A Mathematical Model for Coinfection of Listeriosis and Anthrax Diseases

Home , Anthrax, Sporadic disease

Hindawi International Journal of Mathematics and Mathematical Sciences Volume 2018, Article ID 1725671, 14 pages https://doi.org/10.1155/2018/1725671

Research Article A Mathematical Model for Coinfection of Listeriosis and Anthrax Diseases

Shaibu Osman 1 and Oluwole Daniel Makinde 2

1 Department of Mathematics, Pan African University, Institute for Basic Sciences, Technology and Innovations, Box 62000-00200, Nairobi, Kenya 2Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa

Correspondence should be addressed to Shaibu Osman; [email protected]

Received 8 April 2018; Accepted 9 July 2018; Published 2 August 2018

Academic Editor: Ram N. Mohapatra

Copyright © 2018 Shaibu Osman and Oluwole Daniel Makinde. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Listeriosis and Anthrax are fatal zoonotic diseases caused by Listeria monocytogene and Bacillus Anthracis, respectively. In this paper, we proposed and analysed a compartmental Listeriosis-Anthrax coinfection model describing the transmission dynamics of Listeriosis and Anthrax epidemic in human population using the stability theory of diferential equations. Our model revealed that the disease-free equilibrium of the Anthrax model only is locally stable when the basic reproduction number is less than one. Sensitivity analysis was carried out on the model parameters in order to determine their impact on the disease dynamics. Numerical simulation of the coinfection model was carried out and the results are displayed graphically and discussed. We simulate the Listeriosis-Anthrax coinfection model by varying the human contact rate to see its efects on infected Anthrax population, infected Listeriosis population, and Listeriosis-Anthrax coinfected population.

1. Introduction in animals cases only. Anthrax disease is caused by bacteria infections and it afects both humans and animals. Our model Listeriosis and Anthrax are fatal zoonotic diseases caused by is an improvement of the two models as we considered Listeria monocytogene and Bacillus Anthracis, respectively. Anthrax as a zoonotic disease and also looked at sensitivity Listeriosis in infants can be acquired in two forms. Mothers analysis and the efects of the contact rate on the disease usually acquire it afer eating foods that are contaminated transmissions. with Listeria monocytogenes and can develop sepsis resulting Authorsin[4]publishedapaperontheefectiveness in chorioamnionitis and delivering a septic infant or fetus. of constant and pulse vaccination policies using SIR model. Moreover, mothers carrying the pathogens in the gastroin- Te analysis of their results under constant vaccination testinal tract can infect the skin and respiratory tract of showed that the dynamics of the disease model is similar their babies during childbirth. Listeria monocytogenes are to the dynamics without vaccination [5, 6]. Tere are some among the commonest pathogens responsible for bacterial fndings on the spread of zoonotic diseases but a number meningitis among neonates. Responsible factors for the of these researches focused on the efect of vaccination on disease include induced immune suppression linked with the spread and transmission of the diseases as in the case HIV infection, hemochromatosis hematologic malignancies, of the authors in [7]. Moreover, authors in [8] investigated cirrhosis, diabetes, and renal failure with hemodialysis [1]. a disease transmission model by considering the impact of Authorsin[2]developedamodelforAnthraxtransmis- a protective vaccine and came up with the optimal vaccine sion but never considered the transmissions in both animal coveragethresholdrequiredfordiseaseeradication.However, and human populations. Our model is an improvement of authors in [9] employed optimal control to study a nonlinear the work done by authors in [2, 3]. Both formulated Anthrax SIR epidemic model with a vaccination strategy. Several models but only concentrated on the disease transmissions mathematical modeling techniques have been employed to 2 International Journal of Mathematics and Mathematical Sciences

