axioms Article Qualitative and Quantitative Analyses of COVID-19 Dynamics Taye Samuel Faniran 1,* , Leontine Nkague Nkamba 2 and Thomas Timothee Manga 3 1 Department of Computer Science, Lead City University, Ibadan P.O. Box 30678, Nigeria 2 Department of Mathematics, Higher Teacher Training College, University of Yaounde, Yaounde P.O. Box 47, Cameroon; [email protected] 3 Department of Mechanical Engineering, National Advanced Polytechnic School of Yaounde, Yaounde P.O. Box 8390, Cameroon; [email protected] * Correspondence: [email protected]; Tel.: +234-8037628618 Abstract: COVID-19 is a highly contagious disease which has spread across the world. A deter- ministic model that considers an important component of individuals with vertically transmitted underlying diseases (high-risk susceptible individuals), rather than the general public, is formulated in this paper. We also consider key parameters that are concerned with the disease. An epidemio- logical threshold, R0, is computed using next-generation matrix approach. This is used to establish the existence and global stability of equilibria. We identify the most sensitive parameters which effectively contribute to change the disease dynamics with the help of sensitivity analysis. Our results reveal that increasing contact tracing of the exposed individuals who are tested for COVID-19 and hospitalizing them, largely has a negative impact on R0. Results further reveal that transmission rate between low-risk/high-risk susceptible individuals and symptomatic infectious individuals b and incubation rate of the exposed individuals s have positive impact on R0. Numerical simulations show that there are fewer high-risk susceptible individuals than the general public when R0 < 1. This may be due to the fact that high-risk susceptible individuals may prove a bit more difficult to Citation: Faniran, T.S.; Nkamba, control than the low-risk susceptible individuals as a result of inherited underlying diseases present L.N.; Manga, T.T. Qualitative and in them. We thus conclude that high level of tracing and hospitalizing the exposed individuals, as Quantitative Analyses of COVID-19 well as adherence to standard precautions and wearing appropriate Personal Protective Equipment Dynamics. Axioms 2021, 10, 210. (PPE) while handling emergency cases, are needed to flatten the epidemic curve. https://doi.org/10.3390/ axioms10030210 Keywords: high-risk susceptible individuals; contact tracing; Lyapunov functions; global stability; Academic Editors: Delfim F. M. sensitivity analysis Torres and Palle E. T. Jorgensen MSC: 34D23; 93C15; 93D20 Received: 11 June 2021 Accepted: 22 July 2021 Published: 31 August 2021 1. Introduction Publisher’s Note: MDPI stays neutral Until now, the COVID-19 pandemic remains a global concern due to the continuous with regard to jurisdictional claims in increase in the number of infectious individuals. The number of confirmed cases has published maps and institutional affil- been growing very fast on a daily basis and it has been declared a worldwide pandemic iations. disease. The disease started in Wuhan, Hubei Province, China, in 2020 and since then, it has spread across the world [1]. Presently, the pandemic has disrupted economies all over the world. Since December 2019, the world has reported 148,329,348 confirmed cases and 3,128,962 death cases [2]. High fever, severe chest pain, body aches, headache, difficulty in Copyright: © 2021 by the authors. breathing, fatigue, pains, sore throat, skin rashes, running nose, taste loss, and diarrhea are Licensee MDPI, Basel, Switzerland. the symptoms of COVID-19, but the main and most common symptoms are dry cough and This article is an open access article fatigue [1]. distributed under the terms and With ongoing community transmission from asymptomatic infectious individuals, conditions of the Creative Commons disease burden is expected to rise. Consequently, there will be an ongoing need for people Attribution (CC BY) license (https:// with underlying diseases who are front-line health care workers in patient-facing roles. creativecommons.org/licenses/by/ This is due to the fact that their work requires close personal exposure to patients with 4.0/). Axioms 2021, 10, 210. https://doi.org/10.3390/axioms10030210 https://www.mdpi.com/journal/axioms Axioms 2021, 10, 210 2 of 18 SARS-CoV-2. They are at high risk of infection contributing to further spread [3]. SARS- Cov-2 infects all susceptible individuals. However, evidence to date suggests that a group of people is at a higher risk of contacting the virus quickly than the general public, due to compromised immune system. These are people with underlying medical conditions. An underlying disease affects the immunity, nutritional status and general well-being