefoundin[17], but we have to verify (i) and (ii) for our system. Without

loss of generality, we start our investigation with functions F1(t, S, I, IA, ID, IR, IT , R, V)and

G1(t, S). For the function F, the proof will be performed for (t, S). Thus 2 2 F1(t, S)–F1(t, S1) = δ(t) αI + w(βID + γ IA + δ1IR + δ2IT )+γ1 + μ1 (S – S1) . (32)

We deﬁne the following norm:

2 ϕ∞ = sup |ϕ| , (33) t∈[0,T]

then 2 2 F1(S, t)–F1(S1, t) ≤ sup δ(t) αI + w(βID + γ IA + δ1IR + δ2IT ) (S – S1) t∈[0,T] ≤ 2 | |2 δ(t) αI + w(βID + γ IA + δ1IR + δ2IT ) ∞ S – S1 (34)

2 ≤ K1|S – S1|

and 2 2 G1(S, t)–G1(S1, t) = (11S + 12)S –(11S1 + 12)S1 2 2 2 = 11 S – S1 – 12(S – S1) 2 2 = 11(S + S1)+12 |S – S1| 2 2 2 | |2 = 11(S + S1) +21112(S + S1)+12 S – S1 2 2 2 2 | |2 = 11 S +2SS1 + S1 +21112(S + S1)+12 S – S1 (35) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 18 of 107

⎧ ⎫ ⎪ sup |S2(t)| +2sup |S(t)| sup |S (t)| ⎪ ⎨2 t∈[0,T] t∈[0,T] t∈[0,T] 1 ⎬ ≤ 11 + sup |S2(t)| ⎪ t∈[0,T] 1 ⎪ ⎩⎪ 2 ⎭⎪ +21112 supt∈[0,T] S(t) + supt∈[0,T] S1(t) + 12

×|S – S |2 1 2 2 (S2∞ +2S∞S ∞ + S ∞) ≤ 11 1 1 |S – S |2 2 1 +21112 S ∞ S1 ∞ + 12

2 ≤ K 1|S – S1| ,

where 2 2 2 2 K 1 = 11 S ∞ +2 S ∞ S1 ∞ + S1 ∞ +21112 S ∞ S1 ∞ + 12 (36) 2 2 2 = 11 S ∞ + S1 ∞ +21112 S ∞ S1 ∞ + 12.

Similarly, 2 2 2 K 2 = 21 I ∞ + I1 ∞ +22122 I ∞ I1 ∞ + 22, 2 2 2 K 3 = 31 IA ∞ + IA1 ∞ +23132 IA ∞ IA1 ∞ + 32, 2 2 2 K 4 = 41 ID ∞ + ID1 ∞ +24142 ID ∞ ID1 ∞ + 42, 2 2 2 K 5 = 51 IR ∞ + IR1 ∞ +25152 IR ∞ IR1 ∞ + 52, (37) 2 2 2 K 6 = 61 IT ∞ + IT1 ∞ +26162 IT ∞ IT1 ∞ + 62, 2 2 2 K 7 = 71 R ∞ + R1 ∞ +27172 R ∞ R1 ∞ + 72, 2 2 2 K 8 = 81 V ∞ + V1 ∞ +28182 V ∞ V1 ∞ + 82.

Also 2 2 F2(I, t)–F2(I1, t) = δ(t)α(I – I1)–(ε + ξ + λ + μ1)(I – I1) 2 = δ(t)α –(ε + ξ + λ + μ1) (I – I1) 2 2 ≤ sup δ(t)α –(ε + ξ + λ + μ1) |I – I1| (38) t∈[0,T] ≤ 2| |2 δ(t) ∞ α –(ε + ξ + λ + μ1) I – I1 2 ≤ K2|I – I1| ,

where 2 K2 = δ(t) ∞ α –(ε + ξ + λ + μ1) . (39)

Also 2 2 F3(IA, t)–F3(IA1, t) = –(θ + μ + χ + μ1)(IA – IA1) 2 2 ≤ 2 (θ + μ + χ + μ1) |IA – IA1| (40) 2 ≤ K3|IA – IA1| , Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 19 of 107

where 2 K3 =2(θ + μ + χ + μ1) . (41)

Similarly, we evaluate 2 2 2 F4(ID, t)–F4(ID1, t) = |η + ϕ + μ1| |ID – ID1| (42)

≤ K6|IT – IT1|, 2 2 2 F7(R, t)–F7(R1, t) = | + μ1| |R – R1|

≤ K7|R – R1|, 2 2 2 F8(V, t)–F8(V1, t) = |μ1| |V – V1|

2 ≤ K8|V – V1| .

For both classes Gi and Fi, we have veriﬁed condition (i). Now we verify the second condition. 2 2 F1(S, t) = – δ(t) αI + w(βID + γ IA + δ1IR + δ2IT )+γ1 + μ1 S 2 ≤ S – δ(t) αI + w(βID + γ IA + δ1IR + δ2IT )+γ1 + μ1 S 2 2 ≤|S| – δ(t) αI + w(βID + γ IA + δ1IR + δ2IT )+γ1 + μ1 2 2 < |S| +1 – δ(t) αI + w(βID + γ IA + δ1IR + δ2IT )+γ1 + μ1 (43) 2 2 < |S| +1 – δ(t) αI + w(βID + γ IA + δ1IR + δ2IT )+γ1 + μ1 2 2 < |S| +1 sup – δ(t) αI + w(βID + γ IA + δ1IR + δ2IT )+γ1 + μ1 t∈[0,T] < K 1 |S|2 +1 ,

where 1 2 K = sup – δ(t) αI + w(βID + γ IA + δ1IR + δ2IT )+γ1 + μ1 . (44) t∈[0,T]

Then 2 2 G1(S, t)–G1(S1, t) = (11S + 12)S 2 22 ≤ 11S + 12S 2 22 ≤ (11 + 12) S (45) 2 2 2 ≤ (11 + 12) sup S |S| t∈[0,T] Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 20 of 107

≤ 2 2 | |2 (11 + 12) S ∞ S +1 1 ≤ K |S|2 +1 ,

where

1 2 2 K =(11 + 12) S ∞. (46)

Similarly,

2 2 2 K =(21 + 22) I ∞, 3 2 2 K =(31 + 32) IA ∞, 4 2 2 K =(41 + 42) ID ∞, 5 2 2 K =(51 + 52) IR ∞, (47) 6 2 2 K =(61 + 62) IT ∞, 7 2 2 K =(71 + 72) R ∞, 8 2 2 K =(81 + 82) V ∞.

Also, we have 2 2 F2(I, t) = δ(t) w(βID + γ IA + δ1IR + δ2IT ) S + δ(t)αIS –(ε + ξ + λ + μ1)I 2 ≤ δ(t) w(βID + γ IA + δ1IR + δ2IT ) S + δ(t)αS –(ε + ξ + λ + μ1) |I| | |2 < S +1 sup δ(t) w(βID + γ IA + δ1IR + δ2IT ) S + δ(t)αS (48) t∈[0,T] 2 –(ε + ξ + λ + μ1) < K 2 |I|2 +1 , 2 2 F3(IA, t) = ξI –(θ + μ + χ + μ1)IA 2 2 ≤ ξI –(θ + μ + χ + μ1) |IA| 2 2 ≤ |IA| +1 sup ξI –(θ + μ + χ + μ1) t∈[0,T] 3 2 ≤ K |IA| +1 , 2 2 F4(IA, t) = εI –(η + ϕ + μ1)ID 2 2 ≤ |ID| +1 sup εI –(η + ϕ + μ1) t∈[0,T] 4 2 ≤ K |ID| +1 , 2 2 2 F5(IR, t) ≤ |IR| +1 sup ηID + θIA –(v + ξ + μ1) t∈[0,T] 5 2 ≤ K |IR| +1 , Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 21 of 107

2 2 2 F6(IT , t) ≤ |IT | +1 sup μIA + vIR –(σ + τ + μ1) t∈[0,T] 6 2 ≤ K |IT | +1 , 2 2 2 F6(IT , t) ≤ |IT | +1 sup μIA + vIR –(σ + τ + μ1) t∈[0,T] 6 2 ≤ K |IT | +1 , 2 2 2 F7(R, t) ≤ |R| +1 sup λI + ϕID + χIA + ξIR + σ IT –( + μ1) t∈[0,T] ≤ K 7 |R|2 +1 .

Finally, we have 2 2 2 F8(V, t) ≤ |V| +1 sup |γ1S + R – μ1| t∈[0,T] ≤ K 8 |V|2 +1 .

Both Gi and Fi verify the second condition. Therefore, according to the above theorem, the system has a unique system solution.

7.2 Numerical simulation for the stochastic model Numerical solutions of the suggested stochastic model are presented in Figs. 22–25.The numerical solution depicts the future stochastic behavior of the susceptible class, ﬁve sub- classes of the infected population, the recovered class, the death class, and the vaccination class. These are depicted in ﬁgures below.

8 Atangana–Seda modified scheme The mathematical model considered in this work has the ability to depict two to three waves of COVID-19 spread. The model is subjected to a system of initial conditions. Ad- ditionally, the model is nonlinear, thus it is impossible to obtain exact solutions to the system, thus numerical schemes are needed. We present a numerical scheme based on the Newton polynomial [18]. However, one needs the initial condition and two additional components for the scheme to be implemented. In this section, we present a modified version that will not need the two additional components, and then the scheme will be used later to provide numerical solutions for the suggested COVID-19 model with different differential operators. We start with the classical case, the following is considered:

dy(t) = f t, y(t) . (49) dt

Then 5 4 5 yn+1 = yn + f t , yn–2 – f t , yn–1 + f t , yn t. (50) 12 n–2 3 n–1 12 n

To reduce these requirements, we proceed as follows:

yn – yn–1 = f t , yn ⇒ yn–1 = yn – f t , yn t. (51) t n n Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 22 of 107

Figure 22 Stochastic behavior of S(t)andI(t) classes

On the other hand,

yn–1 – yn–2 = f t , yn–1 (52) t n–1 or n–2 n–1 n–1 y = y – f tn–1, y t (53) n n n n = y – tf tn, y – tf tn–1, y – f tn, y t .

Replacing yn–2 and yn–1 with their values, we obtain

5 yn+1 = yn + tf t , yn – tf t , yn – tf t , yn – f t , yn t (54) 12 n–2 n n–1 n 4 23 – f t , yn – f t , yn t + f t , yn t. 3 n–1 n 12 n

The above does not need y1 and y2, only the initial condition. With the Caputo–Fabrizio derivative, we consider the following: CF α 0 Dt y(t)=f t, y(t) . (55) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 23 of 107

Figure 23 Stochastic behavior of IA(t)andID(t) classes

From the deﬁnition of the Caputo–Fabrizio integral, we can reformulate the above equation as follows: 1–α α t y(t)–y(0) = f t, y(t) + f τ, y(τ) dτ. (56) M(α) M(α) 0

We have, at the point tn+1 =(n +1) t, 1–α α tn+1 y(tn+1)–y(0) = f tn+1, y(tn+1) + f τ, y(τ) dτ, (57) M(α) M(α) 0

and at the point tn = n t, 1–α α tn y(tn)–y(0) = f tn, y(tn) + f τ, y(τ) dτ. (58) M(α) M(α) 0

Taking the diﬀerence of these equations, we can write the following:

1–α y(t )–y(t )= f t , y(t ) – f t , y(t ) (59) n+1 n M(α) n+1 n+1 n n α tn+1 + f τ, y(τ) dτ M(α) tn Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 24 of 107

Figure 24 Stochastic behavior of IR(t)andIT (t) classes 1–α f (t , y(t )– tf (t , y(t ))) = n+1 n n n M(α) – f (tn, y(tn)) ⎧ ⎫ ⎨⎪ 5 f (t , yn – tf (t , yn)– tf (t , yn – f (t , yn) t)) t⎬⎪ α 12 n–2 n n–1 n + – 4 f (t , yn – f (t , yn) t) t . α ⎪ 3 n–1 n ⎪ M( ) ⎩ 23 n ⎭ + 12 f (tn, y ) t

With the Caputo derivative, we write ⎧ ⎨C α 0 Dt y(t)=f (t, y(t)), ⎩ (60) y(0) = y0.

We convert the above into 1 t y(t)–y(0) = f τ, y(τ) (t – τ)α–1 dτ. (61) (α) 0

At the point tn+1 =(n +1) t, we have the following: tn+1 1 α–1 y(tn+1)–y(0) = f τ, y(τ) (tn+1 – τ) dτ, (α) 0 Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 25 of 107

Figure 25 Stochastic behavior of R(t)andV(t) classes

and we write n 1 tj+1 y(t )=y(0) + f τ, y(τ) (t – τ)α–1 dτ. n+1 (α) n+1 j=2 tj

After putting the Newton polynomial into the above equation, the above equation can be written as follows: ( t)α n yn+1 = y0 + f t , yj–2 (n – j +1)α –(n – j)α (α +1) j–2 j=2 ( t)α n + f t , yj–1 – f t , yj–2 (α +2) j–1 j–2 j=2 (n – j +1)α(n – j +3+2α) × (62) α α –(n – j) (n– j +3+3 ) n α( t)α f (t , yj)–2f (t , yj–1) + j j–1 2(α +3) + f (t , yj–2) ⎡ j=2 j–2 ⎤ 2(n – j)2 +(3α + 10)(n – j) ⎢(n – j +1)α ⎥ ⎢ 2 ⎥ ⎢ +2α +9α +12 ⎥ × ⎢ ⎥ , ⎣ 2(n – j)2 +(5α + 10)(n – j) ⎦ –(n – j)α +6α2 +18α +12 Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 26 of 107

where j–1 j j f tj–1, y = f tj–1, y – f tj, y t , (63) j–2 j j j j f tj–2, y = f tj–2, y – tf tj, y – tf tj–1, y – f tj, y t .

With Atangana–Baleanu, we have ⎧ ⎨ABC α 0 Dt y(t)=f (t, y(t)), ⎩ (64) y(0) = y0.

We transform the above equation into 1–α α t y(t)–y(0) = f t, y(t) + f τ, y(τ) (t – τ)α–1 dτ. (65) AB(α) AB(α)(α) 0

At the point tn+1 =(n +1) t, we have the following:

1–α y(t )–y(0) = f t, y(t) (66) n+1 AB(α) tn+1 α α–1 + f τ, y(τ) (tn+1 – τ) dτ, AB(α)(α) 0

and we write

1–α y(t )=y(0) + f t , yn+1 (67) n+1 AB(α) n+1 n α tj+1 + f τ, y(τ) (t – τ)α–1 dτ. AB(α)(α) n+1 j=2 tj

After putting the Newton polynomial into the above equation, the above equation can be written as follows:

1–α yn+1 = y0 + f t , y(t )– tf t , y(t ) (68) AB(α) n+1 n n n n α( t)α t , yj – tf (t , yj) + f j–2 j AB(α)(α +1) – tf (t , yj – f (t , yj) t) j=2 j–1 j × (n – j +1)α –(n – j)α n α( t)α f (t , yj – f (t , yj) t) + j–1 j AB(α)(α +2) – f (t , yj – tf (t , yj)– tf (t , yj – f (t , yj) t)) j=2 j–2 j j–1 j α × (n – j +1) (n – j +3+2α) –(n – j)α(n – j +3+3α) ⎡ ⎤ j j–1 n f (tj, y )–2f (tj–1, y ) α( t)α ⎢ ⎥ + ⎣ t , yj – tf (t , yj) ⎦ 2AB(α)(α +3) + f j–2 j j=2 j j – tf (tj–1, y – f (tj, y ) t) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 27 of 107

⎡ ⎤ 2(n – j)2 +(3α + 10)(n – j) ⎢(n – j +1)α ⎥ ⎢ 2 ⎥ ⎢ +2α +9α +12 ⎥ × ⎢ ⎥ . ⎣ 2(n – j)2 +(5α + 10)(n – j) ⎦ –(n – j)α +6α2 +18α +12

With the Caputo–Fabrizio fractal-fractional derivative, we consider FFE α,β 0 Dt y(t)=f t, y(t) , (69)

y(0) = y0.