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Figure 1: Flowchart for the coinfection model. study the role of optimal control using SIR epidemic model infections determines the efects of the contact rate on the [10–12]. Authors in [13] formulated an SIR epidemic model disease transmissions. by considering vaccination as a control measure in their model analysis. Authors in [14] developed a mathematical model for the transmission dynamics of Listeriosis in animal 2. Model Formulation and human populations but did not use optimal control as In this section, we divide the model into subcompartments a control measure in fghting the disease. Tey divided the (groups)asshowninFigure1.Tetotalhumanpopula- animal population into four compartments by introducing tion (�ℎ) is divided into subcompartments consisting of the vaccination compartment. susceptible humans (�ℎ),individualsthatareinfectedwith Authors in [15] formulated a model and employed opti- Anthrax(��), individuals that are infected with Listeriosis mal control to investigate the impact of chemotherapy on (��),individualsthatareinfectedwithbothAnthraxand malaria disease with infection immigrants and [16] applied Listeriosis (��), and those that have recovered from Anthrax, optimal control methods associated with preventing exoge- Listeriosis, and both Anthrax and Listeriosis, respectively, nous reinfection based on a exogenous reinfection tubercu- (��), (��),and(��). Te total vector population is repre- losis model. Authors in [17] conducted a research on the sented by �V;thisisdividedintosubcompartmentsthat identifcation and reservoirs of pathogens for efective control consist of susceptible animals (�V) and animals infected with of sporadic disease and epidemics. Listeria monocytogenes Anthrax (�V),where(��) is population of carcasses of animals is among the major zoonotic food borne pathogen that is in the soil that may have diet of Anthrax. Carcasses of responsible for approximately twenty-eight percent of most animals which may have not been properly disposed of have food-related deaths in the United States annually and a major the tendency of generating pathogens. Te total vector and cause of serious product recalls worldwide. Te dairy farm humanpopulationsarerepresentedas has been observed as a potential point and reservoir for Listeria monocytogenes. Models are widely used in the study of transmission �ℎ =�ℎ +�� +�� +�� +�� +�� +��. (1) dynamics of infectious diseases. In recent times, the appli- � =� +�, cation of mathematical models in the study of infectious ℎ V V diseases has increased tremendously. Hence the emergence of a branch called mathematical epidemiology. Frequent where �=��V/(�+��). diagnostic tests, the availability of clinical data, and electronic Te concentration of carcasses and ingestion rate are surveillance have facilitated the applications of mathematical denoted as � and V, respectively. Listeriosis related death rates models to critical examining of scientifc hypotheses and the are � and �, respectively, and Anthrax related death rates are design of real-life strategies of controlling diseases [18, 19]. � and �, respectively. Waning immunity rates are given by Authors in [20] constructed a coinfection model of �, �,and�.�, �,and� are the recovery rates, respectively, and malaria and cholera diseases with optimal control but never �(1 − �) are the bi-infected persons who have recovered from considered sensitivity analysis and analysis of the force of Anthrax only. Te natural death rates of human and vector infection. Sensitivity analysis determines the most sensitive populations are �ℎ and �V, respectively, and the modifcation parameters to the model and the analysis of the force of parameter is given by �.Tecoinfectedpersonswhohave International Journal of Mathematics and Mathematical Sciences 3 recovered from Listeriosis are denoted by (1 − �)(1 − �).Tis 3.2. Basic Reproduction Number. In this section, the concept implies that of the Next-Generation Matrix would be employed in com- puting the basic reproduction number. Using the theorem �+�(1−�) + (1−�)(1−�) =1. (2) in Van den Driessche and Watmough [21] on the Listeriosis Te following diferential equations were obtained from the model in (4), the basic reproduction number of the Listeriosis fowchart diagram of the coinfection model in Figure 1: only model, (R0�),isgivenby

��ℎ V�Ωℎ =Ωℎ +�� +�� +�� −�ℎ�V�ℎ −��ℎ −�ℎ�ℎ R0� = (6) �� ℎ�(�+�ℎ +�) �� =�� −�� −(�+� +�)� �� ℎ V ℎ � ℎ � 3.3. Existence of the Disease-Free Equilibrium �� =�� −�� −(�+� +�+�)� 3.4. Endemic Equilibrium. Te endemic equilibrium points �� ℎ � V � ℎ � are computed by setting the system of diferential equations �� in the Listeriosis only model (4) to zero. Te endemic =�ℎ�V�� +�� +(�+�ℎ +�+�)�� equilibrium points are as follows: �� ∗ � ∗ Ωℎ +�� =�� −(�+�ℎ)�� + (1−�) �� = , �� ℎ ∗ �ℎ +� �� (3) �∗�∗ =�� −(�+�ℎ)�� + (1−�) (1 − �) �� ∗ ℎ �� = , (� + �ℎ +�) �� ∗ =�� −(�+�ℎ)�� ∗ � �� = , �+�ℎ �� (7) =�� +�� −�� ∗ �� ∗ �� = . �� V =Ω −� (� +� )� −�� V V � �� V V V ∗ ∗ ∗ ∗ �0� =(�ℎ ,�� ,�� ,��) �� V ∗ ∗ ∗ ∗ ∗ =�V (�� +��)�V −�V�V Ω +�� =( ℎ � , ℎ , � , � ). ∗ �ℎ +� (� + �ℎ +�) �+�ℎ �� 3. Analysis of Listeriosis Only Model Ω +��∗ �∗ = ℎ � ℎ ∗ In this section, only the Listeriosis model is considered in the �ℎ +� analysis of the transmission dynamics. �∗�∗ �∗ = ℎ ��ℎ � =Ωℎ +�� −��ℎ −�ℎ�ℎ (� + �ℎ +�) �� (8) ��∗ �� ∗ � � =�� −(�+� +�)� �� = �� ℎ ℎ � �+�ℎ (4) ∗ �� ∗ �� =�� −(�+�ℎ)�� = ��

�� =��−� � 3.5. Existence of the Endemic Equilibrium �� Lemma 1. Te Listeriosis only model has a unique endemic 3.1. Disease-Free Equilibrium. We obtain the disease-free equilibrium if and only if the basic reproduction number R0� > equilibrium of the Listeriosis only model by setting the sys- 1. tem of equations in (4) to zero. At disease-free equilibrium, there are no infections and recovery. Proof. Te Listeriosis force of infection, (� = ��V/(�+��)), satisfes the polynomial; Ωℎ +�� −��ℎ −�ℎ�ℎ =0 ∗ ∗ 2 ∗ Ω �(� )=�(� ) +�(� )=0 (9) � = ℎ ℎ �=Ω �(� + � )+� �(�(� + � )+� (� + � +�)) �ℎ (5) where ℎ ℎ � ℎ ℎ ℎ , and Ω � =(�∗,�∗,�∗,�∗)=( ℎ ,0,0,0). 0� ℎ � � � �=(�+�ℎ)(1−�0�) . (10) �ℎ 4 International Journal of Mathematics and Mathematical Sciences

