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A Mathematical Model for Coinfection of Listeriosis and Anthrax Diseases

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A Mathematical Model for Coinfection of Listeriosis and Anthrax Diseases 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 ℎIa Ω I R I a Ra ℎ a a Ia S S Ia kRa ℎSℎ Ia +Ial (1-)ΥIal Ral Ωℎ S I S R ℎ al I Ial al ℎRal ℎSℎ +ℎIal (1-)(1-Υ)Ial I Ial Sh R l Rl Cp Il Il Il R C ℎ l p p ℎIl 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 +�� �
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