Applying the associated integral operator with exponential kernel, we can reformulate equation (69) as follows: 1–α α t y(t)= t1–β f t, y(t) + f τ, y(τ) τ 1–β dτ. (70) M(α) M(α) 0

At the point tn+1 =(n +1) t, tn+1 1–α 1–β α 1–β y(tn+1)= tn+1 f tn+1, y(tn+1) + f τ, y(τ) τ dτ, (71) M(α) M(α) 0

and at the point tn = n t,wehave tn 1–α 1–β α 1–β y(tn)= tn f tn, y(tn) + f τ, y(τ) τ dτ. (72) M(α) M(α) 0

If we take the diﬀerence of these equations, we obtain the following equation: 1–β 1–α tn+1 f (tn+1, y(tn+1)) y(tn+1)–y(tn)= 1–β (73) M(α) – tn f (tn, y(tn)) α tn+1 + f τ, y(τ) τ 1–β dτ. M(α) tn

For brevity, we consider

1–α y(t )–y(t )= F t , y(t ) – F t , y(t ) (74) n+1 n M(α) n+1 n+1 n n α tn+1 + F τ, y(τ) dτ, M(α) tn

where F t, y(t) = f t, y(t) t1–β . (75)

We can rearrange the above scheme as follows: 1–α F(t , y(t )– tf (t , y(t ))) yn+1 – yn = n+1 n n n (76) M(α) – F(tn, y(tn)) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 28 of 107

⎧ ⎫ ⎨⎪ 5 F(t , yn – tf (t , yn)– tF(t , yn – f (t , yn) t)) t⎬⎪ α 12 n–2 n n–1 n + – 4 F(t , yn – f (t , yn) t) t . α ⎪ 3 n–1 n ⎪ M( ) ⎩ 23 n ⎭ + 12 F(tn, y ) t

If we replace F(t, y(t)) with its value, we can solve our equation numerically with the following scheme: 1–β n+1 n 1–α tn+1 f (tn+1, y(tn)– tf (tn, y(tn))) y – y = 1–β (77) M(α) – tn f (tn, y(tn)) ⎧ ⎫ ⎨⎪t1–β 5 F(t , yn – tf (t , yn)– tf (t , yn – f (t , yn) t)) t⎬⎪ α n–2 12 n–2 n n–1 n + – 4 t1–β f (t , yn – f (t , yn) t) t . α ⎪ 3 n–1 n–1 n ⎪ M( ) ⎩ 23 1–β n ⎭ + 12 tn f (tn, y ) t

With the Atangana–Baleanu fractal-fractional derivative, we write FFM α,β 0 Dt y(t)=f t, y(t) , (78)

y(0) = y0.

Applying the new fractional integral with Mittag-Leﬄer kernel, we transform the above equation into

1–α y(t)=y(0) + t1–β f t, y(t) (79) AB(α) α t + f τ, y(τ) (t – τ)α–1τ 1–β dτ. AB(α)(α) 0

At the point tn+1 =(n +1) t, we obtain the following:

1–α y(t )=y(0) + t1–β f t , y(t ) (80) n+1 AB(α) n+1 n+1 n+1 tn+1 α α–1 1–β + f τ, y(τ) (tn+1 – τ) τ dτ. AB(α)(α) 0

For simplicity, we shall take F t, y(t) = f t, y(t) t1–β . (81)

We also have

1–α y(t )=y(0) + F t , y(t ) (82) n+1 AB(α) n+1 n+1 n α tj+1 + F τ, y(τ) (t – τ)α–1 dτ. AB(α)(α) n+1 j=2 tj

Replacing them into the above equation and substituting F(t, y(t)) = f (t, y(t))t1–β ,wecan get the following numerical scheme:

1–α yn+1 = t1–β f t , y(t ) (83) AB(α) n+1 n+1 n+1 Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 29 of 107

n α( t)α t , yj – tf (t , yj) + t1–β f j–2 j AB(α)(α +1) j–2 – tf (t , yj – f (t , yj) t) j=2 j–1 j × (n – j +1)α –(n – j)α ⎡ ⎤ 1–β j j n tj–1 f (tj–1, y – f (tj, y ) t) α( t)α ⎢ ⎥ + ⎣ t , yj – tf (t , yj) ⎦ AB(α)(α +2) – t1–β f j–2 j j=2 j–2 j j – tf (tj–1, y – f (tj, y ) t) α × (n – j +1) (n – j +3+2α) –(n – j)α(n – j +3+3α) ⎡ ⎤ t1–β g(t , yj) ⎢ j j ⎥ α n ⎢ 1–β j j ⎥ α( t) ⎢ –2tj–1 f (tj–1, y – f (tj, y ) t) ⎥ + ⎢ ⎥ 2AB(α)(α +3) ⎣ j j ⎦ j=2 1–β tj–2, y – tf (tj, y ) + tj–2 f j j – tf (tj–1, y – f (tj, y ) t) ⎡ ⎤ 2(n – j)2 +(3α + 10)(n – j) ⎢(n – j +1)α ⎥ ⎢ 2 ⎥ ⎢ +2α +9α +12 ⎥ × ⎢ ⎥ . ⎣ 2(n – j)2 +(5α + 10)(n – j) ⎦ –(n – j)α +6α2 +18α +12

With the Caputo fractal-fractional derivative, we consider the following:

FFP α,β 0 Dt y(t)=f t, y(t) , (84)

y(0) = y0.

Applying the new fractional integral with power-law kernel, we transform the above equation into 1 t y(t)=y(0) + f τ, y(τ) (t – τ)α–1τ 1–β dτ. (85) (α) 0

At the point tn+1 =(n +1) t, we obtain the following:

tn+1 1 α–1 1–β y(tn+1)=y(0) + f τ, y(τ) (tn+1 – τ) τ dτ. (86) (α) 0

For simplicity, we shall take

F t, y(t) = f t, y(t) t1–β . (87)

We also have

n 1 tj+1 y(t )=y(0) + F τ, y(τ) (t – τ)α–1 dτ. (88) n+1 (α) n+1 j=2 tj Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 30 of 107

Replacing them into the above equation and substituting F(t, y(t)) = f (t, y(t))t1–β ,wecan get the following numerical scheme: n ( t)α t , yj – tf (t , yj) yn+1 = t1–β f j–2 j (89) (α +1) j–2 – tf (t , yj – f (t , yj) t) j=2 j–1 j × (n – j +1)α –(n – j)α ⎡ ⎤ 1–β j j n tj–1 f (tj–1, y – f (tj, y ) t) ( t)α ⎢ ⎥ + ⎣ t , yj – tf (t , yj) ⎦ (α +2) – t1–β f j–2 j j=2 j–2 j j – tf (tj–1, y – f (tj, y ) t) α × (n – j +1) (n – j +3+2α) –(n – j)α(n – j +3+3α) ⎡ ⎤ t1–β g(t , yj) ⎢ j j ⎥ α n ⎢ 1–β j j ⎥ ( t) ⎢ –2tj–1 f (tj–1, y – f (tj, y ) t) ⎥ + ⎢ ⎥ 2(α +3) ⎣ j j ⎦ j=2 1–β tj–2, y – tf (tj, y ) + tj–2 f j j – tf (tj–1, y – f (tj, y ) t) ⎡ ⎤ 2(n – j)2 +(3α + 10)(n – j) ⎢(n – j +1)α ⎥ ⎢ 2 ⎥ ⎢ +2α +9α +12 ⎥ × ⎢ ⎥ . ⎣ 2(n – j)2 +(5α + 10)(n – j) ⎦ –(n – j)α +6α2 +18α +12

Finally, we present the numerical scheme with fractal-fractional derivative with variable order. We start with the Caputo–Fabrizio case: FFE α,β(t) 0 Dt y(t)=f t, y(t) , (90)

y(0) = y0.

The above equation can be reformulated as follows: 1–α 2–β(t) y(t)= t2–β(t) –β (t) ln(t)+ f t, y(t) (91) M(α) t α t 2–β(τ) + f τ, y(τ) β (τ) ln(τ)+ τ 2–β(τ) dτ. M(α) 0 τ

We write the above equation as follows: 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) t (– ln tn+1 + )f (tn+1, y(tn+1)) 1–α n+1 t tn+1 y(tn+1)–y(tn)= (92) M(α) – t2–β(tn)(– β(tn+1)–β(tn) ln t + 2–β(tn) )f (t , y(t )) n t n tn n n α tn+1 2–β(τ) + f τ, y(τ) β (τ) ln(τ)+ τ 2–β(τ) dτ. M(α) tn τ

For simplicity, we take 2–β(t) F t, y(t) = f t, y(t) –β (t) ln(t)+ t2–β(t), (93) t Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 31 of 107

and we have

1–α y(t )–y(t )= F t , y(t ) – F t , y(t ) (94) n+1 n M(α) n+1 n+1 n n α tn+1 + F τ, y(τ) dτ. M(α) tn

If we do the same routine and replace F(t, y(t)) with its value, we have the following numerical approximation: ⎡ ⎤ 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) t (– ln tn+1 + ) ⎢ n+1 t tn+1 ⎥ ⎢ ⎥ 1–α ⎢ × f (tn+1, y(tn)– tf (tn, y(tn))) ⎥ yn+1 = yn + ⎢ ⎥ (95) ⎢ 2–β(t ) β(t )–β(t ) 2–β(t ) ⎥ M(α) – t n (– n+1 n ln t + n ) ⎣ n t n tn ⎦

× f (tn, y(tn)) ⎧ ⎫ ⎪ 23 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎪ ⎪ tn (– ln tn + ) ⎪ ⎪ 12 t tn ⎪ ⎪ × 23 n ⎪ ⎪ f (tn, y ) t ⎪ ⎪ 12 ⎪ ⎪ 4 2–β(tn–1) β(tn)–β(tn–1) 2–β(tn–1) ⎪ ⎨⎪ – 3 tn–1 (– t ln tn–1 + t ) ⎬⎪ α n–1 × n n + ⎪ f (tn–1, y – f (tn, y ) t) t ⎪ . M(α) ⎪ ⎪ ⎪ 5 2–β(tn–2) β(tn–1)–β(tn–2) 2–β(tn–2) ⎪ ⎪ + t (– ln tn–2 + )⎪ ⎪ 12 n–2 t tn–2 ⎪ ⎪ n n ⎪ ⎪ tn–2, y – tf (tn, y ) ⎪ ⎪ × 5 f t ⎪ ⎩ 12 n n ⎭ – tf (tn–1, y – f (tn, y ) t)

We deal with our problem involving the new constant fractional order and variable fractal dimension FFM α,β(t) 0 Dt y(t)=f t, y(t) , (96)

y(0) = y0,

where the kernel is the Mittag-Leﬄer kernel. If we integrate the above equation with the new integral operator including the Mittag-Leﬄer kernel, the above equation can be converted to 1–α 2–β(t) y(t)= t2–β(t) –β (t) ln(t)+ f t, y(t) AB(α) t α t + f τ, y(τ) (t – τ)α–1 (97) AB(α)(α) 0 2–β(τ) × –β (τ) ln(τ)+ τ 2–β(τ) dτ. τ

At the point tn+1 =(n +1) t, we have the following: % & 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) y(tn+1)= tn+1 – ln tn+1 + AB(α) t tn+1 × f tn+1, y(tn+1) (98) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 32 of 107

α tn+1 + f τ, y(τ) (t – s)α–1 AB(α)(α) n+1 0 2–β(τ) × –β (τ) ln(τ)+ τ 2–β(τ) dτ. τ

For brevity, we consider 2–β(τ) F τ, y(τ) = f τ, y(τ) –β (τ) ln(τ)+ τ 2–β(τ), (99) τ

and we can write the following: % & 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) y(tn+1)= tn+1 – ln tn+1 + (100) AB(α) t tn+1 × f tn+1, y(tn+1) n α tj+1 + F τ, y(τ) (t – τ)α–1 dτ. AB(α)(α) n+1 j=2 tj

One can replace the Newton polynomial in the above equation as follows. Thus, we have the following scheme: % & n+1 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) y = tn+1 – ln tn+1 + (101) AB(α) t tn+1 × f tn+1, y(tn+1) α( t)α n + F t , yj–2 (n – j +1)α –(n – j)α AB(α)(α +1) j–2 j=2

α( t)α n + F t , yj–1 – F t , yj–2 AB(α)(α +2) j–1 j–2 j=2 α × (n – j +1) (n – j +3+2α) –(n – j)α(n – j +3+3α)

α( t)α n + F t , yj –2F t , yj–1 + F t , yj–2 2AB(α)(α +3) j j–1 j–2 j=2 ⎡ ⎤ 2(n – j)2 +(3α + 10)(n – j) ⎢(n – j +1)α ⎥ ⎢ 2 ⎥ ⎢ +2α +9α +12 ⎥ × ⎢ ⎥ . ⎣ 2(n – j)2 +(5α + 10)(n – j) ⎦ –(n – j)α +6α2 +18α +12

Replacing the function G(t, y(t)) with its value, we can present the following scheme for numerical solution of our equation: % & n+1 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) y = tn+1 – ln tn+1 + AB(α) t tn+1 n n × f tn+1, y + f tn, y t (102) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 33 of 107

α n α( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) + t j–2 – ln t + AB(α)(α +1) j–2 t j–2 t j=2 j–2 j j × tj–2, y – tf (tj, y ) α α f j j (n – j +1) –(n – j) – tf (tj–1, y – f (tj, y ) t) ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t [– ln tj–1 + ] ⎢ j–1 t tj–1 ⎥ ⎢ j j ⎥ ⎢ × f (tj–1, y – f (tj, y ) t) ⎥ α n ⎢ ⎥ α( t) ⎢ ⎥ + 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎢ – t [– ln tj–2 + ]⎥ AB(α)(α +2) ⎢ j–2 t tj–2 ⎥ j=2 ⎢ ⎥ ⎣ j j ⎦ × tj–2, y – tf (tj, y ) f j j – tf (tj–1, y – f (tj, y ) t) α × (n – j +1) (n – j +3+2α) –(n – j)α(n – j +3+3α) ⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t [– ln tj + ] ⎢ j t tj ⎥ ⎢ ⎥ ⎢ × f (t , yj) t ⎥ ⎢ j ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t [– ln tj–1 + ]⎥ α n ⎢ j–1 t tj–1 ⎥ α( t) ⎢ ⎥ + ⎢ × f (t , yj – f (t , yj) t) ⎥ 2AB(α)(α +3) ⎢ j–1 j ⎥ j=2 ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ + t [– ln tj–2 + ]⎥ ⎢ j–2 t tj–2 ⎥ ⎢ ⎥ ⎣ j j ⎦ × tj–2, y – tf (tj, y ) f j j – tf (tj–1, y – f (tj, y ) t) ⎡ ⎤ 2(n – j)2 +(3α + 10)(n – j) ⎢(n – j +1)α ⎥ ⎢ 2 ⎥ ⎢ +2α +9α +12 ⎥ × ⎢ ⎥ . ⎣ 2(n – j)2 +(5α + 10)(n – j) ⎦ –(n – j)α +6α2 +18α +12

We deal with our problem involving the new constant fractional order and variable fractal dimension

FFP α,β(t) 0 Dt y(t)=f t, y(t) , (103)

y(0) = y0,

wherethekernelisthepower-lawkernel.Ifweintegrateequation(103)withthenew integral operator including the power-law kernel, the above equation can be converted to 1 t 2–β(τ) y(t)= f τ, y(τ) (t – τ)α–1 –β (τ) ln(τ)+ τ 2–β(τ) dτ. (104) (α) 0 τ

At the point tn+1 =(n +1) t, we have the following: 1 tn+1 y(t )= f τ, y(τ) (t – s)α–1 (105) n+1 (α) n+1 0 2–β(τ) × –β (τ) ln(τ)+ τ 2–β(τ) dτ. τ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 34 of 107

For brevity, we consider

2–β(τ) F τ, y(τ) = f τ, y(τ) –β (τ) ln(τ)+ τ 2–β(τ), (106) τ

and we can write the following:

n 1 tj+1 y(t )= F τ, y(τ) (t – τ)α–1 dτ. (107) n+1 (α) n+1 j=2 tj

Thus, we have the following scheme:

( t)α n yn+1 = F t , yj–2 (n – j +1)α –(n – j)α (108) (α +1) j–2 j=2

( t)α n + F t , yj–1 – F t , yj–2 (α +2) j–1 j–2 j=2 α × (n – j +1) (n – j +3+2α) –(n – j)α(n – j +3+3α)

( t)α n + F t , yj –2F t , yj–1 + F t , yj–2 2(α +3) j j–1 j–2 j=2 ⎡ ⎤ 2(n – j)2 +(3α + 10)(n – j) ⎢(n – j +1)α ⎥ ⎢ 2 ⎥ ⎢ +2α +9α +12 ⎥ × ⎢ ⎥ . ⎣ 2(n – j)2 +(5α + 10)(n – j) ⎦ –(n – j)α +6α2 +18α +12

Replacing the function G(t, y(t)) with its value, we can present the following scheme for numerical solution of our equation:

α n ( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) yn+1 = t j–2 – ln t + (α +1) j–2 t j–2 t j=2 j–2 j j × tj–2, y – tf (tj, y ) α α f j j (n – j +1) –(n – j) – tf (tj–1, y – f (tj, y ) t) ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t [– ln tj–1 + ] ⎢ j–1 t tj–1 ⎥ ⎢ j j ⎥ ⎢ × f (tj–1, y – f (tj, y ) t) ⎥ α n ⎢ ⎥ ( t) ⎢ ⎥ + 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) (109) ⎢ – t [– ln tj–2 + ]⎥ (α +2) ⎢ j–2 t tj–2 ⎥ j=2 ⎢ ⎥ ⎣ j j ⎦ × tj–2, y – tf (tj, y ) f j j – tf (tj–1, y – f (tj, y ) t) α × (n – j +1) (n – j +3+2α) –(n – j)α(n – j +3+3α) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 35 of 107

⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t [– ln tj + ] ⎢ j t tj ⎥ ⎢ ⎥ ⎢ × f (t , yj) t ⎥ ⎢ j ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t [– ln tj–1 + ]⎥ α n ⎢ j–1 t tj–1 ⎥ ( t) ⎢ ⎥ + ⎢ × f (t , yj – f (t , yj) t) ⎥ 2(α +3) ⎢ j–1 j ⎥ j=2 ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ + t [– ln tj–2 + ]⎥ ⎢ j–2 t tj–2 ⎥ ⎢ ⎥ ⎣ j j ⎦ × tj–2, y – tf (tj, y ) f j j – tf (tj–1, y – f (tj, y ) t) ⎡ ⎤ 2(n – j)2 +(3α + 10)(n – j) ⎢(n – j +1)α ⎥ ⎢ 2 ⎥ ⎢ +2α +9α +12 ⎥ × ⎢ ⎥ . ⎣ 2(n – j)2 +(5α + 10)(n – j) ⎦ –(n – j)α +6α2 +18α +12

9 Application to COVID-19 model In this section, using the suggested numerical scheme, we present its application to solve the mathematical model of COVID-19 with possibility of waves. The numerical scheme will be applied for all cases where the differential operators are with classical differential operators, modern fractional differential operators, and variable orders, although only few examples will be used for numerical simulations. Firstly, we shall use the Caputo–Fabrizio fractional derivative CF α ∗ ∗ ∗ ∗ ∗ 0 Dt S = – δ(t) αI + wβID + γ wIA + wδ1IR + wδ2IT + γ1 + μ1 S, CF α ∗ ∗ ∗ ∗ ∗ 0 Dt I = δ(t) αI + wβID + γ wIA + wδ1IR + wδ2IT S –(ε + ξ + λ + μ1)I, CF α 0 Dt IA = ξI –(θ + μ + χ + μ1)IA, CF α 0 Dt ID = εI –(η + ϕ + μ1)ID, CF α 0 Dt IR = ηID + θIA –(v + ξ + μ1)IR, (110) CF α 0 Dt IT = μIA + vIR –(σ + τ + μ1)IT , CF α 0 Dt R = λI + ϕID + χIA + ξIR + σ IT –( + μ1)R, CF α 0 Dt D = τIT , CF α 0 Dt V = γ1S + R – μ1V.