By mathematical induction, �>0and �>0whenever Watmough [21] on the systems of equations in model (11), the basic reproduction number is less than one (R0� <1). the linear stability of the disease-free equilibrium, (�0�),can ∗ Tis implies that � =−�/�≤0.In conclusion, the be ascertained. Te disease-free equilibrium is locally asymp- Listeriosis model has no endemic equilibrium and the basic totically stable whenever the basic reproduction number is reproductive number is less than one (R0� <1). less than one (R0� <1).And it is unstable whenever the basic reproduction number is greater than one (R0� >1). Te analysis illustrates the impossibility of backward Te disease-free equilibrium is the state at which there are no bifurcation in the Listeriosis model, because there is no infections in the system. At disease-free equilibrium, there are existence of endemic equilibrium whenever the basic repro- no infections in the system. duction number is less than one (R0� <1). 4.4. Endemic Equilibrium. Te endemic equilibrium points 4. Analysis of Anthrax Only Model arecomputedbysettingthesystemofdiferentialequationsin the Anthrax only model (11) to zero. Te endemic equilibrium In this section, only the Anthrax model is considered in the points are as follows: analysis of the transmission dynamics. Ω +��∗ �� = ℎ � , ℎ =Ω +�� −� � � −� � ℎ � +� �∗ �� ℎ � ℎ V ℎ ℎ ℎ ℎ ℎ V �� ∗�∗ � �∗ = V ℎ V , =��V�ℎ −(�+�ℎ +�)�� (� + �ℎ +�) �� ∗ � ∗ �� =�� −(�+�ℎ)�� (11) � = , �� (15) �+�ℎ ��V =Ω −�� −�� ∗ Ω V V � V V V � = V , �� V ∗ �V +�V�� V =�� −�� ∗�∗ �� V � V V V ∗ V V � �V = . �V 4.1. Disease-Free Equilibrium. Te disease-free equilibrium of the Anthrax only model is obtained by setting the system of Te endemic equilibrium of the Anthrax only model is given equations in model (11) to zero. At disease-free equilibrium, by there are no infections and recovery. Ω +��∗ � �∗�∗ � =(�∗,�∗,�∗,�∗,�∗)=( ℎ � , V ℎ V , 0� ℎ � � V V � +� �∗ (�+� +�) Ωℎ +�� −�ℎ�V�ℎ −�ℎ�ℎ =0 ℎ ℎ V ℎ (16) Ω ��∗ Ω � �∗�∗ � = ℎ . � , V , V V � .) ℎ ∗ �ℎ �+�ℎ �V +�V�� V (12) Ω −�� −�� =0 Ω +��∗ � �∗�∗ ��∗ Ω V V � V V V � =( ℎ � , V ℎ V , � , V , 0� � +� �∗ (�+� +�) �+� � +��∗ Ω ℎ ℎ V ℎ ℎ V V � � = V . (17) V � � �∗�∗ V V V � .) � ∗ ∗ ∗ ∗ ∗ Ωℎ ΩV V �0� =(�ℎ ,�� ,�� ,�V ,�V )=( ,0,0, ,0). (13) �ℎ �V 4.5. Existence of the Endemic Equilibrium 4.2. Basic Reproduction Number. In this section, the concept Lemma 2. Te Anthrax only model has a unique endemic of the Next-Generation Matrix would be employed in com- equilibrium whenever the basic reproduction number (R0�) is puting the basic reproduction number. Using the theorem in greater than one (R0� >1). Van den Driessche and Watmough [21] on the Anthrax model in (11), the basic reproduction number of the Anthrax only Proof. Considering the endemic equilibrium points of the model, (R0�),isgivenby Anthrax only model, Ω Ω � � Ω +��∗ � �∗�∗ ��∗ Ω R = √ ℎ V ℎ V � =( ℎ � , V ℎ V , � , V , 0� 2 (14) 0� ∗ ∗ �ℎ�V (� + �ℎ +�) �ℎ +�ℎ�V (� + �ℎ +�) �+�ℎ �V +�V�� (18) � �∗�∗ 4.3. Stability of the Disease-Free Equilibrium. Using the next- V V � ). � generation operator concept in Van den Driessche and V International Journal of Mathematics and Mathematical Sciences 5

Te endemic equilibrium point satisfes the given polynomial 5.2. Basic Reproduction Number. Te concept of the next- generation operator method in Van den Driessche and ∗ ∗ 2 ∗ Watmough [21] was employed on the system of diferential �(�� )=�1 (�� ) +�1 (�� )=0 (19) equations in model (3) to compute the basic reproduction number of the Anthrax-Listeriosis coinfection model. Te where Anthrax-Listeriosis coinfection model has a reproduction number (R��) given by �1 =�V (ΩV�ℎ (�� + �ℎ (�+�+�+�ℎ)) (20) R�� = max {R�, R�} (25) +�ℎ (� + �ℎ)(�+�+�ℎ)�V) where R� and R� are the basic reproduction numbers of and Anthrax and Listeriosis, respectively.