of patients-factors that play a critical role in protecting individuals against COVID-19. The World Health Organization emphasizes that this group of individuals (high-risk susceptible individuals) must protect themselves from COVID-19 in order to protect others. Furthermore, the highly susceptible individuals experience severe symptoms that may lead to death when they contact COVID-19. In response to the pandemic, government authorities at various levels have put in place various control measures such as imposing strict mandatory lockdown, use of a face mask in public, regular hand washing, and social distancing. Therefore, in order to reduce COVID-19 spread, contact tracing of suspected infected individuals has been stepped up in several countries and detected cases are quickly placed in isolation for prompt treatment [4]. The use of mathematical models to describe the transmission and spread of COVID-19, so as to gain insights into the disease behaviour and develop strategies for its curtailment, has been extensively studied by researchers. (see in [5–18] and the references cited therein.) Our model has a unique difference from the models developed in [10,18] as follows: the authors of [18] developed a mathematical model of COVID-19 to investigate the impact of non-pharmaceutical interventions for possible control of the disease. In their model, they only considered symptomatically infected and asymptomatically infected individ- uals without considering some individuals in the susceptible population who are more vulnerable to the disease than the rest of the susceptible population. These are front-line health care workers, pregnant women, children and family of COVID-19 patients. For in- stance, any COVID-19 patient who visits hospital for medical treatment will be attended to by these health care workers and this puts them in higher risk of contacting the virus. Classifying these individuals as a risk and treating them as a genuine threat is a national priority. Furthermore, in [10], they modeled the effects of non-pharmaceutical interventions on COVID-19 spread in Kenya. Their model excludes the class of high risk susceptible individuals which is taken into account in our model. The work in [19] investigated a class of SEIRS epidemic models with a general nonlinear incidence function. Systems of differential equations are also used as HIV infection model with transitions between uninfected cells and infected cells. The classical HIV infection divided the cells population into following three main compartments depending on cell status, target cells, infected cells producing viruses, matured virus particles, and its concentrations are denoted by x(t), y(t), and v(t), respectively. The authors of [20] developed a nonlinear mathematical model of MERS-COV to study the dynamical behaviour of the disease with two discrete-time delays. We highlight the main objectives of the study to achieve our aim in what follows. (1) It is noteworthy to mention that the high-risk susceptible individuals that we refer to, in this paper, are individuals with vertically transmitted or inherited underlying diseases, i.e., HIV, asthma and so on. These individuals are at higher-risk of con- tacting COVID-19 than the rest of individuals in the population and thus the need for the inclusion of high-risk susceptible individuals in order to understand the disease dynamics. (2) Global stability analyses of the disease-free and endemic equilibrium points are derived (3) Robust sensitivity analysis of the model is performed to determine the contributory effects of the factors/parameters on the spread of COVID-19. The rest of the paper is organized as follows. Section2 has the model formulation and computation of basic reproduction number. In Section3, global stability of disease-free and endemic equilibrium points are established. Sensitivity analysis of the model parameters is performed in Section4. Some simulations are done in order to confirm our theoretical results, regarding the global stability of equilibriums, in Section5. Section6 wraps the modeling work with conclusion. Axioms 2021, 10, 210 3 of 18 2. Materials and Methods The total population size, N, is divided into eight mutually exclusive compartments; S, J, E, A, B, H, C, and R, which is interpreted as (i) low-risk susceptible individuals, (ii) high-risk susceptible individuals, (iii) exposed individuals, (iv) asymptomatic infectious individuals, (v) symptomatic infectious individuals, (vi) hospitalized individuals, (vii) death class, and (viii) recovered individuals. People with vertically transmitted or inherited disease are assumed as the high-risk susceptible individuals. Individuals in S and J can be exposed to Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2)