For simplicity, we rearrange the above equation as follows:

CF α ∗ 0 Dt S = S (t, S, I, IA, ID, IR, IT , R, D, V), CF α ∗ 0 Dt I = I (t, S, I, IA, ID, IR, IT , R, D, V), CF α ∗ 0 Dt IA = IA(t, S, I, IA, ID, IR, IT , R, D, V), CF α ∗ 0 Dt ID = ID(t, S, I, IA, ID, IR, IT , R, D, V), CF α ∗ 0 Dt IR = IR(t, S, I, IA, ID, IR, IT , R, D, V), (111) CF α ∗ 0 Dt IT = IT (t, S, I, IA, ID, IR, IT , R, D, V), Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 36 of 107

CF α ∗ 0 Dt R = R (t, S, I, IA, ID, IR, IT , R, D, V),

CF α ∗ 0 Dt D = D (t, S, I, IA, ID, IR, IT , R, D, V),

CF α ∗ 0 Dt V = V (t, S, I, IA, ID, IR, IT , R, D, V).

Thus, we can have the following scheme for our model:

⎡ ⎛ ⎞⎤ t , Sn + tSn∗, In + tIn∗, In + tIn∗, ⎢ ⎜ n+1 A A ⎟⎥ ⎢ ∗ ⎜ n n∗ n n∗ n n∗ ⎟⎥ 1–α ⎢S ⎝ I + tI , I + tI , I + tI , ⎠⎥ Sn+1 = Sn + ⎢ D D R R T T ⎥ (112) M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV ∗ n n n n n n n n n – S tn, S , I , IA, ID, IR, IT , R , D , V ⎧ ⎫ ⎪ 23 S∗(t , Sn, In, In, In , In, In , Rn, Dn, V n) t ⎪ ⎪ 12⎛ n A D R T ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – 4 S In – tIn , In – tIn , In – tIn , t ⎪ ⎪ 3 ⎝ D D R R T T ⎠ ⎪ ⎪ ∗ ∗ ∗ ⎪ ⎨⎪ Rn – tRn , Dn – tDn , V n – tV n ⎬⎪ α ⎛ ⎞ + t , Sn – tSn∗ – tS(n–1)∗, In – tIn∗ – tI(n–1)∗, , ⎪ ⎜ n–2 ⎟ ⎪ M(α) ⎪ ∗ ∗ ∗ ∗ ⎪ ⎪ ⎜ In – tIn – tI(n–1) , In – tIn – tI(n–1) , ⎟ ⎪ ⎪ ⎜ A A A D D D ⎟ ⎪ ⎪ 5 ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + S ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 ⎜ R R R T T T ⎟ ⎪ ⎪ ⎜ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎟ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ t , Sn + tSn∗, In + tIn∗, In + tIn∗, ⎢ ⎜ n+1 A A ⎟⎥ ⎢ ∗ ⎜ n n∗ n n∗ n n∗ ⎟⎥ 1–α ⎢I ⎝ I + tI , I + tI , I + tI , ⎠⎥ In+1 = Sn + ⎢ D D R R T T ⎥ (113) M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV ∗ n n n n n n n n n – I tn, S , I , IA, ID, IR, IT , R , D , V ⎧ ⎫ ⎪ 23 I∗(t , Sn, In, In, In , In, In , Rn, Dn, V n) t ⎪ ⎪ 12⎛ n A D R T ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – 4 I In – tIn , In – tIn , In – tIn , t ⎪ ⎪ 3 ⎝ D D R R T T ⎠ ⎪ ⎪ ∗ ∗ ∗ ⎪ ⎨⎪ Rn – tRn , Dn – tDn , V n – tV n ⎬⎪ α ⎛ ⎞ + t , Sn – tSn∗ – tS(n–1)∗, In – tIn∗ – tI(n–1)∗, , ⎪ ⎜ n–2 ⎟ ⎪ M(α) ⎪ ∗ ∗ ∗ ∗ ⎪ ⎪ ⎜ In – tIn – tI(n–1) , In – tIn – tI(n–1) , ⎟ ⎪ ⎪ ⎜ A A A D D D ⎟ ⎪ ⎪ 5 ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + I ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 ⎜ R R R T T T ⎟ ⎪ ⎪ ⎜ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎟ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ t , Sn + tSn∗, In + tIn∗, In + tIn∗, ⎢ ⎜ n+1 A A ⎟⎥ ⎢ ∗ ⎜ n n∗ n n∗ n n∗ ⎟⎥ 1–α ⎢IA ⎝ I + tI , I + tI , I + tI , ⎠⎥ In+1 = In + ⎢ D D R R T T ⎥ (114) A A M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV ∗ n n n n n n n n n – IA tn, S , I , IA, ID, IR, IT , R , D , V Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 37 of 107

⎧ ⎫ ⎪ 23 I∗ (t , Sn, In, In, In , In, In , Rn, Dn, V n) t ⎪ ⎪ 12⎛A n A D R T ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – 4 I In – tIn , In – tIn , In – tIn , t ⎪ ⎪ 3 A ⎝ D D R R T T ⎠ ⎪ ⎪ ∗ ∗ ∗ ⎪ ⎨⎪ Rn – tRn , Dn – tDn , V n – tV n ⎬⎪ α ⎛ ⎞ + t , Sn – tSn∗ – tS(n–1)∗, In – tIn∗ – tI(n–1)∗, , ⎪ ⎜ n–2 ⎟ ⎪ M(α) ⎪ ∗ ∗ ∗ ∗ ⎪ ⎪ ⎜ In – tIn – tI(n–1) , In – tIn – tI(n–1) , ⎟ ⎪ ⎪ ⎜ A A A D D D ⎟ ⎪ ⎪ 5 ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + I ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 A ⎜ R R R T T T ⎟ ⎪ ⎪ ⎜ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎟ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ t , Sn + tSn∗, In + tIn∗, In + tIn∗, ⎢ ⎜ n+1 A A ⎟⎥ ⎢ ∗ ⎜ n n∗ n n∗ n n∗ ⎟⎥ 1–α ⎢ID ⎝ I + tI , I + tI , I + tI , ⎠⎥ In+1 = In + ⎢ D D R R T T ⎥ (115) D D M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV ∗ n n n n n n n n n – ID tn, S , I , IA, ID, IR, IT , R , D , V ⎧ ⎫ ⎪ 23 I∗ (t , Sn, In, In, In , In, In , Rn, Dn, V n) t ⎪ ⎪ 12⎛D n A D R T ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – 4 I In – tIn , In – tIn , In – tIn , t ⎪ ⎪ 3 D ⎝ D D R R T T ⎠ ⎪ ⎪ ∗ ∗ ∗ ⎪ ⎨⎪ Rn – tRn , Dn – tDn , V n – tV n ⎬⎪ α ⎛ ⎞ + t , Sn – tSn∗ – tS(n–1)∗, In – tIn∗ – tI(n–1)∗, , ⎪ ⎜ n–2 ⎟ ⎪ M(α) ⎪ ∗ ∗ ∗ ∗ ⎪ ⎪ ⎜ In – tIn – tI(n–1) , In – tIn – tI(n–1) , ⎟ ⎪ ⎪ ⎜ A A A D D D ⎟ ⎪ ⎪ 5 ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + I ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 D ⎜ R R R T T T ⎟ ⎪ ⎪ ⎜ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎟ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ t , Sn + tSn∗, In + tIn∗, In + tIn∗, ⎢ ⎜ n+1 A A ⎟⎥ ⎢ ∗ ⎜ n n∗ n n∗ n n∗ ⎟⎥ 1–α ⎢IR ⎝ I + tI , I + tI , I + tI , ⎠⎥ In+1 = In + ⎢ D D R R T T ⎥ (116) R R M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV ∗ n n n n n n n n n – IR tn, S , I , IA, ID, IR, IT , R , D , V ⎧ ⎫ ⎪ 23 I∗(t , Sn, In, In, In , In, In , Rn, Dn, V n) t ⎪ ⎪ 12⎛R n A D R T ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – 4 I In – tIn , In – tIn , In – tIn , t ⎪ ⎪ 3 R ⎝ D D R R T T ⎠ ⎪ ⎪ ∗ ∗ ∗ ⎪ ⎨⎪ Rn – tRn , Dn – tDn , V n – tV n ⎬⎪ α ⎛ ⎞ + t , Sn – tSn∗ – tS(n–1)∗, In – tIn∗ – tI(n–1)∗, , ⎪ ⎜ n–2 ⎟ ⎪ M(α) ⎪ ∗ ∗ ∗ ∗ ⎪ ⎪ ⎜ In – tIn – tI(n–1) , In – tIn – tI(n–1) , ⎟ ⎪ ⎪ ⎜ A A A D D D ⎟ ⎪ ⎪ 5 ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + I ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 R ⎜ R R R T T T ⎟ ⎪ ⎪ ⎜ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎟ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ t , Sn + tSn∗, In + tIn∗, In + tIn∗, ⎢ ⎜ n+1 A A ⎟⎥ ⎢ ∗ ⎜ n n∗ n n∗ n n∗ ⎟⎥ 1–α ⎢IT ⎝ I + tI , I + tI , I + tI , ⎠⎥ In+1 = In + ⎢ D D R R T T ⎥ (117) T T M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV ∗ n n n n n n n n n – IT tn, S , I , IA, ID, IR, IT , R , D , V Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 38 of 107

⎧ ⎫ ⎪ 23 I∗ (t , Sn, In, In, In , In, In , Rn, Dn, V n) t ⎪ ⎪ 12⎛T n A D R T ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – 4 I In – tIn , In – tIn , In – tIn , t ⎪ ⎪ 3 T ⎝ D D R R T T ⎠ ⎪ ⎪ ∗ ∗ ∗ ⎪ ⎨⎪ Rn – tRn , Dn – tDn , V n – tV n ⎬⎪ α ⎛ ⎞ + t , Sn – tSn∗ – tS(n–1)∗, In – tIn∗ – tI(n–1)∗, , ⎪ ⎜ n–2 ⎟ ⎪ M(α) ⎪ ∗ ∗ ∗ ∗ ⎪ ⎪ ⎜ In – tIn – tI(n–1) , In – tIn – tI(n–1) , ⎟ ⎪ ⎪ ⎜ A A A D D D ⎟ ⎪ ⎪ 5 ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + I ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 T ⎜ R R R T T T ⎟ ⎪ ⎪ ⎜ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎟ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ t , Sn + tSn∗, In + tIn∗, In + tIn∗, ⎢ ⎜ n+1 A A ⎟⎥ ⎢ ∗ ⎜ n n∗ n n∗ n n∗ ⎟⎥ 1–α ⎢R ⎝ I + tI , I + tI , I + tI , ⎠⎥ Rn+1 = Rn + ⎢ D D R R T T ⎥ (118) M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV ∗ n n n n n n n n n – R tn, S , I , IA, ID, IR, IT , R , D , V ⎧ ⎫ ⎪ 23 R∗(t , Sn, In, In, In , In, In , Rn, Dn, V n) t ⎪ ⎪ 12⎛ n A D R T ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – 4 R In – tIn , In – tIn , In – tIn , t ⎪ ⎪ 3 ⎝ D D R R T T ⎠ ⎪ ⎪ ∗ ∗ ∗ ⎪ ⎨⎪ Rn – tRn , Dn – tDn , V n – tV n ⎬⎪ α ⎛ ⎞ + t , Sn – tSn∗ – tS(n–1)∗, In – tIn∗ – tI(n–1)∗, , ⎪ ⎜ n–2 ⎟ ⎪ M(α) ⎪ ∗ ∗ ∗ ∗ ⎪ ⎪ ⎜ In – tIn – tI(n–1) , In – tIn – tI(n–1) , ⎟ ⎪ ⎪ ⎜ A A A D D D ⎟ ⎪ ⎪ 5 ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + R ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 ⎜ R R R T T T ⎟ ⎪ ⎪ ⎜ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎟ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ t , Sn + tSn∗, In + tIn∗, In + tIn∗, ⎢ ⎜ n+1 A A ⎟⎥ ⎢ ∗ ⎜ n n∗ n n∗ n n∗ ⎟⎥ 1–α ⎢D ⎝ I + tI , I + tI , I + tI , ⎠⎥ Dn+1 = Dn + ⎢ D D R R T T ⎥ (119) M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV ∗ n n n n n n n n n – D tn, S , I , IA, ID, IR, IT , R , D , V ⎧ ⎫ ⎪ 23 D∗(t , Sn, In, In, In , In, In , Rn, Dn, V n) t ⎪ ⎪ 12⎛ n A D R T ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – 4 D In – tIn , In – tIn , In – tIn , t ⎪ ⎪ 3 ⎝ D D R R T T ⎠ ⎪ ⎪ ∗ ∗ ∗ ⎪ ⎨⎪ Rn – tRn , Dn – tDn , V n – tV n ⎬⎪ α ⎛ ⎞ + t , Sn – tSn∗ – tS(n–1)∗, In – tIn∗ – tI(n–1)∗, , ⎪ ⎜ n–2 ⎟ ⎪ M(α) ⎪ ∗ ∗ ∗ ∗ ⎪ ⎪ ⎜ In – tIn – tI(n–1) , In – tIn – tI(n–1) , ⎟ ⎪ ⎪ ⎜ A A A D D D ⎟ ⎪ ⎪ 5 ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + D ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 ⎜ R R R T T T ⎟ ⎪ ⎪ ⎜ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎟ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ t , Sn + tSn∗, In + tIn∗, In + tIn∗, ⎢ ⎜ n+1 A A ⎟⎥ ⎢ ∗ ⎜ n n∗ n n∗ n n∗ ⎟⎥ 1–α ⎢V ⎝ I + tI , I + tI , I + tI , ⎠⎥ V n+1 = V n + ⎢ D D R R T T ⎥ (120) M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV ∗ n n n n n n n n n – V tn, S , I , IA, ID, IR, IT , R , D , V Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 39 of 107

⎧ ⎫ ⎪ 23 ∗ n n n n n n n n n ⎪ ⎪ V (tn, S , I , I , ID, IR, IT , R , D , V ) t ⎪ ⎪ 12⎛ A ⎞ ⎪ ⎪ t , Sn – tSn∗, In – tIn∗, In – tIn∗, ⎪ ⎪ n A A ⎪ ⎪ 4 ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – V ⎝ In – tIn , In – tIn , In – tIn , ⎠ t ⎪ ⎪ 3 D D R R T T ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎨ ⎛ R – tR , D – tD , V – tV ⎞ ⎬ α n n∗ (n–1)∗ n n∗ (n–1)∗ + ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ . M(α) ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + 5 V ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 ⎜ R R R T T T ⎟ ⎪ ⎪ ⎝ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎠ ⎪ ⎪ R – tR – tR , D – tD – tD , ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗

With the Atangana–Baleanu fractional derivative, we can solve numerically our model as follows: ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ Sn+1 = S ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ (121) AB(α) D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ j j (j–1) j j (j–1) ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ × S ⎜ I – tI – tI , I – tI – tI , ⎟ × ⎜ R R R T T T ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ α( t)α + AB(α)(α +2) ⎡ ⎛ ∗ ⎞ ⎤ t , Sj – tSj∗, Ij – tIj∗, Ij – tIj , ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ S ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ⎥ ⎢ Rj – tRj∗, Dj – tDj∗, V j – tV j∗ ⎥ ⎢ ⎛ ⎞⎥ n ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ × ⎢ ⎜ ∗ ∗ ∗ ∗ ⎟⎥ × ⎢ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – S ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ R R R T T T ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ α( t)α + 2AB(α)(α +3) ⎡ ⎤ S∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2S∗ ⎝ j j j j j j j j∗ ⎠ ⎥ ⎢ ID – tID , IR – tIR , IT – tIT , R – tR , ⎥ ⎢ ∗ ∗ ⎥ ⎢ Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ × ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 40 of 107

⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ In+1 = I ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ AB(α) D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ I ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ ∗ j j j j j j j j j I (tj, S , I , I , I , I , I , R , D , V ) ⎢ A D R T ⎥ ⎢ j j∗ j j∗ j j∗ j j∗ ⎥ ⎢ ∗ tj–1, S – tS , I – tI , IA – tIA , ID – tID , ⎥ ⎢ –2I j j∗ j j∗ ∗ ∗ ∗ ⎥ ⎢ I – tI , I – tI , Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ R⎛ R T T ⎞ ⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟ ⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟ ⎥ j=2 ⎢ ⎜ A A A D D D ⎟ ⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎥ ⎢ + I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎥ ⎢ ⎜ ⎟ ⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ ⎦ V j – tV j∗ – tV (j–1)∗

× , ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ In+1 = I ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ A AB(α) A D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 41 of 107

⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ IA ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ I∗ (t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛A j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2I∗ ⎝ Ij tIj Ij tIj Ij tIj ⎠ ⎥ ⎢ A D – D , R – R , T – T , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ In+1 = I ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ D AB(α) D D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 42 of 107

⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ ID ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ I∗ (t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛D j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2I∗ ⎝ j j j j j j ⎠ ⎥ ⎢ D ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ In+1 = I ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ R AB(α) R D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ IR ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 43 of 107

⎡ ⎤ I∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛R j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2I∗ ⎝ j j j j j j ⎠ ⎥ ⎢ R ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ In+1 = I ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ T AB(α) T D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ IT ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ I∗ (t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛T j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2I∗ ⎝ Ij tIj Ij tIj Ij tIj ⎠ ⎥ ⎢ T D – D , R – R , T – T , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 44 of 107

⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ Rn+1 = R ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ AB(α) D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ R ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ R∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2R∗ ⎝ j j j j j j ⎠ ⎥ ⎢ ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ Dn+1 = D ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ AB(α) D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 45 of 107

⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ D ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ D∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2D∗ ⎝ Ij tIj Ij tIj Ij tIj ⎠ ⎥ ⎢ D – D , R – R , T – T , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ V n+1 = V ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ AB(α) D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 46 of 107

⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ V ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ V ∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2V ∗ ⎝ j j j j j j ⎠ ⎥ ⎢ ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

where

⎡ ⎤ 2(n – j)2 +(3α + 10)(n – j) ⎢(n – j +1)α ⎥ ⎢ 2 ⎥ ⎢ +2α +9α +12 ⎥ = ⎢ ⎥ , (122) ⎣ 2(n – j)2 +(5α + 10)(n – j) ⎦ –(n – j)α +6α2 +18α +12 (n – j +1)α(n – j +3+2α) = , = (n – j +1)α –(n – j)α . –(n – j)α(n – j +3+3α) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 47 of 107

With the Caputo fractional derivative, we can obtain the following:

⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 ∗ ⎜ j j (j–1) j j (j–1) ⎟ × S = S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (123) (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ S ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ α n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ( t) ⎢ ⎜ j–2 ⎟⎥ + ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ (α +2) ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× ⎡ ⎤ S∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2S∗ ⎝ j j j j j j ⎠ ⎥ ⎢ ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ( t)α ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ + ⎢ ⎜ j–2 ⎟⎥ 2(α +3) ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× , ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 ∗ ⎜ j j (j–1) j j (j–1) ⎟ × I = I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ I ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ α n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ( t) ⎢ ⎜ j–2 ⎟⎥ × + ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ (α +2) ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 48 of 107

⎡ ⎤ I∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2I∗ ⎝ j j j j j j ⎠ ⎥ ⎢ ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ( t)α ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ + ⎢ ⎜ j–2 ⎟⎥ 2(α +3) ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× , ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IA = IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ IA ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ α n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ( t) ⎢ ⎜ j–2 ⎟⎥ + ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ (α +2) ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× ⎡ ⎤ I∗ (t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛A j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2I∗ ⎝ j j j j j j ⎠ ⎥ ⎢ A ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ( t)α ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ + ⎢ ⎜ j–2 ⎟⎥ 2(α +3) ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× , ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 ∗ ⎜ j j (j–1) j j (j–1) ⎟ × ID = ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 49 of 107

( t)α + (α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ ID ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

( t)α + 2(α +3) ⎡ ⎤ ∗ j j j j j j j j j I (tj, S , I , I , I , I , I , R , D , V ) ⎢ D A D R T ⎥ ⎢ j j∗ j j∗ j j∗ j j∗ ⎥ ⎢ ∗ tj–1, S – tS , I – tI , IA – tIA , ID – tID , ⎥ ⎢ –2ID j j∗ j j∗ ∗ ∗ ∗ ⎥ ⎢ I – tI , I – tI , Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ R⎛ R T T ⎞ ⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟ ⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟ ⎥ j=2 ⎢ ⎜ A A A D D D ⎟ ⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎥ ⎢ + ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎥ ⎢ ⎜ ⎟ ⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ ⎦ V j – tV j∗ – tV (j–1)∗

× , ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IR = IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ IR ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ α n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ( t) ⎢ ⎜ j–2 ⎟⎥ × + ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ (α +2) ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 50 of 107

⎡ ⎤ I∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛R j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2I∗ ⎝ j j j j j j ⎠ ⎥ ⎢ R ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ( t)α ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ + ⎢ ⎜ j–2 ⎟⎥ 2(α +3) ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× , ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IT = IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ IT ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ α n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ α( t) ⎢ ⎜ j–2 ⎟⎥ × + ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ (α +2) ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎡ ⎤ I∗ (t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛T j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2I∗ ⎝ Ij tIj Ij tIj Ij tIj ⎠ ⎥ ⎢ T D – D , R – R , T – T , ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ( t)α ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ + ⎢ ⎜ j–2 ⎟⎥ 2(α +3) ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× , ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 ∗ ⎜ j j (j–1) j j (j–1) ⎟ × R = R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 51 of 107

⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ R ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ α n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ( t) ⎢ ⎜ j–2 ⎟⎥ + ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ (α +2) ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× ⎡ ⎤ R∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2R∗ ⎝ j j j j j j ⎠ ⎥ ⎢ ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ( t)α ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ + ⎢ ⎜ j–2 ⎟⎥ 2(α +3) ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× , ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 ∗ ⎜ j j (j–1) j j (j–1) ⎟ × D = D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ D ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ α n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ( t) ⎢ ⎜ j–2 ⎟⎥ + ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ (α +2) ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 52 of 107

⎡ ⎤ D∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2D∗ ⎝ j j j j j j ⎠ ⎥ ⎢ ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ( t)α ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ + ⎢ ⎜ j–2 ⎟⎥ 2(α +3) ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× , ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 ∗ ⎜ j j (j–1) j j (j–1) ⎟ × V = V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ V ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ α n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ( t) ⎢ ⎜ j–2 ⎟⎥ + ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ (α +2) ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× ⎡ ⎤ V ∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2V ∗ ⎝ Ij tIj Ij tIj Ij tIj ⎠ ⎥ ⎢ D – D , R – R , T – T , ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ( t)α ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ + ⎢ ⎜ j–2 ⎟⎥ 2(α +3) ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× .

We now do the same routine for fractal-fractional derivatives. We start with the Caputo– Fabrizio fractal-fractional derivative

FFE α ∗ 0 Dt S = S (t, S, I, IA, ID, IR, IT , R, D, V),

FFE α ∗ 0 Dt I = I (t, S, I, IA, ID, IR, IT , R, D, V), Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 53 of 107

FFE α ∗ 0 Dt IA = IA(t, S, I, IA, ID, IR, IT , R, D, V), FFE α ∗ 0 Dt ID = ID(t, S, I, IA, ID, IR, IT , R, D, V), FFE α ∗ 0 Dt IR = IR(t, S, I, IA, ID, IR, IT , R, D, V), (124) FFE α ∗ 0 Dt IT = IT (t, S, I, IA, ID, IR, IT , R, D, V), FFE α ∗ 0 Dt R = R (t, S, I, IA, ID, IR, IT , R, D, V), FFE α ∗ 0 Dt D = D (t, S, I, IA, ID, IR, IT , R, D, V), FFE α ∗ 0 Dt V = V (t, S, I, IA, ID, IR, IT , R, D, V).

After applying the fractional integral with exponential kernel and putting the Newton polynomial into these equations, we can solve our model as follows: ⎡ ⎛ ⎞⎤ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎢ ⎜ A A ⎟⎥ 1–α ⎢t1–β S∗ ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠⎥ Sn+1 = Sn + ⎢ n+1 D D R R T T ⎥ (125) M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV 1–β ∗ n n n n n n n n n – tn S tn, S , I , IA, ID, IR, IT , R , D , V α + M(α) ⎧ ⎫ ⎪ 23 t1–β S∗(t , Sn, In, In, In , In, In , Rn, Dn, V n) t ⎪ ⎪ 12 n ⎛ n A D R T ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn–1, S – tS , I – tI , IA – tIA , ⎪ ⎪ 1–β ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – 4 t S ⎝ In – tIn , In – tIn , In – tIn , ⎠ t ⎪ ⎪ 3 n–1 D D R R T T ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎨ ⎛ R – tR , D – tD , V – tV ⎞ ⎬ × t , Sn – tSn∗ – tS(n–1)∗, In – tIn∗ – tI(n–1)∗, , ⎪ ⎜ n–2 ⎟ ⎪ ⎪ ⎜ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ ∗ ∗ ⎪ ⎪ + 5 t1–β S∗ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ t⎪ ⎪ 12 n–2 ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎪ ⎪ ⎜ ∗ ∗ ∗ ∗ ⎟ ⎪ ⎪ ⎝ Rn – tRn – tR(n–1) , Dn – tDn – tD(n–1) , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ 1–α In+1 = Sn + (126) M(α) ⎡ ⎤ t , Sn + tSn∗, In + tIn∗, In + tIn∗, In + tIn∗, ⎢t1–β I∗ n+1 A A D D ⎥ ⎣ n+1 n n∗ n n∗ n n∗ n n∗ n n∗ ⎦ IR + tIR , IT + tIT , R + tR , D + tD ,V + tV 1–β ∗ n n n n n n n n n – tn I tn, S , I , IA, ID, IR, IT , R , D , V α + M(α) ⎧ ⎫ ⎪ 23 t1–β I∗(t , Sn, In, In, In , In, In , Rn, Dn, V n) t ⎪ ⎪ 12 n ⎛ n A D R T ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn–1, S – tS , I – tI , IA – tIA , ⎪ ⎪ 1–β ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – 4 t I ⎝ In – tIn , In – tIn , In – tIn , ⎠ t ⎪ ⎪ 3 n–1 D D R R T T ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎨ ⎛ R – tR , D – tD , V – tV ⎞ ⎬ × t , Sn – tSn∗ – tS(n–1)∗, In – tIn∗ – tI(n–1)∗, , ⎪ ⎜ n–2 ⎟ ⎪ ⎪ ⎜ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ ∗ ∗ ⎪ ⎪ + 5 t1–β I∗ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ t⎪ ⎪ 12 n–2 ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎪ ⎪ ⎜ ∗ ∗ ∗ ∗ ⎟ ⎪ ⎪ ⎝ Rn – tRn – tR(n–1) , Dn – tDn – tD(n–1) , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 54 of 107

⎡ ⎛ ⎞⎤ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎢ ⎜ A A ⎟⎥ ⎢ 1–β ∗ ⎝ n n∗ n n∗ n n∗ ⎠⎥ 1–α ⎢tn+1 IA I + tI , I + tI , I + tI , ⎥ In+1 = In + ⎢ D D R R T T ⎥ (127) A A M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV 1–β ∗ n n n n n n n n n – tn IA tn, S , I , IA, ID, IR, IT , R , D , V α + M(α) ⎧ ⎫ ⎪ 23 1–β ∗ n n n n n n n n n ⎪ ⎪ tn IA(tn, S , I , IA, ID, IR, IT , R , D , V ) t ⎪ ⎪ 12 ⎛ ⎞ ⎪ ⎪ t , Sn – tSn∗, In – tIn∗, In – tIn∗, ⎪ ⎪ n–1 A A ⎪ ⎪ 4 1–β ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – t I ⎝ In – tIn , In – tIn , In – tIn , ⎠ t ⎪ ⎪ 3 n–1 A D D R R T T ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎨ ⎛ R – tR , D – tD , V – tV ⎞ ⎬ × n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ , ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ 1–β ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + 5 t I ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 n–2 A ⎜ R R R T T T ⎟ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎢ ⎜ A A ⎟⎥ ⎢ 1–β ∗ ⎝ n n∗ n n∗ n n∗ ⎠⎥ 1–α ⎢tn+1 ID I + tI , I + tI , I + tI , ⎥ In+1 = In + ⎢ D D R R T T ⎥ (128) D D M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV 1–β ∗ n n n n n n n n n – tn ID tn, S , I , IA, ID, IR, IT , R , D , V α + M(α) ⎧ ⎫ ⎪ 23 1–β ∗ n n n n n n n n n ⎪ ⎪ tn ID(tn, S , I , IA, ID, IR, IT , R , D , V ) t ⎪ ⎪ 12 ⎛ ⎞ ⎪ ⎪ t , Sn – tSn∗, In – tIn∗, In – tIn∗, ⎪ ⎪ n–1 A A ⎪ ⎪ 4 1–β ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – t I ⎝ In – tIn , In – tIn , In – tIn , ⎠ t ⎪ ⎪ 3 n–1 D D D R R T T ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎨ ⎛ R – tR , D – tD , V – tV ⎞ ⎬ × n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ , ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ 1–β ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + 5 t I ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 n–2 D ⎜ R R R T T T ⎟ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎢ ⎜ A A ⎟⎥ ⎢ 1–β ∗ ⎝ n n∗ n n∗ n n∗ ⎠⎥ 1–α ⎢tn+1 IR I + tI , I + tI , I + tI , ⎥ In+1 = In + ⎢ D D R R T T ⎥ (129) R R M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV 1–β ∗ n n n n n n n n n – tn IR tn, S , I , IA, ID, IR, IT , R , D , V α + M(α) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 55 of 107

⎧ ⎫ ⎪ 23 1–β ∗ n n n n n n n n n ⎪ ⎪ tn IR(tn, S , I , IA, ID, IR, IT , R , D , V ) t ⎪ ⎪ 12 ⎛ ⎞ ⎪ ⎪ t , Sn – tSn∗, In – tIn∗, In – tIn∗, ⎪ ⎪ n–1 A A ⎪ ⎪ 4 1–β ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – t I ⎝ In – tIn , In – tIn , In – tIn , ⎠ t ⎪ ⎪ 3 n–1 R D D R R T T ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎨ ⎛ R – tR , D – tD , V – tV ⎞ ⎬ × n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ , ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ 1–β ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + 5 t I ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 n–2 R ⎜ R R R T T T ⎟ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎢ ⎜ A A ⎟⎥ ⎢ 1–β ∗ ⎝ n n∗ n n∗ n n∗ ⎠⎥ 1–α ⎢tn+1 IT I + tI , I + tI , I + tI , ⎥ In+1 = In + ⎢ D D R R T T ⎥ (130) T T M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV 1–β ∗ n n n n n n n n n – tn IT tn, S , I , IA, ID, IR, IT , R , D , V α + M(α) ⎧ ⎫ ⎪ 23 1–β ∗ n n n n n n n n n ⎪ ⎪ tn IT (tn, S , I , IA, ID, IR, IT , R , D , V ) t ⎪ ⎪ 12 ⎛ ⎞ ⎪ ⎪ t , Sn – tSn∗, In – tIn∗, In – tIn∗, ⎪ ⎪ n–1 A A ⎪ ⎪ 4 1–β ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – t I ⎝ In – tIn , In – tIn , In – tIn , ⎠ t ⎪ ⎪ 3 n–1 T D D R R T T ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎨ ⎛ R – tR , D – tD , V – tV ⎞ ⎬ × n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ , ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ 1–β ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + 5 t I ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 n–2 T ⎜ R R R T T T ⎟ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎢ ⎜ A A ⎟⎥ ⎢ 1–β ∗ ⎝ n n∗ n n∗ n n∗ ⎠⎥ 1–α ⎢tn+1 R I + tI , I + tI , I + tI , ⎥ Rn+1 = Rn + ⎢ D D R R T T ⎥ (131) M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV 1–β ∗ n n n n n n n n n – tn R tn, S , I , IA, ID, IR, IT , R , D , V α + M(α) ⎧ ⎫ ⎪ 23 1–β ∗ n n n n n n n n n ⎪ ⎪ tn R (tn, S , I , IA, ID, IR, IT , R , D , V ) t ⎪ ⎪ 12 ⎛ ⎞ ⎪ ⎪ t , Sn – tSn∗, In – tIn∗, In – tIn∗, ⎪ ⎪ n–1 A A ⎪ ⎪ 4 1–β ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – t R ⎝ In – tIn , In – tIn , In – tIn , ⎠ t ⎪ ⎪ 3 n–1 D D R R T T ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎨ ⎛ R – tR , D – tD , V – tV ⎞ ⎬ × n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ , ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ 1–β ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + 5 t R ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 n–2 ⎜ R R R T T T ⎟ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 56 of 107