2 ΩℎΩV�ℎ�V �1 =(�+�ℎ)(1−�0�). (21) R = √ � 2 (26) �ℎ�V (� + �ℎ +�)

By mathematical induction, �1 >0and �1 >0whenever the and basicreproductionnumberislessthanone(R0� <1).Tis �∗ =−�/� ≤0. implies that � 1 1 In conclusion, the Anthrax V�Ωℎ (� + �ℎ +�+�)+�(�+�ℎ +�) only model has no endemic any time the basic reproductive R� = ( ) (27) ��ℎ� (� + �ℎ +�)(�+�ℎ +�+�) number is less than one (R0� <1).

Teorem 3. Te disease-free equilibrium (�0��) is locally Te analysis illustrates the impossibility of backward asymptotically stable whenever the basic reproduction number (R <1) bifurcation in the Anthrax only model. Because there is is less than one �� and unstable otherwise. no existence of endemic equilibrium whenever the basic reproduction number is less than one (R0� <1). 5.3. Impact of Listeriosis on Anthrax. In this section, the impact of Listeriosis on Anthrax and vice versa is analysed. 5. Anthrax-Listeriosis Coinfection Model Tis is done by expressing the reproduction number of one in terms of the other by expressing the basic reproduction In this section, the dynamics of the Anthrax-Listeriosis number of Listeriosis on Anthrax, that is, expressing R� in coinfection model in (3) is considered in the analysis of the terms of R� 2 transmission dynamics. from R� = √ΩℎΩV�ℎ�V/�ℎ�V (� + �ℎ +�). Solving for �ℎ in the above, 5.1. Disease-Free Equilibrium. Te disease-free equilibrium 2 2 of the Anthrax-Listeriosis model is obtained by setting the −�1R� + √�1R� +4�2 system of equations of model (3) to zero. At disease-free � = , (28) ℎ 2� R equilibrium, there are no infections and recovery. V � where Ωℎ +�� +�� +�� −�ℎ�V�ℎ −��ℎ −�ℎ�ℎ =0 � =� (� + �) Ω 1 V �∗ = ℎ (29) ℎ � =Ω Ω � � �ℎ and 2 ℎ V ℎ V (22) ΩV −�V (�� +��)�V −�V�V =0 Also, letting Ω ∗ V √ 2 2 �V = �1R� +4�2 =�3R� +�4, (30) �V this implies Te disease-free equilibrium is given by R (� −� )+� � = � 3 1 4 � = �∗,�∗,�∗,�∗ ,�∗,�∗,�∗ ,�∗,�∗,�∗ ℎ (31) 0�� ( ℎ � � �� V V ) (23) 2�VR� Ω Ω � � =( ℎ ,0,0,0,0,0,0,0, V ,0) By substituting ℎ into the basic reproduction number of 0�� (24) (R ) �ℎ �V Listeriosis � ,

R0� (�4 +(�3 −�1) R� +2(�+�+�)�VR� +�(�4 +(�3 −�1) R� +2(�+�) �VR�)) R� = (32) �4 +(�3 −�1) R� +2(�+�+�)�VR� 6 International Journal of Mathematics and Mathematical Sciences where the basic reproduction number of Listeriosis only cases in the environment. Tat is Anthrax enhances model (�0�) is given in the relation Listeriosis infections in the environment.

V�Ωℎ However, by expressing the basic reproduction number of R0� = . (33) R R ��ℎ�(�+�ℎ +�) Anthrax on Listeriosis, that is expressing � in terms of �,

2 Now, taking the partial derivative of R� with respect to R� in �1 −�2R� + √�3R +�4R� +�5 � (36) (32) gives �ℎ = , 2R� �R 2� �(�+�−(�+�+�))�R � = 4 V 0� . where �R 2 (34) � [�4 +(�3 −�1 +2(�+�+�)�VR�)] �1 = (1+�) R0�, If (�+�)≥(�+�+�), the derivative (�R�/�R�),isstrictly �2 =(�+�+�+�+�) positive. Two scenarios can be deduced from the derivative (�R /�R ) � � , depending on the values of the parameters: �3 =(�+�+�−�−�), (37)

�R �4 =2(�−1) (�+�−�−�−�)R0� � =0, �R � � = (1+�)2 R2 . (35) 1 0� �R� and ≥0. �R� By letting

√ 2 �3R� +�4R� +�5 =�6R� +�7, (38) (1) If �R�/�R� =0,itimpliesthat(�+�) = (�+�+�) and the epidemiological implication is that Anthrax has it implies that no signifcance efect on the transmission dynamics of Listeriosis. (� −�) R +� +� � = 6 2 � 7 1 . �R /�R >0 (�+�) ≥ (�+�+�) ℎ (39) (2) If � � ,itimpliesthat ,and 2R� the epidemiological implication is that an increase in Anthrax cases would result in an increase Listeriosis Terefore,

2 2 4ΩℎΩV�ℎ�VR� R� = (40) [(�6 −�2) R� +�7 +�1][�7 +�1 +2(�+�)R� +(�6 −�2) R�]�V

Now, taking the partial derivative of R� with respect to R� in equation (40) gives