⎡ ⎛ ⎞⎤ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎢ ⎜ A A ⎟⎥ ⎢ 1–β ∗ ⎝ n n∗ n n∗ n n∗ ⎠⎥ 1–α ⎢tn+1 D I + tI , I + tI , I + tI , ⎥ Dn+1 = Dn + ⎢ D D R R T T ⎥ (132) M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV 1–β ∗ n n n n n n n n n – tn D tn, S , I , IA, ID, IR, IT , R , D , V α + M(α) ⎧ ⎫ ⎪ 23 1–β ∗ n n n n n n n n n ⎪ ⎪ tn D (tn, S , I , IA, ID, IR, IT , R , D , V ) t ⎪ ⎪ 12 ⎛ ⎞ ⎪ ⎪ t , Sn – tSn∗, In – tIn∗, In – tIn∗, ⎪ ⎪ n–1 A A ⎪ ⎪ 4 1–β ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – t D ⎝ In – tIn , In – tIn , In – tIn , ⎠ t ⎪ ⎪ 3 n–1 D D R R T T ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎨ ⎛ R – tR , D – tD , V – tV ⎞ ⎬ × n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ , ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ 1–β ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + 5 t D ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 n–2 ⎜ R R R T T T ⎟ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎛ ⎞⎤ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎢ ⎜ A A ⎟⎥ ⎢ 1–β ∗ ⎝ n n∗ n n∗ n n∗ ⎠⎥ 1–α ⎢tn+1 V I + tI , I + tI , I + tI , ⎥ V n+1 = V n + ⎢ D D R R T T ⎥ (133) M(α) ⎣ n n∗ n n∗ n n∗ ⎦ R + tR , D + tD , V + tV 1–β ∗ n n n n n n n n n – tn V tn, S , I , IA, ID, IR, IT , R , D , V α + M(α) ⎧ ⎫ ⎪ 23 1–β ∗ n n n n n n n n n ⎪ ⎪ tn V (tn, S , I , IA, ID, IR, IT , R , D , V ) t ⎪ ⎪ 12 ⎛ ⎞ ⎪ ⎪ t , Sn – tSn∗, In – tIn∗, In – tIn∗, ⎪ ⎪ n–1 A A ⎪ ⎪ 4 1–β ∗ ⎜ ∗ ∗ ∗ ⎟ ⎪ ⎪ – t V ⎝ In – tIn , In – tIn , In – tIn , ⎠ t ⎪ ⎪ 3 n–1 D D R R T T ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎨ ⎛ R – tR , D – tD , V – tV ⎞ ⎬ × n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ . ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ 1–β ∗ ⎜ ∗ (n–1)∗ ∗ (n–1)∗ ⎟ ⎪ ⎪ + 5 t V ⎜ In – tIn – tI , In – tIn – tI , ⎟ t⎪ ⎪ 12 n–2 ⎜ R R R T T T ⎟ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ ⎝ R – tR – tR , D – tD – tD , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗

For the Atangana–Baleanu fractal-fractional derivative, we can have the following numerical scheme:

⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ Sn+1 = t1–β S ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ (134) AB(α) n+1 D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 57 of 107

⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × tj–2 S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 S ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ t1–β S∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β S∗ ⎝ Ij tIj Ij tIj Ij tIj ⎠ ⎥ ⎢ j–1 D – D , R – R , T – T , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ In+1 = t1–β I ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ AB(α) n+1 D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × tj–2 I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 58 of 107

⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 I ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ t1–β I∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β I∗ ⎝ j j j j j j ⎠ ⎥ ⎢ j–1 ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ In+1 = t1–β I ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ A AB(α) n+1 A D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × tj–2 IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 IA ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 59 of 107

⎡ ⎤ t1–β I∗ (t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛A j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β I∗ ⎝ j j j j j j ⎠ ⎥ ⎢ j–1 A ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ In+1 = t1–β I ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ D AB(α) n+1 D D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × tj–2 ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 ID ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ t1–β I∗ (t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛D j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β I∗ ⎝ Ij tIj Ij tIj Ij tIj ⎠ ⎥ ⎢ j–1 D D – D , R – R , T – T , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 60 of 107

⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ In+1 = t1–β I ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ R AB(α) n+1 R D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × tj–2 IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 IR ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ t1–β I∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛R j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β I∗ ⎝ j j j j j j ⎠ ⎥ ⎢ j–1 R ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA 1–α ∗ ⎜ ⎟ In+1 = t1–β I ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ T AB(α) n+1 T D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 61 of 107

⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × tj–2 IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 IT ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ t1–β I∗ (t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛T j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β I∗ ⎝ Ij tIj Ij tIj Ij tIj ⎠ ⎥ ⎢ j–12 T D – D , R – R , T – T , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ Rn+1 = t1–β R ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ AB(α) n+1 D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × tj–2 R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 62 of 107

⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 R ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ t1–β R∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β R∗ ⎝ j j j j j j ⎠ ⎥ ⎢ j–1 ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ Dn+1 = t1–β D ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ AB(α) n+1 D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × tj–2 D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 D ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 63 of 107

⎡ ⎤ t1–β D∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β D∗ ⎝ j j j j j j ⎠ ⎥ ⎢ j–1 ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , IA + tIA , 1–α ∗ ⎜ ⎟ V n+1 = t1–β V ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠ AB(α) n+1 D D R R T T Rn + tRn∗, Dn + tDn∗, V n + tV n∗ α( t)α + AB(α)(α +1) ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × tj–2 V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 V ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) ⎡ ⎤ t1–β V ∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β V ∗ ⎝ Ij tIj Ij tIj Ij tIj ⎠ ⎥ ⎢ j–1 D – D , R – R , T – T , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ . ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 64 of 107

For the power-law kernel, we can have the following:

( t)α Sn+1 = (135) (α +1) ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ j j (j–1) j j (j–1) ⎟ n ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × tj–2 S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

( t)α + (α +2) ⎛ ⎞ ⎡ ∗ ⎤ t , Sj – tSj∗, Ij – tIj∗, Ij – tIj , ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ 1–β ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ tj–1 S ⎝ I – tI , I – tI , I – tI , ⎠ ⎥ ⎢ D D R R T T ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ⎢ ∗ ∗ ∗ ∗ ⎥ n ⎢ t , Sj – tSj – tS(j–1) , Ij – tIj – tI(j–1) , ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , ID – tID – tID , ⎟⎥ j=2 ⎢ ⎜ A A A ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

( t)α + 2(α +3) ⎡ ⎤ t1–β S∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ –2tj–1 S ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ × ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ × , ⎢ ⎜ ⎟⎥ j=2 ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ∗ ∗ ∗ ∗ ⎟⎥ ⎢ + t1–β S∗ ⎜ j j (j–1) j j (j–1) ⎟⎥ ⎢ j–2 ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × I = tj–2 I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

( t)α + (α +2) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 65 of 107

⎛ ⎞ ⎡ ∗ ⎤ t , Sj – tSj∗, Ij – tIj∗, Ij – tIj , ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ 1–β ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ tj–1 I ⎝ I – tI , I – tI , I – tI , ⎠ ⎥ ⎢ D D R R T T ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ⎢ ∗ ∗ ∗ ∗ ⎥ n ⎢ t , Sj – tSj – tS(j–1) , Ij – tIj – tI(j–1) , ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , ID – tID – tID , ⎟⎥ j=2 ⎢ ⎜ A A A ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

( t)α + 2(α +3) ⎡ ⎤ t1–β I∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ –2tj–1 I ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ × ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ × , ⎢ ⎜ ⎟⎥ j=2 ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ∗ ∗ ∗ ∗ ⎟⎥ ⎢ + t1–β I∗ ⎜ j j (j–1) j j (j–1) ⎟⎥ ⎢ j–2 ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IA = tj–2 IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

( t)α + (α +2) ⎛ ⎞ ⎡ ∗ ⎤ t , Sj – tSj∗, Ij – tIj∗, Ij – tIj , ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ 1–β ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ tj–1 IA ⎝ I – tI , I – tI , I – tI , ⎠ ⎥ ⎢ D D R R T T ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ⎢ ∗ ∗ ∗ ∗ ⎥ n ⎢ t , Sj – tSj – tS(j–1) , Ij – tIj – tI(j–1) , ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , ID – tID – tID , ⎟⎥ j=2 ⎢ ⎜ A A A ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

( t)α + 2(α +3) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 66 of 107

⎡ ⎤ t1–β I∗ (t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛A j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ –2tj–1 IA ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ × ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ × , ⎢ ⎜ ⎟⎥ j=2 ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ∗ ∗ ∗ ∗ ⎟⎥ ⎢ + t1–β I∗ ⎜ j j (j–1) j j (j–1) ⎟⎥ ⎢ j–2 A ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × ID = tj–2 ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

( t)α + (α +2) ⎛ ⎞ ⎡ ∗ ⎤ t , Sj – tSj∗, Ij – tIj∗, Ij – tIj , ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ 1–β ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ tj–1 ID ⎝ I – tI , I – tI , I – tI , ⎠ ⎥ ⎢ D D R R T T ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ⎢ ∗ ∗ ∗ ∗ ⎥ n ⎢ t , Sj – tSj – tS(j–1) , Ij – tIj – tI(j–1) , ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , ID – tID – tID , ⎟⎥ j=2 ⎢ ⎜ A A A ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

( t)α + 2(α +3) ⎡ ⎤ t1–β I∗ (t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛D j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ –2tj–1 ID ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ × ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ × , ⎢ ⎜ ⎟⎥ j=2 ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ∗ ∗ ∗ ∗ ⎟⎥ ⎢ + t1–β I∗ ⎜ j j (j–1) j j (j–1) ⎟⎥ ⎢ j–2 D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 67 of 107

⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IR = tj–2 IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

( t)α + (α +2) ⎡ ⎤ j j∗ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , I – tI , ⎢t1–β I∗ A A D D ⎥ ⎢ j–12 R j j∗ j j∗ j j∗ j j∗ j j∗ ⎥ ⎢ IR –⎛ tIR , IT – tIT , R – tR , D – tD , V – tV⎞ ⎥ ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ n ⎢ ⎜ j–2 ⎟ ⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎥ × ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟ ⎥ ⎢ ⎜ A A A D D D ⎟ ⎥ j=2 ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎥ ⎢ – tj–2 IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎥ ⎢ ⎜ ⎟ ⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ ⎦ V j – tV j∗ – tV (j–1)∗

( t)α + 2(α +3) ⎡ ⎤ t1–β I∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛R j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β I∗ ⎝ j j j j j j ⎠ ⎥ ⎢ j–1 R ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IT = tj–2 IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + (α +2) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 68 of 107

⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 IT ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

( t)α + 2(α +3) ⎡ ⎤ t1–β I∗ (t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛T j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β I∗ ⎝ j j j j j j ⎠ ⎥ ⎢ j–1 T ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × R = tj–2 R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

( t)α + (α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 R ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

( t)α + 2(α +3) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 69 of 107

⎡ ⎤ t1–β R∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β R∗ ⎝ j j j j j j ⎠ ⎥ ⎢ j–1 ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × D = tj–2 D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

( t)α + (α +2) ⎡ ⎛ ⎞ ⎤ j j∗ j j∗ j j∗ tj–1, S – tS , I – tI , I – tI , ⎢ ⎜ A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 D ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎛ ⎞⎥ n ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ × ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

( t)α + 2(α +3) ⎡ ⎤ t1–β D∗(t , Sj, Ij, Ij , Ij , Ij , Ij , Rj, Dj, V j) ⎢ j ⎛ j A D R T ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ –2t1–β D∗ ⎝ j j j j j j ⎠ ⎥ ⎢ j–1 ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ n ⎢ ⎛ ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ × ⎢ ⎜ j–2 ⎟⎥ , ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ j=2 ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + tj–2 D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ⎛ ⎞ j j∗ (j–1)∗ j j∗ (j–1)∗ tj–2, S – tS – tS , I – tI – tI , ⎜ ∗ ∗ ∗ ∗ ⎟ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟ α n ⎜ A A A D D D ⎟ ( t) ⎜ ∗ ∗ ∗ ∗ ⎟ n+1 1–β ∗ ⎜ j j (j–1) j j (j–1) ⎟ × V = tj–2 V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ (α +1) ⎜ ⎟ j=2 ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 70 of 107

( t)α + (α +2) ⎡ ⎤ ⎛ ∗ ⎞ t , Sj – tSj∗, Ij – tIj∗, Ij – tIj , ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ 1–β ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ tj–1 V ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ⎥ ⎢ Rj – tRj∗, Dj – tDj∗, V j – tV j∗ ⎥ ⎢ ⎛ ⎞⎥ n ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ × ⎢ ⎜ ∗ ∗ ∗ ∗ ⎟⎥ × ⎢ ⎜ Ij – tIj – tI(j–1) , Ij – tIj – tI(j–1) , ⎟⎥ j=2 ⎢ ⎜ A A A D D D ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ – tj–2 V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ ( t)α + 2(α +3) ⎡ ⎤ 1–β ∗ j j j j j j j j j tj V (tj, S , I , IA, ID, IR, IT , R , D , V ) ⎢ ⎛ ∗ ⎞ ⎥ ⎢ j j∗ j j∗ j j ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ 1–β ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ –2t V ⎝ I – tI , I – tI , I – tI , ⎠ ⎥ ⎢ j–1 D D R R T T ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ n ⎢ ⎛ R – tR , D – tD , V – tV ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ × ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ × . ⎢ ⎜ ∗ ∗ ∗ ∗ ⎟⎥ j=2 ⎢ ⎜ j j (j–1) j j (j–1) ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ 1–β ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ + t V ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ j–2 ⎜ R R R T T T ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

Now we apply

FFE α,β(t) ∗ 0 Dt S = S (t, S, I, IA, ID, IR, IT , R, D, V), FFE α,β(t) ∗ 0 Dt I = I (t, S, I, IA, ID, IR, IT , R, D, V), FFE α,β(t) ∗ 0 Dt IA = IA(t, S, I, IA, ID, IR, IT , R, D, V), FFE α,β(t) ∗ 0 Dt ID = ID(t, S, I, IA, ID, IR, IT , R, D, V), FFE α,β(t) ∗ 0 Dt IR = IR(t, S, I, IA, ID, IR, IT , R, D, V), (136) FFE α,β(t) ∗ 0 Dt IT = IT (t, S, I, IA, ID, IR, IT , R, D, V), FFE α,β(t) ∗ 0 Dt R = R (t, S, I, IA, ID, IR, IT , R, D, V), FFE α,β(t) ∗ 0 Dt D = D (t, S, I, IA, ID, IR, IT , R, D, V), FFE α,β(t) ∗ 0 Dt V = V (t, S, I, IA, ID, IR, IT , R, D, V).

After applying the fractional integral with exponential kernel and putting the Newton polynomial into these equations, we can solve our model as follows: ⎡ ⎤ 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) t (– ln tn+1 + ) ⎢ n+1⎛ t tn+1 ⎞⎥ ⎢ n n∗ n n∗ n n∗ ⎥ ⎢ tn+1, S + tS , I + tI , I + tI , ⎥ ⎢ ⎜ A A ⎟⎥ 1–α ⎢ × S∗ ⎝ In + tIn∗, In + tIn∗, In + tIn∗, ⎠⎥ Sn+1 = Sn + ⎢ D D R R T T ⎥ (137) ⎢ n n∗ n n∗ n n∗ ⎥ M(α) ⎢ R + tR , D + tD , V + tV ⎥ ⎢ 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎥ ⎣ – tn (– ln tn + ) ⎦ t tn ∗ n n n n n n n n n S tn, S , I , IA, ID, IR, IT , R , D , V Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 71 of 107

⎧ ⎫ 23 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎪ tn (– ln tn + ) ⎪ ⎪ 12 t tn ⎪ ⎪ ∗ n n n n n n n n n ⎪ ⎪ × S (tn, S , I , I , I , I , I , R , D , V ) t ⎪ ⎪ A D R T ⎪ ⎪ 4 2–β(tn–1) β(tn)–β(tn–1) 2–β(tn–1) ⎪ ⎪ – t (– ln tn–1 + ) ⎪ ⎪ ⎛3 n–1 t tn–1 ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn–1, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ ⎪ ⎪ S ID – tID , IR – tIR , IT – tIT , t ⎪ ⎨⎪ ⎬⎪ α Rn – tRn∗, Dn – tDn∗, V n – tV n∗ + , ⎪ 5 2–β(tn–2) β(tn–1)–β(tn–2) 2–β(tn–2) ⎪ M(α) ⎪ + t (– ln tn–2 + ) ⎪ ⎪ ⎛ 12 n–2 t tn–2 ⎞ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ ∗ ∗ ⎪ ⎪ × ∗ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ t⎪ ⎪ ⎜ ∗ ∗ ∗ ∗ ⎟ ⎪ ⎪ ⎝ Rn – tRn – tR(n–1) , Dn – tDn – tD(n–1) , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ 1–α In+1 = In + M(α) ⎡ ⎤ 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) t (– ln tn+1 + ) ⎢ n+1 t tn+1 ⎥ ⎢ t , Sn + tSn∗, In + tIn∗, In + tIn∗, In + tIn∗, ⎥ ⎢ × ∗ n+1 A A D D ⎥ ⎢ I ∗ ∗ ∗ ∗ ∗ ⎥ × ⎢ In + tIn , In + tIn , Rn + tRn , Dn + tDn , V n + tV n ⎥ ⎢ R R T T ⎥ 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎣ – tn (– ln tn + ) ⎦ t tn ∗ n n n n n n n n n I tn, S , I , IA, ID, IR, IT , R , D , V ⎧ ⎫ 23 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎪ tn (– ln tn + ) ⎪ ⎪ 12 t tn ⎪ ⎪ ∗ n n n n n n n n n ⎪ ⎪ × I (tn, S , I , I , I , I , I , R , D , V ) t ⎪ ⎪ A D R T ⎪ ⎪ 4 2–β(tn–1) β(tn)–β(tn–1) 2–β(tn–1) ⎪ ⎪ – t (– ln tn–1 + ) ⎪ ⎪ ⎛3 n–1 t tn–1 ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn–1, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ ⎪ ⎪ I ID – tID , IR – tIR , IT – tIT , t ⎪ ⎨⎪ ⎬⎪ α Rn – tRn∗, Dn – tDn∗, V n – tV n∗ + , ⎪ 5 2–β(tn–2) β(tn–1)–β(tn–2) 2–β(tn–2) ⎪ M(α) ⎪ + t (– ln tn–2 + ) ⎪ ⎪ ⎛ 12 n–2 t tn–2 ⎞ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ ∗ ∗ ⎪ ⎪ × ∗ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ t⎪ ⎪ ⎜ ∗ ∗ ∗ ∗ ⎟ ⎪ ⎪ ⎝ Rn – tRn – tR(n–1) , Dn – tDn – tD(n–1) , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎤ 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) t (– ln tn+1 + ) ⎢ n⎛+1 t tn+1 ⎞⎥ ⎢ n n∗ n n∗ n n∗ ⎥ ⎢ tn+1, S + tS , I + tI , IA + tIA , ⎥ ⎢ ⎜ ⎟⎥ 1–α ⎢ × I∗ ⎝ In tIn∗ In tIn∗ In tIn∗ ⎠⎥ n+1 n ⎢ A D + D , R + R , T + T , ⎥ IA = IA + ⎢ ∗ ∗ ∗ ⎥ M(α) ⎢ Rn + tRn , Dn + tDn , V n + tV n ⎥ ⎢ 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎥ ⎣ – tn (– ln tn + ) ⎦ t tn × ∗ n n n n n n n n n IA tn, S , I , IA, ID, IR, IT , R , D , V Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 72 of 107

⎧ ⎫ 23 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎪ tn (– ln tn + ) ⎪ ⎪ 12 t tn ⎪ ⎪ ∗ n n n n n n n n n ⎪ ⎪ × I (tn, S , I , I , I , I , I , R , D , V ) t ⎪ ⎪ A A D R T ⎪ ⎪ 4 2–β(tn–1) β(tn)–β(tn–1) 2–β(tn–1) ⎪ ⎪ – t (– ln tn–1 + ) ⎪ ⎪ ⎛3 n–1 t tn–1 ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn–1, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ ⎪ ⎪ IA ID – tID , IR – tIR , IT – tIT , t ⎪ ⎨⎪ ⎬⎪ α Rn – tRn∗, Dn – tDn∗, V n – tV n∗ + , ⎪ 5 2–β(tn–2) β(tn–1)–β(tn–2) 2–β(tn–2) ⎪ M(α) ⎪ + t (– ln tn–2 + ) ⎪ ⎪ ⎛ 12 n–2 t tn–2 ⎞ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ ∗ ∗ ⎪ ⎪ × ∗ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ t⎪ ⎪ ⎜ ∗ ∗ ∗ ∗ ⎟ ⎪ ⎪ ⎝ Rn – tRn – tR(n–1) , Dn – tDn – tD(n–1) , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ 1–α In+1 = In + D D M(α) ⎡ ⎤ 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) t (– ln tn+1 + ) ⎢ n+1 t tn+1 ⎥ ⎢ t , Sn + tSn∗, In + tIn∗, In + tIn∗, In + tIn∗, ⎥ ⎢ × ∗ n+1 A A D D ⎥ ⎢ ID ∗ ∗ ∗ ∗ ∗ ⎥ × ⎢ In + tIn , In + tIn , Rn + tRn , Dn + tDn , V n + tV n ⎥ ⎢ R R T T ⎥ 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎣ – tn (– ln tn + ) ⎦ t tn × ∗ n n n n n n n n n ID tn, S , I , IA, ID, IR, IT , R , D , V ⎧ ⎫ 23 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎪ tn (– ln tn + ) ⎪ ⎪ 12 t tn ⎪ ⎪ ∗ n n n n n n n n n ⎪ ⎪ × I (tn, S , I , I , I , I , I , R , D , V ) t ⎪ ⎪ D A D R T ⎪ ⎪ 4 2–β(tn–1) β(tn)–β(tn–1) 2–β(tn–1) ⎪ ⎪ – t (– ln tn–1 + ) ⎪ ⎪ ⎛3 n–1 t tn–1 ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn–1, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ ⎪ ⎪ ID ID – tID , IR – tIR , IT – tIT , t ⎪ ⎨⎪ ⎬⎪ α Rn – tRn∗, Dn – tDn∗, V n – tV n∗ + , ⎪ 5 2–β(tn–2) β(tn–1)–β(tn–2) 2–β(tn–2) ⎪ M(α) ⎪ + t (– ln tn–2 + ) ⎪ ⎪ ⎛ 12 n–2 t tn–2 ⎞ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ ∗ ∗ ⎪ ⎪ × ∗ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ t⎪ ⎪ ⎜ ∗ ∗ ∗ ∗ ⎟ ⎪ ⎪ ⎝ Rn – tRn – tR(n–1) , Dn – tDn – tD(n–1) , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ ⎡ ⎤ 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) t (– ln tn+1 + ) ⎢ n⎛+1 t tn+1 ⎞⎥ ⎢ n n∗ n n∗ n n∗ ⎥ ⎢ tn+1, S + tS , I + tI , IA + tIA , ⎥ ⎢ ⎜ ⎟⎥ 1–α ⎢ × I∗ ⎝ In tIn∗ In tIn∗ In tIn∗ ⎠⎥ n+1 n ⎢ R D + D , R + R , T + T , ⎥ IR = IR + ⎢ ∗ ∗ ∗ ⎥ M(α) ⎢ Rn + tRn , Dn + tDn , V n + tV n ⎥ ⎢ 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎥ ⎣ – tn (– ln tn + ) ⎦ t tn × ∗ n n n n n n n n n IR tn, S , I , IA, ID, IR, IT , R , D , V Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 73 of 107

⎧ ⎫ 23 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎪ tn (– ln tn + ) ⎪ ⎪ 12 t tn ⎪ ⎪ ∗ n n n n n n n n n ⎪ ⎪ × I (tn, S , I , I , I , I , I , R , D , V ) t ⎪ ⎪ D A D R T ⎪ ⎪ 4 2–β(tn–1) β(tn)–β(tn–1) 2–β(tn–1) ⎪ ⎪ – t (– ln tn–1 + ) ⎪ ⎪ ⎛3 n–1 t tn–1 ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn–1, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ ⎪ ⎪ IR ID – tID , IR – tIR , IT – tIT , t ⎪ ⎨⎪ ⎬⎪ α Rn – tRn∗, Dn – tDn∗, V n – tV n∗ + , ⎪ 5 2–β(tn–2) β(tn–1)–β(tn–2) 2–β(tn–2) ⎪ M(α) ⎪ + t (– ln tn–2 + ) ⎪ ⎪ ⎛ 12 n–2 t tn–2 ⎞ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ ∗ ∗ ⎪ ⎪ × ∗ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ t⎪ ⎪ ⎜ ∗ ∗ ∗ ∗ ⎟ ⎪ ⎪ ⎝ Rn – tRn – tR(n–1) , Dn – tDn – tD(n–1) , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ 1–α In+1 = In + T T M(α) ⎡ ⎤ 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) t (– ln tn+1 + ) ⎢ n+1 t tn+1 ⎥ ⎢ t , Sn + tSn∗, In + tIn∗, In + tIn∗, In + tIn∗, ⎥ ⎢ × ∗ n+1 A A D D ⎥ ⎢ IT ∗ ∗ ∗ ∗ ∗ ⎥ × ⎢ In + tIn , In + tIn , Rn + tRn , Dn + tDn , V n + tV n ⎥ ⎢ R R T T ⎥ 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎣ – tn (– ln tn + ) ⎦ t tn × ∗ n n n n n n n n n IT tn, S , I , IA, ID, IR, IT , R , D , V ⎧ ⎫ 23 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎪ tn (– ln tn + ) ⎪ ⎪ 12 t tn ⎪ ⎪ ∗ n n n n n n n n n ⎪ ⎪ × I (tn, S , I , I , I , I , I , R , D , V ) t ⎪ ⎪ T A D R T ⎪ ⎪ 4 2–β(tn–1) β(tn)–β(tn–1) 2–β(tn–1) ⎪ ⎪ – t (– ln tn–1 + ) ⎪ ⎪ ⎛3 n–1 t tn–1 ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn–1, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ ⎪ ⎪ IT ID – tID , IR – tIR , IT – tIT , t ⎪ ⎨⎪ ⎬⎪ α Rn – tRn∗, Dn – tDn∗, V n – tV n∗ + , ⎪ 5 2–β(tn–2) β(tn–1)–β(tn–2) 2–β(tn–2) ⎪ M(α) ⎪ + t (– ln tn–2 + ) ⎪ ⎪ ⎛ 12 n–2 t tn–2 ⎞ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ ∗ ∗ ⎪ ⎪ × ∗ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ t⎪ ⎪ ⎜ ∗ ∗ ∗ ∗ ⎟ ⎪ ⎪ ⎝ Rn – tRn – tR(n–1) , Dn – tDn – tD(n–1) , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ 1–α Rn+1 = Rn + M(α) ⎡ ⎤ 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) t (– ln tn+1 + ) ⎢ n+1 t tn+1 ⎥ ⎢ t , Sn + tSn∗, In + tIn∗, In + tIn∗, In + tIn∗, ⎥ ⎢ × ∗ n+1 A A D D ⎥ ⎢ R ∗ ∗ ∗ ∗ ∗ ⎥ × ⎢ In + tIn , In + tIn , Rn + tRn , Dn + tDn , V n + tV n ⎥ ⎢ R R T T ⎥ 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎣ – tn (– ln tn + ) ⎦ t tn ∗ n n n n n n n n n R tn, S , I , IA, ID, IR, IT , R , D , V Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 74 of 107

⎧ ⎫ 23 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎪ tn (– ln tn + ) ⎪ ⎪ 12 t tn ⎪ ⎪ ∗ n n n n n n n n n ⎪ ⎪ × R (tn, S , I , I , I , I , I , R , D , V ) t ⎪ ⎪ A D R T ⎪ ⎪ 4 2–β(tn–1) β(tn)–β(tn–1) 2–β(tn–1) ⎪ ⎪ – t (– ln tn–1 + ) ⎪ ⎪ ⎛3 n–1 t tn–1 ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn–1, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ ⎪ ⎪ R ID – tID , IR – tIR , IT – tIT , t ⎪ ⎨⎪ ⎬⎪ α Rn – tRn∗, Dn – tDn∗, V n – tV n∗ + , ⎪ 5 2–β(tn–2) β(tn–1)–β(tn–2) 2–β(tn–2) ⎪ M(α) ⎪ + t (– ln tn–2 + ) ⎪ ⎪ ⎛ 12 n–2 t tn–2 ⎞ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ ∗ ∗ ⎪ ⎪ × ∗ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ t⎪ ⎪ ⎜ ∗ ∗ ∗ ∗ ⎟ ⎪ ⎪ ⎝ Rn – tRn – tR(n–1) , Dn – tDn – tD(n–1) , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ 1–α Dn+1 = Dn + M(α) ⎡ ⎤ 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) t (– ln tn+1 + ) ⎢ n+1 t tn+1 ⎥ ⎢ t , Sn + tSn∗, In + tIn∗, In + tIn∗, In + tIn∗, ⎥ ⎢ × ∗ n+1 A A D D ⎥ ⎢ D ∗ ∗ ∗ ∗ ∗ ⎥ × ⎢ In + tIn , In + tIn , Rn + tRn , Dn + tDn , V n + tV n ⎥ ⎢ R R T T ⎥ 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎣ – tn (– ln tn + ) ⎦ t tn × ∗ n n n n n n n n n D tn, S , I , IA, ID, IR, IT , R , D , V ⎧ ⎫ 23 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎪ tn (– ln tn + ) ⎪ ⎪ 12 t tn ⎪ ⎪ ∗ n n n n n n n n n ⎪ ⎪ × D (tn, S , I , I , I , I , I , R , D , V ) t ⎪ ⎪ A D R T ⎪ ⎪ 4 2–β(tn–1) β(tn)–β(tn–1) 2–β(tn–1) ⎪ ⎪ – t (– ln tn–1 + ) ⎪ ⎪ ⎛3 n–1 t tn–1 ⎞ ⎪ ⎪ n n∗ n n∗ n n∗ ⎪ ⎪ tn–1, S – tS , I – tI , I – tI , ⎪ ⎪ ⎜ A A ⎟ ⎪ ⎪ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ ⎪ ⎪ D ID – tID , IR – tIR , IT – tIT , t ⎪ ⎨⎪ ⎬⎪ α Rn – tRn∗, Dn – tDn∗, V n – tV n∗ + , ⎪ 5 2–β(tn–2) β(tn–1)–β(tn–2) 2–β(tn–2) ⎪ M(α) ⎪ + t (– ln tn–2 + ) ⎪ ⎪ ⎛ 12 n–2 t tn–2 ⎞ ⎪ ⎪ n n∗ (n–1)∗ n n∗ (n–1)∗ ⎪ ⎪ tn–2, S – tS – tS , I – tI – tI , ⎪ ⎪ ⎜ ∗ ∗ ⎟ ⎪ ⎪ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎪ ⎪ ∗ ∗ ⎪ ⎪ × ∗ ⎜ n n∗ (n–1) n n∗ (n–1) ⎟ ⎪ ⎪ D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ t⎪ ⎪ ⎜ ∗ ∗ ∗ ∗ ⎟ ⎪ ⎪ ⎝ Rn – tRn – tR(n–1) , Dn – tDn – tD(n–1) , ⎠ ⎪ ⎩⎪ ⎭⎪ V n – tV n∗ – tV (n–1)∗ 1–α V n+1 = V n + (138) M(α) ⎡ ⎤ 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) t (– ln tn+1 + ) ⎢ n+1 t tn+1 ⎥ ⎢ t , Sn + tSn∗, In + tIn∗, In + tIn∗, In + tIn∗, ⎥ ⎢ × ∗ n+1 A A D D ⎥ ⎢ V ∗ ∗ ∗ ∗ ∗ ⎥ × ⎢ In + tIn , In + tIn , Rn + tRn , Dn + tDn , V n + tV n ⎥ ⎢ R R T T ⎥ 2–β(tn) β(tn+1)–β(tn) 2–β(tn) ⎣ – tn (– ln tn + ) ⎦ t tn × ∗ n n n n n n n n n V tn, S , I , IA, ID, IR, IT , R , D , V Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 75 of 107

For the Atangana–Baleanu fractal-fractional derivative, we can have the following numerical scheme:

% & n+1 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) S = tn+1 – ln tn+1 + (139) AB(α) t tn+1 ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎜ A A ⎟ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ S ID + tID , IR + tIR , IT + tIT , Rn + tRn∗, Dn + tDn∗, V n + tV n∗ % & α n α( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) + t j–2 – ln t + AB(α)(α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ S ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ % & ⎥ n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ – t – ln tj–2 + ⎥ × ⎢ ⎛ j–2 t tj–2 ⎞⎥ × ⎢ ∗ ∗ ∗ ∗ ⎥ j=2 ⎢ t , Sj – tSj – tS(j–1) , Ij – tIj – tI(j–1) , ⎥ ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 76 of 107

α( t)α + 2AB(α)(α +3) ⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t (– ln tj + ) ⎢ j t tj ⎥ ⎢ ∗ j j j j ⎥ ⎢ × S (t , Sj, Ij, I , I , I , I , Rj, Dj, V j) ⎥ ⎢ j A D R T ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t (– ln tj–1 + ) ⎥ ⎢ ⎛j–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ × S∗ ⎝ Ij tIj Ij tIj Ij tIj ⎠ ⎥ ⎢ D – D , R – R , T – T , ⎥ n ∗ ∗ ∗ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ × ⎢ ⎥ × , ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ + t (– ln tj–2 + ) ⎥ j=2 ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ % & n+1 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) I = tn+1 – ln tn+1 + AB(α) t tn+1 ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎜ A A ⎟ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ I ID + tID , IR + tIR , IT + tIT , Rn + tRn∗, Dn + tDn∗, V n + tV n∗ % & α n α( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) + t j–2 – ln t + AB(α)(α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j⎛–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ I ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎥ n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ – tj–2 (– ln tj–2 + ) ⎥ × ⎢ ⎛ t tj–2 ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ j=2 ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