�R 4(� +�)[� +� +(�+�+� −�) R ]Ω Ω � � R � = 7 1 7 1 6 2 � ℎ V ℎ V � �R 2 2 (41) � [(�6 −�2) R� +�7 +�1] [�7 +�1 +(2(�+�)+�6 −�2) R�] �V

If the partial derivative of R� with respect to R� is greater 5.4. Analysis of Backward Bifurcation. In this section, the than zero, (�R�/�R� >0), the biological implication is that phenomenon of backward bifurcation is carried out by an increase in the number of cases of Listeriosis would result employing the center manifold theory on the system of difer- in an increase in the number of cases of Anthrax in the ential equations in model (3). Bifurcation analysis was carried environment. Moreover, the impact of Anthrax treatment on out by employing the center manifold theory in Castillo- Listeriosis can also be analysed by taking the partial derivative Chavez and Song [22]. Considering the human transmission of R� with respect to �, (�R�/��). rate (�ℎ) and V as the bifurcation parameters, it implies that R� =1and R� =1if and only if �R� � =− . (42) �� +�+�ℎ

Clearly, R� is a decreasing function of �; the epidemiological 2 ∗ �ℎ� (�+�+�ℎ) implication is that the treatment of Listeriosis would have an � =� = V , (43) ℎ ℎ Ω Ω � impact on the transmission dynamics of Anthrax. ℎ V V International Journal of Mathematics and Mathematical Sciences 7

�� and 5 =�� −(�+� )� + (1−�) �� 2 ℎ 5 4 � � �(�+� +�)(�+� +�+�) V = V∗ = � ℎ ℎ ℎ . ��6 (44) =��3 −(�+�ℎ)�6 + (1−�) (1 − �) ��4 �Ωℎ (� + �ℎ +�+�+�(�+�+�ℎ)) �� 7 =�� −(�+� )� By considering the following change of variables, �� 4 ℎ 7

��8 �ℎ =�1, =�� +�� −� � �� 3 4 � 8 �� =�2, ��9 =ΩV −�V (�2 +�4)�9 −�V�9 �� =�3, �� =�4, 10 =� (� +� )� −�� V 2 4 9 V 10 � =�, � 5 (49) (45) �� =�6, Backward bifurcation is carried out by employing the center � =�, �� 7 manifold theory on the system of diferential equations in � =�, model (3). Tis concept involves the computation of the � 8 Jacobian of the system of diferential equations in (49) at the disease-free equilibrium (�0). Te Jacobian matrix at disease- �V =�9, free equilibrium is given by �V =�10. �(�0) Tis would give the total population as −�ℎ 00�1 ��2 0�3 [ ] [ 0−� 0000 000�] [ 4 3 ] �=�1 +�2 +�3 +�4 +�5 +�6 +�7 +�8 +�9 [ ] [ 00−�� 00 0� 00] (46) [ 5 1 2 ] +� . [ ] 10 [ 000−�00 0000] [ 6 ] [ ] [ 0�0�7 −�8 00000] (50) = [ ] By applying vector notation [ ] [ 00��9 0−�10 0000] [ ] [ 000�00−�000] � [ 11 ] �=(�,� ,� ,� ,� ,� ,� ,� ,� ,� ) . (47) [ ] 1 2 3 4 5 6 7 8 9 10 [ 00��000−�00] [ � ] [ ] [ 0−� 0−� 00 00−�0 ] Te Anthrax-Listeriosis coinfection model can be expressed 12 12 V as [ 0�12 0�12 00 000−�V]

�� =�(�) , where �� (48) �Ω � ℎ �=(�,�,�,�,�,�,�,�,�,� ) . �1 = , where 1 2 3 4 5 6 7 8 9 10 �ℎ

�� (� + �ℎ +�)(�+�ℎ +�+�) Te following system of diferential equations is obtained: �2 = , �(�+�ℎ +�+�+�(�+�ℎ +�))

��1 3 =Ωℎ +��5 +��6 +��7 −�ℎ�10�1 −��1 �V (�+�+�ℎ) �� 3 = , ΩV�V −�ℎ�1 �4 =(�+�+�ℎ), �� 2 =�� −�� −(�+� +�)� �� ℎ 10 1 2 ℎ 2 �5 =(�+�ℎ +�), �� =(�+� +�+�), 3 =�� −�� −(�+� +�+�)� 6 ℎ �� 1 � 10 3 ℎ 3 �7 = (1−�) ��, ��4 =��10�3 +��2 +(�+�ℎ +�+�)�4 �� 8 =(�+�ℎ), 8 International Journal of Mathematics and Mathematical Sciences

� = (1−�) (1 − �) �, 9 0.4

�10 =(�+�ℎ), 0.35

�11 =(�+�ℎ) 0.3 ΩV�V and �12 = . �V 0.25 (51) 0.2 Clearly, the Jacobian matrix at disease-free equilibrium has a case of simple zero eigenvalue as well as other eigenvalues 0.15 with negative real parts. Tis is an indication that the center manifold theorem is applicable. By applying the center man- 0.1 ifold theorem in Castillo-Chavez and Song [22], the lef and �(� ) only Anthrax with infected Population righteigenvectorsoftheJacobianmatrix 0 are computed 0.05 frst. Letting the lef and right eigenvector represented by