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⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t (– ln tj + ) ⎢ j t tj ⎥ ⎢ ∗ j j j j ⎥ ⎢ × I (t , Sj, Ij, I , I , I , I , Rj, Dj, V j) ⎥ ⎢ j A D R T ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t (– ln tj–1 + ) ⎥ ⎢ ⎛j–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ × I∗ ⎝ Ij – tIj , Ij – tIj , Ij – tIj , ⎠ ⎥ ⎢ D D R R T T ⎥ n ∗ ∗ ∗ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ × ⎢ ⎥ × , ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ + t (– ln tj–2 + ) ⎥ j=2 ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ % & n+1 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) IA = tn+1 – ln tn+1 + AB(α) t tn+1 ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎜ A A ⎟ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ IA ID + tID , IR + tIR , IT + tIT , Rn + tRn∗, Dn + tDn∗, V n + tV n∗ % & α n α( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) + t j–2 – ln t + AB(α)(α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ IA ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎥ n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ – tj–2 (– ln tj–2 + ) ⎥ × ⎢ ⎛ t tj–2 ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ j=2 ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

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⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t (– ln tj + ) ⎢ j t tj ⎥ ⎢ ∗ j j j j ⎥ ⎢ × I (t , Sj, Ij, I , I , I , I , Rj, Dj, V j) ⎥ ⎢ A j A D R T ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t (– ln tj–1 + ) ⎥ ⎢ ⎛j–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ × I∗ ⎝ Ij – tIj , Ij – tIj , Ij – tIj , ⎠ ⎥ ⎢ A D D R R T T ⎥ n ∗ ∗ ∗ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ × ⎢ ⎥ × , ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ + t (– ln tj–2 + ) ⎥ j=2 ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ % & n+1 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) ID = tn+1 – ln tn+1 + AB(α) t tn+1 ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎜ A A ⎟ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ ID ID + tID , IR + tIR , IT + tIT , Rn + tRn∗, Dn + tDn∗, V n + tV n∗ % & α n α( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) + t j–2 – ln t + AB(α)(α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ ID ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎥ n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ – tj–2 (– ln tj–2 + ) ⎥ × ⎢ ⎛ t tj–2 ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ j=2 ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

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⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t (– ln tj + ) ⎢ j t tj ⎥ ⎢ ∗ j j j j ⎥ ⎢ × I (t , Sj, Ij, I , I , I , I , Rj, Dj, V j) ⎥ ⎢ D j A D R T ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t (– ln tj–1 + ) ⎥ ⎢ ⎛j–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ × I∗ ⎝ Ij – tIj , Ij – tIj , Ij – tIj , ⎠ ⎥ ⎢ D D D R R T T ⎥ n ∗ ∗ ∗ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ × ⎢ ⎥ × , ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ + t (– ln tj–2 + ) ⎥ j=2 ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ % & n+1 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) IR = tn+1 – ln tn+1 + AB(α) t tn+1 ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎜ A A ⎟ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ IR ID + tID , IR + tIR , IT + tIT , Rn + tRn∗, Dn + tDn∗, V n + tV n∗ % & α n α( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) + t j–2 – ln t + AB(α)(α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j⎛–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ IR ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎥ n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ – tj–2 (– ln tj–2 + ) ⎥ × ⎢ ⎛ t tj–2 ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ j=2 ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

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⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t (– ln tj + ) ⎢ j t tj ⎥ ⎢ ∗ j j j j ⎥ ⎢ × I (t , Sj, Ij, I , I , I , I , Rj, Dj, V j) ⎥ ⎢ R j A D R T ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t (– ln tj–1 + ) ⎥ ⎢ ⎛j–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ × I∗ ⎝ Ij – tIj , Ij – tIj , Ij – tIj , ⎠ ⎥ ⎢ R D D R R T T ⎥ n ∗ ∗ ∗ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ × ⎢ ⎥ × , ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ + t (– ln tj–2 + ) ⎥ j=2 ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ % & n+1 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) IT = tn+1 – ln tn+1 + AB(α) t tn+1 ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎜ A A ⎟ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ IT ID + tID , IR + tIR , IT + tIT , Rn + tRn∗, Dn + tDn∗, V n + tV n∗ % & α n α( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) + t j–2 – ln t + AB(α)(α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ IT ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎥ n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ – tj–2 (– ln tj–2 + ) ⎥ × ⎢ ⎛ t tj–2 ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ j=2 ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

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⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t (– ln tj + ) ⎢ j t tj ⎥ ⎢ ∗ j j j j ⎥ ⎢ × I (t , Sj, Ij, I , I , I , I , Rj, Dj, V j) ⎥ ⎢ T j A D R T ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t (– ln tj–1 + ) ⎥ ⎢ ⎛j–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ × I∗ ⎝ Ij – tIj , Ij – tIj , Ij – tIj , ⎠ ⎥ ⎢ T D D R R T T ⎥ n ∗ ∗ ∗ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ × ⎢ ⎥ × , ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ + t (– ln tj–2 + ) ⎥ j=2 ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ % & n+1 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) R = tn+1 – ln tn+1 + AB(α) t tn+1 ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎜ A A ⎟ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ R ID + tID , IR + tIR , IT + tIT , Rn + tRn∗, Dn + tDn∗, V n + tV n∗ % & α n α( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) + t j–2 – ln t + AB(α)(α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ R ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎥ n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ – tj–2 (– ln tj–2 + ) ⎥ × ⎢ ⎛ t tj–2 ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ j=2 ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

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⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t (– ln tj + ) ⎢ j t tj ⎥ ⎢ ∗ j j j j ⎥ ⎢ × R (t , Sj, Ij, I , I , I , I , Rj, Dj, V j) ⎥ ⎢ j A D R T ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t (– ln tj–1 + ) ⎥ ⎢ ⎛j–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ × R∗ ⎝ Ij – tIj , Ij – tIj , Ij – tIj , ⎠ ⎥ ⎢ D D R R T T ⎥ n ∗ ∗ ∗ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ × ⎢ ⎥ × , ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ + t (– ln tj–2 + ) ⎥ j=2 ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ % & n+1 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) D = tn+1 – ln tn+1 + AB(α) t tn+1 ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎜ A A ⎟ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ D ID + tID , IR + tIR , IT + tIT , Rn + tRn∗, Dn + tDn∗, V n + tV n∗ % & α n α( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) + t j–2 – ln t + AB(α)(α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ D ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎥ n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ – tj–2 (– ln tj–2 + ) ⎥ × ⎢ ⎛ t tj–2 ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ j=2 ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

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⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t (– ln tj + ) ⎢ j t tj ⎥ ⎢ ∗ j j j j ⎥ ⎢ × D (t , Sj, Ij, I , I , I , I , Rj, Dj, V j) ⎥ ⎢ j A D R T ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t (– ln tj–1 + ) ⎥ ⎢ ⎛j–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , IA – tIA , ⎥ ⎢ ⎜ ∗ ∗ ∗ ⎟ ⎥ ⎢ × D∗ ⎝ Ij – tIj , Ij – tIj , Ij – tIj , ⎠ ⎥ ⎢ D D R R T T ⎥ n ∗ ∗ ∗ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ × ⎢ ⎥ × , ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ + t (– ln tj–2 + ) ⎥ j=2 ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ ⎥ ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗ % & n+1 1–α 2–β(tn+1) β(tn+2)–β(tn+1) 2–β(tn+1) V = tn+1 – ln tn+1 + AB(α) t tn+1 ⎛ ⎞ n n∗ n n∗ n n∗ tn+1, S + tS , I + tI , I + tI , ⎜ A A ⎟ × ∗ ⎝ n n∗ n n∗ n n∗ ⎠ V ID + tID , IR + tIR , IT + tIT , Rn + tRn∗, Dn + tDn∗, V n + tV n∗ % & α n α( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) + t j–2 – ln t + AB(α)(α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗

α( t)α + AB(α)(α +2) ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎝ j j∗ j j∗ j j∗ ⎠ ⎥ ⎢ V ID – tID , IR – tIR , IT – tIT , ⎥ ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ ⎢ ⎥ n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ⎢ – tj–2 (– ln tj–2 + ) ⎥ × ⎢ ⎛ t tj–2 ⎞⎥ × ⎢ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎥ j=2 ⎢ ⎜ j–2 ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ I – tI – tI , I – tI – tI , ⎟⎥ ⎢ ⎜ A A A D D D ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

α( t)α + 2AB(α)(α +3) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 84 of 107

⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t (– ln tj + ) ⎢ j t tj ⎥ ⎢ ∗ j j j j ⎥ ⎢ × V (t , Sj, Ij, I , I , I , I , Rj, Dj, V j) ⎥ ⎢ j A D R T ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t (– ln tj–1 + ) ⎥ ⎢ ⎛j–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ V ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ n ⎢ ∗ ∗ ∗ ⎥ ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ × ⎢ ⎥ × . ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ j ⎢ + t (– ln tj–2 + ) ⎥ =2 ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ∗ ∗ ∗ ∗ ⎟⎥ ⎢ × V ∗ ⎜ j j (j–1) j j (j–1) ⎟⎥ ⎢ ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

For the power-law kernel, we can have the following:

% & α n ( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) Sn+1 = t j–2 – ln t + (140) (α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ ∗ ∗ j j∗ ⎥ ⎢ t , Sj – tSj , Ij – tIj , I – tI , ⎥ ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ S ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ R – tR , D – tD , V – tV ⎥ α n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ( t) ⎢ – t (– ln tj–2 + ) ⎥ + ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ (α +2) ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ j=2 ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ S ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 85 of 107

⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t (– ln tj + ) ⎢ j t tj ⎥ ⎢ ∗ j j j j ⎥ ⎢ × S (t , Sj, Ij, I , I , I , I , Rj, Dj, V j) ⎥ ⎢ j A D R T ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t (– ln tj–1 + ) ⎥ ⎢ ⎛j–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ S ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ n ⎢ ∗ ∗ ∗ ⎥ ( t)α ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ + ⎢ ⎥ 2(α +3) ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ j ⎢ + t (– ln tj–2 + ) ⎥ =2 ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ∗ ∗ ∗ ∗ ⎟⎥ ⎢ × S∗ ⎜ j j (j–1) j j (j–1) ⎟⎥ ⎢ ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× , % & α n ( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) In+1 = t j–2 – ln t + (α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j⎛–1 t tj–1 ⎞ ⎥ ⎢ ∗ ∗ j j∗ ⎥ ⎢ t , Sj – tSj , Ij – tIj , I – tI , ⎥ ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ I ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ R – tR , D – tD , V – tV ⎥ α n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ( t) ⎢ – t (– ln tj–2 + ) ⎥ + ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ (α +2) ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ j=2 ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ I ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 86 of 107

× , % & α n ( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) In+1 = t j–2 – ln t + A (α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ ∗ ∗ j j∗ ⎥ ⎢ t , Sj – tSj , Ij – tIj , I – tI , ⎥ ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ IA ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ R – tR , D – tD , V – tV ⎥ α n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ( t) ⎢ – t (– ln tj–2 + ) ⎥ + ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ (α +2) ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ j=2 ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ IA ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 87 of 107

× , % & α n ( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) In+1 = t j–2 – ln t + D (α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ ∗ ∗ j j∗ ⎥ ⎢ t , Sj – tSj , Ij – tIj , I – tI , ⎥ ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ ID ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ R – tR , D – tD , V – tV ⎥ α n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ( t) ⎢ – t (– ln tj–2 + ) ⎥ + ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ (α +2) ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ j=2 ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ID ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 88 of 107

× , % & α n ( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) In+1 = t j–2 – ln t + R (α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j⎛–1 t tj–1 ⎞ ⎥ ⎢ ∗ ∗ j j∗ ⎥ ⎢ t , Sj – tSj , Ij – tIj , I – tI , ⎥ ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ IR ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ R – tR , D – tD , V – tV ⎥ α n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ( t) ⎢ – t (– ln tj–2 + ) ⎥ + ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ (α +2) ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ j=2 ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ IR ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 89 of 107

× , % & α n ( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) In+1 = t j–2 – ln t + T (α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ ∗ ∗ j j∗ ⎥ ⎢ t , Sj – tSj , Ij – tIj , I – tI , ⎥ ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ IT ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ R – tR , D – tD , V – tV ⎥ α n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ( t) ⎢ – t (– ln tj–2 + ) ⎥ + ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ (α +2) ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ j=2 ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ IT ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 90 of 107

× , % & α n ( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) Rn+1 = t j–2 – ln t + (α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ ∗ ∗ j j∗ ⎥ ⎢ t , Sj – tSj , Ij – tIj , I – tI , ⎥ ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ R ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ R – tR , D – tD , V – tV ⎥ α n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ( t) ⎢ – t (– ln tj–2 + ) ⎥ + ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ (α +2) ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ j=2 ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ R ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 91 of 107

× , % & α n ( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) Dn+1 = t j–2 – ln t + (α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ ∗ ∗ j j∗ ⎥ ⎢ t , Sj – tSj , Ij – tIj , I – tI , ⎥ ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ D ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ R – tR , D – tD , V – tV ⎥ α n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ( t) ⎢ – t (– ln tj–2 + ) ⎥ + ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ (α +2) ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ j=2 ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ D ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 92 of 107

× , % & α n ( t) 2–β(t ) β(tj–1)–β(tj–2) 2–β(tj–2) V n+1 = t j–2 – ln t + (α +1) j–2 t j–2 t j=2 j–2 ⎛ ⎞ t , Sj – tSj∗ – tS(j–1)∗, Ij – tIj∗ – tI(j–1)∗, ⎜ j–2 ⎟ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟ ⎜ ∗ ∗ ∗ ∗ ⎟ × ∗ ⎜ j j (j–1) j j (j–1) ⎟ × V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟ ⎜ ⎟ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠ V j – tV j∗ – tV (j–1)∗ ⎡ ⎤ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) t (– ln tj–1 + ) ⎢ j–1⎛ t tj–1 ⎞ ⎥ ⎢ ∗ ∗ j j∗ ⎥ ⎢ t , Sj – tSj , Ij – tIj , I – tI , ⎥ ⎢ ⎜ j–1 A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ V ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ ⎢ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ R – tR , D – tD , V – tV ⎥ α n ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ ( t) ⎢ – t (– ln tj–2 + ) ⎥ + ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ (α +2) ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ j=2 ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎢ × ∗ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ V ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 93 of 107

⎡ ⎤ 2–β(tj) β(tj+1)–β(tj) 2–β(tj) t (– ln tj + ) ⎢ j t tj ⎥ ⎢ ∗ j j j j ⎥ ⎢ × V (t , Sj, Ij, I , I , I , I , Rj, Dj, V j) ⎥ ⎢ j A D R T ⎥ ⎢ 2–β(tj–1) β(tj)–β(tj–1) 2–β(tj–1) ⎥ ⎢ –2t (– ln tj–1 + ) ⎥ ⎢ ⎛j–1 t tj–1 ⎞ ⎥ ⎢ j j∗ j j∗ j j∗ ⎥ ⎢ tj–1, S – tS , I – tI , I – tI , ⎥ ⎢ ⎜ A A ⎟ ⎥ ⎢ × ∗ ⎜ j j∗ j j∗ j j∗ ⎟ ⎥ ⎢ V ⎝ ID – tID , IR – tIR , IT – tIT , ⎠ ⎥ n ⎢ ∗ ∗ ∗ ⎥ ( t)α ⎢ Rj – tRj , Dj – tDj , V j – tV j ⎥ + ⎢ ⎥ 2(α +3) ⎢ 2–β(tj–2) β(tj–1)–β(tj–2) 2–β(tj–2) ⎥ j ⎢ + t (– ln tj–2 + ) ⎥ =2 ⎢ ⎛ j–2 t tj–2 ⎞⎥ ⎢ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎥ ⎢ tj–2, S – tS – tS , I – tI – tI , ⎥ ⎢ ⎜ ⎟⎥ ⎢ ⎜ j j∗ (j–1)∗ j j∗ (j–1)∗ ⎟⎥ ⎢ ⎜ IA – tIA – tIA , ID – tID – tID , ⎟⎥ ⎢ ⎜ ∗ ∗ ∗ ∗ ⎟⎥ ⎢ × V ∗ ⎜ j j (j–1) j j (j–1) ⎟⎥ ⎢ ⎜ IR – tIR – tIR , IT – tIT – tIT , ⎟⎥ ⎢ ⎜ ⎟⎥ ⎣ ⎝ Rj – tRj∗ – tR(j–1)∗, Dj – tDj∗ – tD(j–1)∗, ⎠⎦ V j – tV j∗ – tV (j–1)∗

× .

10 Numerical simulation In this section, using the numerical solutions obtained in the previous section, we present a numerical method for all cases. The numerical simulations are depicted for diﬀerent values of fractional order and fractal dimension as presented in Figs. 26–37. FFM α,β ∗ ∗ ∗ ∗ ∗ 0 Dt S = – δ(t) αI + wβID + γ wIA + wδ1IR + wδ2IT + γ1 + μ1 S, FFM α,β ∗ ∗ ∗ ∗ ∗ 0 Dt I = δ(t) αI + wβID + γ wIA + wδ1IR + wδ2IT S –(ε + ξ + λ + μ1)I, FFM α,β 0 Dt IA = ξI –(θ + μ + χ + μ1)IA, FFM α,β 0 Dt ID = εI –(η + ϕ + μ1)ID,

Figure 26 Numerical visualization of COVID-19 model for α =0.76 Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 94 of 107

Figure 27 Numerical visualization of COVID-19 model for α =0.85

Figure 28 Numerical visualization of COVID-19 model for α =0.91

FFM α,β 0 Dt IR = ηID + θIA –(v + ξ + μ1)IR, (141)

FFM α,τ 0 Dt IT = μIA + vIR –(σ + τ + μ1)IT ,

FFM α,τ 0 Dt R = λI + ϕID + χIA + ξIR + σ IT –( + μ1)R,

FFM α,τ 0 Dt D = τIT ,

FFM α,τ 0 Dt V = γ1S + R – μ1V, Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 95 of 107

Figure 29 Numerical visualization of COVID-19 model for α =0.76

Figure 30 Numerical visualization of COVID-19 model for α =0.90,β =0.85

where

⎧ ⎫ ⎪d (1 – a ) cos(–b t–t0 ), 0 < t < t ⎪ ⎪ 0 n T 0 ⎪ ⎪ ⎪ ⎨ d0, t0 < t < t1 ⎬ δ(t)= t–t1 . (142) ⎪d1(1 – ar) cos(–b T ), t1 < t < t2⎪ ⎪ ⎪ ⎪ d1, t2 < t < t3 ⎪ ⎩⎪ ⎭⎪ t–t2 d2(1 – at) cos(–b T ), t > t3 Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 96 of 107

Figure 31 Numerical visualization of COVID-19 model for α =0.95,β =0.95

Figure 32 Numerical visualization of COVID-19 model for α =0.72

Also, the initial conditions are

S(0) = 800,000, I(0) = 3, IA(0) = 0, ID(0) = 0, IR(0) = 0, (143)

IT (0) = 0, R(0) = 0, D(0) = 0, V(0) = 0.

Also, the parameters are chosen as follows:

= 810,000, η = 0.12, χ = 0.15, v =0.4, γ = 0.09, (144)

β = 0.75, γ1 =0.4, μ1 =0.3, ε = 0.161, τ = 0.0199,

= 0.015, λ = 0.0345, ϕ = 0.0345, δ1 =0.5, ξ = 0.015, Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 97 of 107

Figure 33 Numerical visualization of COVID-19 model for α =0.82

Figure 34 Numerical visualization of COVID-19 model for α =0.90

σ = 0.015, δ0 = 0.99, t = 900, t0 = 30, δ2 =0.4, w =0.4,

b =0.2, an =0.1, ar =0.2, at =0.3, d0 = 0.02, d1 =0.2,

d2 = 0.15.

11 Likelihood with hyper-Poisson distribution Using the suggested numerical model, we obtain the approximate solution (S∗(t), I∗(t), ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ IA(t), ID(t), IR(t), IT (t), R (t), D (t), V (t)). We are more interested in I (t), R (t), and D (t) t t t and the approximate solution I, R, D becausewehavethecollecteddatazI , zR, zD which represent the number of infections, recovered, and deaths daily. We assume that such fol- low hyper-Poisson distribution with parameters. The hyper-Poisson distribution is given Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 98 of 107

Figure 35 Numerical visualization of COVID-19 model for α =1

Figure 36 Numerical visualization of COVID-19 model for α =0.89,β =0.85

as follows:

(β) P(X = k)= , λ >0,k =0,1,2,...,n, (145) (k + β)(1, β, λ)

where

∞ k (1)kλ (1, β, λ)= ,(β)k = β(β +1)···(β + k) (146) (β)kk! k=0 Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 99 of 107

Figure 37 Numerical visualization of COVID-19 model for α =0.95,β =0.97

with parameters k1, k2, k3

∗ k1 = 1I (t), ∗ k2 = 2R (t), (147) ∗ k3 = 3D (t)

and t ∼ ∗ zI HP k1 = 1I (t) , t ∼ ∗ zR HP k2 = 2R (t) , (148) t ∼ ∗ zD HP k3 = 3D (t) .

Here, the parameters 1, 2,and3 are a combination of collection accuracy and de- tectability of infected, recovered, and dead. Thus the likelihood function is deﬁned as follows:

n t L(k1)= g zI /k1 , t=0 n t L(k2)= g zR/k2 , (149) t=0 n t L(k3)= g zD/k3 . t=0

Thus

n t (β)λzI L(k )= , 1 (zt + β)(1, β, λ) t=0 I Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 100 of 107

n t (β)λzR L(k )= , (150) 2 (zt + β)(1, β, λ) t=0 R n t (β)λzD L(k )= . 3 (zt + β)(1, β, λ) t=0 D

Without loss of generality, we consider L(k1):

n t (β)λzI log L(k )= log (151) 1 (zt + β)(1, β, λ) t=0 I n t ∗ t ∗ = log (β)+zI log 1I – log zI + β – log 1, β, 1I t=0

and

n n n t ∂ log L(k ) ∗ ((z + β)) 1 = log( )+ log I – I ∂zt 1 (zt + β) I t=0 t=0 t=0 I n t ∗ ((z + β)) = n log( )+log I – I (152) 1 (zt + β) t=0 I n t ∗ ((z + β)) = n log I – I , 1 (zt + β) t=0 I ∂ log L(k ) I∗ n (1, β, I∗) 1 = nzt – 1 (153) ∂I∗ I I∗ (1, β, I∗) t=0 1 ∗ ∗ t I (1, β, 1I ) = nzI ∗ – n ∗ , I (1, β, 1I ) ∗ ∂ log L(k1) t 1 (1, β, 1I ) = nzI – n ∗ (154) ∂1 1 (1, β, 1I ) ∗ (1, β, 1I ) =–n ∗ , (1, β, 1I ) n t (β)λzR L(k )= log (155) 2 (zt + β)(1, β, λ) t=0 R n t t = log (β)+zR log λ – log zR + β – log (1, β, λ) , t=0 n n n t ∂ log L(k ) ∗ ((z + β)) 2 = log( )+ log R – R ∂zt 2 (zt + β) R t=0 t=0 t=0 R n t ∗ ((z + β)) = n log( )+log R – R (156) 2 (zt + β) t=0 R n t ∗ ((z + β)) = n log R – R , 2 (zt + β) t=0 R Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 101 of 107

∂ log L(k ) R∗ n (1, β, R∗) 2 = nzt – 2 (157) ∂R∗ R R∗ (1, β, R∗) t=0 2 ∗ ∗ t R (1, β, 2R ) = nzR ∗ – n ∗ , R (1, β, 2R ) ∗ ∂ log L(k2) t 2 (1, β, 2R ) = nzR – n ∗ (158) ∂2 2 (1, β, 2R ) ∗ (1, β, 2R ) =–n ∗ , (1, β, 2R ) n t (β)λzD L(k )= log (159) 3 (zt + β)(1, β, λ) t=0 D n t t = log (β)+zD log λ – log zD + β – log (1, β, λ) , t=0 n n n t ∂ log L(k ) ∗ ((z + β)) 3 = log( )+ log D – D ∂zt 3 (zt + β) D t=0 t=0 t=0 D n t ∗ ((z + β)) = n log( )+log D – D (160) 3 (zt + β) t=0 D n t ∗ ((z + β)) = n log D – D , 3 (zt + β) t=0 D ∂ log L(k ) D∗ n (1, β, D∗) 3 = nzt – 3 (161) ∂R∗ D D∗ (1, β, D∗) t=0 3 ∗ ∗ t D (1, β, 3D ) = nzD ∗ – n ∗ , D (1, β, 3D ) ∗ ∂ log L(k3) t 3 (1, β, 3D ) = nzD – n ∗ (162) ∂3 3 (1, β, 3D ) ∗ (1, β, 3D ) =–n ∗ . (1, β, 3D )

12 Likelihood with Weibull distribution We will do the same routine for the Weibull distribution known as % & k λ k–1 P(X = k)= exp(–λ/α)k, λ, α >0,k =0,1,2,...,n, (163) α α

with parameters k1, k2, k3

∗ k1 = 1I (t), ∗ k2 = 2R (t), (164) ∗ k3 = 3D (t)

and t ∼ ∗ εI W k1 = 1I (t) , Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 102 of 107

t ∼ ∗ εR W k2 = 2R (t) , (165) t ∼ ∗ εD W k3 = 3D (t) .

Thus the likelihood function is given by

n t L(k1)= W εI /k1 , t=0 n t L(k2)= W εR/k2 , (166) t=0 n t L(k3)= W εD/k3 . t=0

Thus

n % & t t εI –1 εI λ εt L(k )= exp(–λ/α) I , 1 α α t=0 n % & t t εR–1 εR λ εt L(k )= exp(–λ/α) R , (167) 2 α α t=0 n % & t t εD–1 εD λ εt L(k )= exp(–λ/α) D . 3 α α t=0

Without loss of generality, we consider L(k1):

% & n t ∗ εt–1 ε I I ∗ εt log L(k )= log I 1 exp – I /α I 1 α α 1 t=0 t t ∗ t ∗ = log εI – log α + εI –1 log 1I – log α – εI –1I /α (168)

and

n t n n ∂ log L(k ) ε ∗ ∗ 1 = I + log I – log α – – I /α ∂εt εt 1 1 I t=0 I t=0 t=0 t εI ∗ ∗ = n t + n log 1I – log α + n 1I /α , (169) εI ∂ log L(k ) I∗ n (– I∗/α) 1 = n εt –1 – 1 (170) ∂I∗ I I∗ (– I∗/α) t=0 1 ∗ ∗ t I (–1I /α) = n εI –1 ∗ – n ∗ , I (–1I /α) ∗ ∂ log L(k1) t 1 (–1I /α) = n εI –1 – n ∗ (171) ∂1 1 (–1I /α) ∗ (–1I /α) =–n ∗ , (–1I /α) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 103 of 107

% & n t ∗ εt –1 ε I R ∗ εt log L(k )= log R 2 exp – R /α I 2 α α 2 t=0 t t ∗ t ∗ = log εR – log α + εR –1 log 2R – log α – εR –2R /α (172)

and

n t n n ∂ log L(k ) ε ∗ ∗ 2 = R + log R – log α – – R /α ∂εt εt 2 2 R t=0 R t=0 t=0 t εR ∗ ∗ = n t + n log 2R – log α + n –2R /α , (173) εR ∂ log L(k ) R∗ n (– R∗/α) 2 = n εt –1 – 2 (174) ∂R∗ R R∗ (– R∗/α) t=0 2 ∗ ∗ t R (–2R /α) = n εR –1 ∗ – n ∗ , R (–2R /α) ∗ ∂ log L(k2) t 2 (–2R /α) = n εR –1 – n ∗ (175) ∂2 2 (–2R /α) ∗ (–2R /α) =–n ∗ , (–2R /α) % & n t ∗ εt –1 ε D D ∗ εt log L(k )= log D 3 exp – D /α D (176) 3 α α 3 t=0 t t ∗ t ∗ = log εD – log α + εD –1 log 3D – log α – εD –3D /α

and

n t n n ∂ log L(k ) ε ∗ ∗ 3 = D + log D – log α – – D /α ∂εt εt 3 3 D t=0 D t=0 t=0 t εD ∗ ∗ = n t + n log 3R – log α + n –3D /α , (177) εD ∂ log L(k ) D∗ n (– D∗/α) 3 = n εt –1 – 3 (178) ∂D∗ D D∗ (– D∗/α) t=0 3 ∗ ∗ t D (–3D /α) = n εI –1 ∗ – n ∗ , D (–3D /α) ∗ ∂ log L(k1) t 3 (–3D /α) = n εD –1 – n ∗ (179) ∂1 3 (–3D /α) ∗ (–3D /α) =–n ∗ . (–3D /α)

13 Likelihood with Mittag-Leffler distribution Finally, we shall use the Mittag-Leffler distribution for similar processes. The Mittag- Leffler distribution is defined by

λk P(X = k)= , λ >0,k =0,1,2,...,n, (180) (αk + β)Eα,β (λ) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 104 of 107

where

∞ λk Eα β (λ)= . (181) , (αk + β) k=0

i, i =1,2,3withparametersk1, k2, k3

∗ k1 = 1I (t), ∗ k2 = 2R (t), (182) ∗ k3 = 3D (t)

and

t ∼ ∗ εI ML k1 = 1I (t) , t ∼ ∗ εR ML k2 = 2R (t) , (183) t ∼ ∗ εD ML k3 = 3D (t) .

Thus the likelihood function is written as

n t L(k1)= ML εI /k1 , t=0 n t L(k2)= ML εR/k2 , (184) t=0 n t L(k3)= ML εD/k3 . t=0

Thus

n εt λ I L(k )= , 1 (αεt + β)E (λ) t=0 I α,β

n εt λ R L(k )= , (185) 2 (αεt + β)E (λ) t=0 R α,β

n εt λ D L(k )= . 3 (αεt + β)E (λ) t=0 D α,β

We write L(k1):

εt λ I log L(k1)=log t (186) (αεI + β)Eα,β (λ) n t ∗ t ∗ = εI log 1I – log αεI + β – log Eα,β 1I t=0 Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 105 of 107

and

n n n t ∂ log L(k ) ∗ ((αε + β)) 1 = log( )+ log I – I ∂εt 1 (αεt + β) I t=0 t=0 t=0 I n t ∗ ((αε + β)) = n log( )+log I – I (187) 1 (αεt + β) t=0 I n t ∗ ((αε + β)) = n log I – I , 1 (αεt + β) t=0 I ∂ log L(k ) I∗ n E ( I∗) 1 = nεt – α,β 1 (188) ∂I∗ I I∗ E ( I∗) t=0 α,β 1 ∗ ∗ t I Eα,β (1I ) = nεI ∗ – n ∗ , I Eα,β (1I ) ∗ ∂ log L(k1) t 1 Eα,β (1I ) = nεI – n ∗ (189) ∂1 1 Eα,β (1I ) ∗ Eα,β (1I ) =–n ∗ . Eα,β (1I )

With the same routine,

n εt λ R log L(k )= log (190) 2 (αεt + β)E (λ) t=0 I α,β n t ∗ t ∗ = εR log 1R – log αεR + β – log Eα,β 2R t=0

and

n n n t ∂ log L(k ) ∗ ((αε + β)) 2 = log( )+ log R – R ∂εt 2 (αεt + β) R t=0 t=0 t=0 R n t ∗ ((αε + β)) = n log( )+log R – R (191) 2 (αεt + β) t=0 R n t ∗ ((αε + β)) = n log R – R , 2 (αεt + β) t=0 R ∂ log L(k ) R∗ n E ( R∗) 2 = nεt – α,β 2 (192) ∂R∗ R R∗ E ( R∗) t=0 α,β 2 ∗ ∗ t R Eα,β (2R ) = nεI ∗ – n ∗ , R Eα,β (2R ) ∗ ∂ log L(k2) t 2 Eα,β (2R ) = nεR – n ∗ (193) ∂2 2 Eα,β (2R ) ∗ Eα,β (2R ) =–n ∗ Eα,β (2R ) Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 106 of 107

and

n εt λ D log L(k )= log (194) 3 (αεt + β)E (λ) t=0 D α,β n t ∗ t ∗ = εD log 3D – log αεD + β – log Eα,β 3D t=0

and

n n n t ∂ log L(k ) ∗ ((αε + β)) 1 = log( )+ log D – D ∂εt 3 (αεt + β) I t=0 t=0 t=0 D n t ∗ ((αε + β)) = n log( )+log D – D (195) 3 (αεt + β) t=0 D n t ∗ ((αε + β)) = n log D – D , 3 (αεt + β) t=0 D ∂ log L(k ) D∗ n E ( D∗) 3 = nεt – α,β 3 (196) ∂D∗ D D∗ E ( D∗) t=0 α,β 3 ∗ ∗ t D Eα,β (3D ) = nεD ∗ – n ∗ , D Eα,β (3D ) ∗ ∂ log L(k3) t 1 Eα,β (3D ) = nεD – n ∗ (197) ∂3 1 Eα,β (3D ) ∗ Eα,β (3D ) =–n ∗ . Eα,β (3D )

14 Conclusion Up to date humans have relied on forecasting with the aim to better control their world, or at least to have an asymptotic idea of their future. They have many ways to achieve this, one way is to use the deterministic approach and another is stochastic one. In this work, we presented a comprehensive analysis ranging from stochastic, fractal to diﬀerentiation with the aim to predict the future behavior of COVID-19 with cases studied in Africa and Europe. With stochastic approach, we were able to detect a possibility of the second wave of COVID-19 spread in Europe and in Africa, a continuous exponential growth could be possible. We presented an extension of the blancmange function to capture more fractal behaviors, and some examples were presented resembling the COVID-19 spread in vari- ous countries in Africa and Europe. A complex and nonlinear mathematical model with wave function was considered and solved numerically with a modiﬁed scheme.

Acknowledgements The authors of this paper would like to thank the referees for their valuable suggestions and comments.

Funding Thereisnofundingforthispaper.

Availability of data and materials There are no data for this paper.

Competing interests The authors declare that they have no competing interests. Atangana and ˙Igret˘ Araz Advances in Diﬀerence Equations (2021) 2021:57 Page 107 of 107

Authors’ contributions All authors have contributed equally in this work. All authors read and approved the ﬁnal manuscript.

Author details 1Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein, South Africa. 2Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan. 3Department of Mathematic Education, Faculty of Education, Siirt University, Siirt 56100, Turkey.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 5 November 2020 Accepted: 4 January 2021

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