�=[�1,�2,�3,�4,�5,�6,�7,�8,�9,�10] 0 (52) 0.8 1 1.2 1.4 � R and �=[�1,�2,�3,�4,�5,�6,�7,�8,�9,�10] , 0a respectively, the following were obtained: Figure 2: Simulation of the coinfection model showing the existence 2 of backward bifurcation. ��5 �2�V (� + � + �ℎ) �1 = + , �ℎ �ℎ It can be deduced that the coefcient � would always be �2 positive. Backward bifurcation will take place in the system � = V , � 2 Ω � of diferential equations in (3) if the coefcient is positive. V V In conclusion, it implies that the disease-free equilibrium is � =� =� =� =� =0, not globally stable. 3 4 6 7 8 (53) Figure 2 shows the simulation of the coinfection model 2 ��V indicating the phenomenon of backward bifurcation as evi- �5 = , ΩV�V (�+�ℎ) dence to the model analysis. Tis phenomenon usually exists in cases where the disease-free equilibrium and the endemic �9 =−�10, equilibrium coexist. Epidemically, the implication is that the concept of whenever the basic reproduction number � =1. 10 is less than unity, the ability to control the disease is no And longer sufcient. Figure 2 confrms the analytical results � =� =� =� =� =� =� =0, which shows that endemic equilibrium exists when the basic 1 3 5 6 7 8 9 reproduction number is greater than unity. V10ΩV�V �2 = , �V (�+�+�ℎ) 6. Sensitivity Analysis of (54) the Coinfection Model �2 =�4, −� (� + � +�+�) In this section, we performed the sensitivity analysis of the � = V ℎ . basic reproduction number of the coinfection model to each 10 Ω � V V of the parameter values. Tis is to determine the signifcance Moreover, by further simplifcations, it can be shown that or contribution of each parameter on the basic reproduction number. Te sensitivity index of the basic reproduction �� 3 (� + � +�+�) �= 10 V ℎ number (R0) to a parameter � is given by the relation ΩV�V R �R � �2 (� + � +�+�) Π 0 =( 0 )( ). V ℎ � �� R (56) −2�10�V [ 0 �ℎΩV�V (55) Sensitivity analysis of the basic reproduction number of ��2 (� + � +�+�) + V ℎ ], Anthrax R0� and Listeriosis R0� to each of the parameter �ℎΩV�V (� + �ℎ)(�+�+�ℎ) values was computed separately, since the basic reproduction Ω number of the coinfection model is usually �=�� ℎ >0. 2 10 � ℎ R0 = max {R0�, R0�}. (57) International Journal of Mathematics and Mathematical Sciences 9

Table 1: Sensitivity indices of R0� to each of the parameter values. analysis of the model which has already been performed. Te variable and parameter values in Table 3 were used in the Parameter Description Sensitivity Index simulation of the coinfection model in (3). For the purposes Ωℎ Human recruitment rate 1.2164 of illustrations, we assumed some of the parameter values. ΩV Vector recruitment rate 0.2433 Table 3 shows the detailed description of parameters and �ℎ Human transmission rate 0.1216 values that were used in the simulations of model (3). We �V Vector transmission rate 0.0243 � Anthraxrecoveryrate −0.0037 used a Range-Kutta fourth-order scheme in the numerical �ℎ Human natural death rate −0.0122 solutions of the system of diferential equations in model (3) �V Vector natural death rate −0.0061 by using matlab program. � Anthrax related death rate −0.0065 � 3.42913 ∗ 10−6 Modifcation parameter 7.1. Simulation of Model Showing the Efects of Increasing Force of Infection on Infectious Anthrax and Listeriosis Populations Only. In this section, we simulate the system of diferential 6.1. Sensitivity Indices of R0�. In this section, we derive the equations in model (3) by varying the human contact rate sensitivity of R0�, to each of the parameters. Table 1 shows to see its efects on infected Anthrax population, infected the detailed sensitivity indices of the basic reproduction Listeriosis population, and Anthrax-Listeriosis coinfected number of Anthrax (R0�) to each of the parameter values. population. Tis was done by setting the values of human From the values in Table 1, it can be observed that the contact rate as �ℎ = 0.01,�ℎ = 0.02,�ℎ =0.03,and�ℎ = most sensitive parameters are human transmission rate, 0.04. Figure 3 shows an increase in the infected Anthrax vector transmission rate, human recruitment rate, and vector population as the value of contact rate increases. Moreover, as recruitment rate. Since the basic reproduction number is less thevalueofthehumancontactratedecreases,thenumberof than one, increasing the human recruitment rate by 10% Anthraxinfectedpopulationdecreaseswithtime.However, would increase the basic reproduction number of Anthrax by an increase or decrease in the human contact rate increases 12.164%. However, decreasing the human recruitment rate or decreases the Listeriosis infected population with time as by 10% would decrease the basic reproduction number of confrmed in Figure 4. Te number of Anthrax-Listeriosis Anthrax by 12.164%. Moreover, decreasing human and vector coinfected population shows a sharp reduction in the number transmission rates by 10%woulddecreasethebasicreproduc- ofindividualsinfectedwithbothdiseasesbutthethereisan tion number of Anthrax by 1.216%and0.243%, respectively. increase in the number of infectious population as shown in However, increasing human and vector transmission rates Figure 5. An increase or decrease in the human contact rate by 10% would increase the basic reproduction number of showsanincreaseordecreaseinthenumberofAnthrax- Anthrax by 1.216%and0.243%, respectively. Te sensitivity Listeriosis coinfected population as indicated in Figure 5. analysis determines the contribution of each parameter to the Analysis of force of infection gives a better understanding of basic reproduction number. Tis is an improvement of the the efects of the contact rate which was not considered by the work done by authors in [2, 3]. work of authors in [2, 3, 20].

6.2. Sensitivity Indices of R0�. In this section, we derive the 7.2. Simulation of Model Showing Infected Anthrax, Listeriosis, sensitivity of R0� to each of the parameters. Te detailed and Coinfected Populations. In this section, we simulate sensitivity indices of the basic reproduction number of the model (3) to see the behaviour of Anthrax infected Listeriosis (R0�) to each of the parameter values are shown population, Listeriosis infected population, and Anthrax- inTable2.WeobservefromthevaluesinTable2that Listeriosis coinfected population. Figure 6 shows an increase the most sensitive parameters are bacteria ingestion rate, in the number of Anthrax infected individuals and a sharp Listeriosis related death, human recruitment rate, and Lis- increase in the number of Listeriosis infected individuals. teriosis contribution to environment. Decreasing the human Figure 7 shows a sharp reduction in the number of Anthrax- recruitment rate by 10% would cause a decrease in the basic Listeriosis coinfected population from the beginning and reproduction number of Listeriosis by 0.201487%. However, it increases steadily at a point in time. Since the number increasing the human recruitment rate by 10% would cause of susceptible human populations increases in the system an increase in the basic reproduction number of Listeriosis by with time, there are higher chances of individuals being 0.201487%. Moreover, decreasing Listeriosis contribution to infected with Anthrax, Listeriosis, and Anthrax-Listeriosis environment and bacteria ingestion rate by 10%wouldcause coinfection. Tis is because the concept of mass action was adecreaseinthebasicreproductionnumberofListeriosis. one of the assumptions that was incorporated in our model. Increasing Listeriosis contribution to environment and bacte- 10 ria ingestion rate by %wouldcauseanincreaseinthebasic 7.3. Simulation of Model Showing Susceptible Human Bacte- reproduction number of Listeriosis. ria Populations. In this section, we simulate model (3), to observe the behaviour of the susceptible human population 7. Numerical Methods and Results and how the bacteria (carcasses) growth behaves with time in the epidemics. Figure 8 shows an increase in both the In this section, we carried out the numerical simulations of susceptible and bacteria growth. An increase in the number the coinfection model to illustrate the results of the qualitative of susceptible from the beginning confrms the increase in 10 International Journal of Mathematics and Mathematical Sciences

Table 2: Sensitivity indices of R0� to each of the parameter values.

Parameter Description Sensitivity Index

Ωℎ Human recruitment rate 0.0201487 − � Co-infected human recovery rate −5.41441 ∗ 10 6

�ℎ Human natural death rate −0.00014638 − � Listeriosis death rate among co-infected −5.41441 ∗ 10 6 − � Modifcation parameter 3.42913 ∗ 10 6 V Bacteria ingestion rate 0.0000402975 � Listeriosis contribution to environment 0.0000309981 − � Concentration of carcasses −2.01487 ∗ 10 10 � Listeriosis recovery rate −0.0000402218

�� Carcasses mortality rate −0.0080595 � Listeriosis related death −0.0000402218

Table 3: Variable and parameter values of the coinfection model.

Parameter Description Value Reference � Anthrax related death rate 0.2 (Health line, Dec., 2015) � Listeriosis related death rate 0.2 Adak et al., 2002. � Anthrax death rate among co-infected 0.04 assumed � Listeriosis death rate among co-infected 0.08 assumed

�ℎ Human transmission rate 0.01 [23]

�V Vector transmission rate 0.05 assumed � Anthrax waning immunity 0.02 assumed

�V Vector natural death rate 0.0004 [23]

Ωℎ Human recruitment rate 0.001 assumed

ΩV Vector recruitment rate 0.005 [23] � Anthraxrecoveryrate 0.33 [24] � Listeriosis recovery rate 0.002 assumed � Anthrax-Listeriosis waning immunity 0.07 assumed � Listeriosis contribution to environment 0.65 assumed � Co-infected recovery rate 0.005 assumed

�� Bacteria death rate 0.0025 assumed

�ℎ Human natural death rate 0.2 [23] � Listeriosis waning immunity 0.001 assumed � Modifcation parameter 0.45 assumed � Co-infected who recover from Anthrax only 0.025 assumed � Concentration of carcasses 10000 [20] V Bacteria ingestion rate 0.5 [20]

the number of Anthrax infection and Listeriosis infection in fully understand the transmission mechanism of Anthrax- Figure 6. Te increase in the number of susceptible human Listeriosis coinfection. Our model revealed that the disease- populations could be attributed to our model being an open free equilibrium of the Anthrax model only is locally system. stable when the basic reproduction number is less than one and a unique endemic equilibrium whenever the basic 8. Conclusion reproduction number is greater than one. Te disease-free equilibrium of the Listeriosis model only is locally stable In this paper, we analysed the transmission dynamics of when the basic reproduction number is less than one and Anthrax-Listeriosis coinfection model. Te compartmen- a unique endemic equilibrium whenever the basic repro- tal model was analysed qualitatively and quantitatively to duction number is greater than one. Our model analysis International Journal of Mathematics and Mathematical Sciences 11

EFFECTS OF INCREASING FORCE OF EFFECTS OF INCREASING FORCE OF INFECTION () ON ANTHRAX INFECTED INFECTION () ON CO-INFECTED POPULATION POPULATION ONLY 100 90

300 80 70

250 60

50 200 40

150 30 20 Anthrax-Listeriosis co-infected Population co-infected Anthrax-Listeriosis Anthrax Infected Population only Population Infected Anthrax 100 10 0 20 40 60 80 100 120 50 Time (Days) New caption 02040 60 80 100 120  = 0.04  = 0.01 Time (Days)  = 0.03  = 0.02  = 0.02  = 0.01  = 0.03  = 0.04 Figure 5: Simulation showing the efects of increasing the force of infection on Anthrax infected population only. Figure 3: Simulation showing the efects of increasing the force of infection on Anthrax infected population only.

unity and unstable otherwise does not apply. Tis means that EFFECTS OF INCREASING FORCE OF INFECTION ON LISTERIOSIS INFECTED the Anthrax-Listeriosis coinfection model shows a case of POPULATION ONLY coexistence of the disease-free equilibrium and the endemic 200 equilibrium whenever the basic reproduction number is less than one. 180 We performed the sensitivity analysis of the basic reproductive number to each of the parameters to determine 160 which parameter is more sensitive. Te sensitivity indices of the basic reproduction number of Anthrax to each of the parameter values revealed that the most sensitive parame- 140 ters are human transmission rate, vector transmission rate, human recruitment rate, and vector recruitment rate. Since 120 the basic reproduction number is less than one, increasing the human recruitment rate would increase the basic reproduction number. Tis analysis is an improvement of

Listeriosis infected population only Listeriosis infected population 100 theworkdoneby[2,3].Teyconsideredthedynamicsof Anthrax in animal population but never considered sensitiv- 80 ity analysis to determine the most sensitive parameter to the 02040 60 80 100 120 model. Time (Days) Te sensitivity indices of the basic reproduction number of Listeriosis to each of the parameter values shows that the  = 0.04  = 0.01 most sensitive parameters are bacteria ingestion rate, Liste-  = 0.03  = 0.02 riosis related death, human recruitment rate, and Listeriosis Figure 4: Simulation showing the efects of increasing the force of contribution to environment. infection on Anthrax infected population only. We simulate the Anthrax-Listeriosis coinfection model by varying the human contact rate to see its efects on also reveals that the disease-free equilibrium of the Anthrax- infected Anthrax population, infected Listeriosis population, Listeriosis coinfection model is locally stable whenever the and Anthrax-Listeriosis coinfected population. Tis analysis basic reproduction number is less than one. Te phenomenon is an improvement of the work done by authors in [2, 3, 20]. of backward bifurcation was exhibited by our model. Te Our simulation shows an increase in the infected Anthrax biological implication is that the idea of the model been population, an increase the number Listeriosis infected pop- locally stable whenever the reproduction number is less than ulation,andanincreaseinthenumberofAnthrax-Listeriosis 12 International Journal of Mathematics and Mathematical Sciences

300 180

170 250 160

200 150

140 150 130 Listeriosis infected only 120 100

Anthrax Infected Population only Population Infected Anthrax 110

50 100 0 20 40 60 80 100 120 02040 60 80 100 120 Time (Days) Time (Days)

Anthrax infected population only Listeriosis infected only

Figure 6: Simulation showing infected Anthrax and infected Listeriosis population.

100

30 Anthrax-Listeriosis co-infected Population co-infected Anthrax-Listeriosis 20

0 20 40 60 80 100 120 Time (Days)

Co-infected population

Figure 7: Simulation of model showing Anthrax-Listeriosis coinfection population.

coinfected population as the value of the human contact rate articles are cited in Table 3 of this paper. Tese published increases. articles are also cited at relevant places within the text as references. Data Availability Conflicts of Interest Te data supporting this deterministic model are from previously published articles and they have been duly cited Te authors declare that there are no conficts of interest in this paper. Tose parameter values taken from published regarding the publication of this paper. International Journal of Mathematics and Mathematical Sciences 13

200000 450 180000

400 160000

140000 350 120000

300 100000

80000 250 Bacteria population Bacteria 60000 Susceptible human population human Susceptible 200 40000

20000 150

0 20 40 60 80 100 120 0 20 40 60 80 100 120 Time (Days) Time (Days) Susceptible human population Bacteria population

Figure 8: Simulation of model showing susceptible human and bacteria populations.

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