International Journal of Mathematics and Mathematical Sciences
Volume 2018 (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 Osman¹ and Oluwole Daniel Makinde²

¹Department 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; shaibuo@yahoo.com

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. This 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.

Abstract.

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 differential 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 effects on infected Anthrax population, infected Listeriosis population, and Listeriosis-Anthrax coinfected population.

1. Introduction

Listeriosis and Anthrax are fatal zoonotic diseases caused by Listeria monocytogene and Bacillus Anthracis, respectively. Listeriosis in infants can be acquired in two forms. Mothers usually acquire it after eating foods that are contaminated with Listeria monocytogenes and can develop sepsis resulting in chorioamnionitis and delivering a septic infant or fetus. Moreover, mothers carrying the pathogens in the gastrointestinal tract can infect the skin and respiratory tract of their babies during childbirth. Listeria monocytogenes are among the commonest pathogens responsible for bacterial meningitis among neonates. Responsible factors for the disease include induced immune suppression linked with HIV infection, hemochromatosis hematologic malignancies, cirrhosis, diabetes, and renal failure with hemodialysis [1].

Authors in [2] developed a model for Anthrax transmission but never considered the transmissions in both animal and human populations. Our model is an improvement of the work done by authors in [2, 3]. Both formulated Anthrax models but only concentrated on the disease transmissions in animals cases only. Anthrax disease is caused by bacteria infections and it affects both humans and animals. Our model is an improvement of the two models as we considered Anthrax as a zoonotic disease and also looked at sensitivity analysis and the effects of the contact rate on the disease transmissions.

Authors in [4] published a paper on the effectiveness of constant and pulse vaccination policies using SIR model. The analysis of their results under constant vaccination showed that the dynamics of the disease model is similar to the dynamics without vaccination [5, 6]. There are some findings on the spread of zoonotic diseases but a number of these researches focused on the effect of vaccination on the spread and transmission of the diseases as in the case of the authors in [7]. Moreover, authors in [8] investigated a disease transmission model by considering the impact of a protective vaccine and came up with the optimal vaccine coverage threshold required for disease eradication. However, authors in [9] employed optimal control to study a nonlinear SIR epidemic model with a vaccination strategy. Several mathematical modeling techniques have been employed to study the role of optimal control using SIR epidemic model [10–12]. Authors in [13] formulated an SIR epidemic model 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 and human populations but did not use optimal control as a control measure in fighting the disease. They divided the animal population into four compartments by introducing the vaccination compartment.

Authors in [15] formulated a model and employed optimal control to investigate the impact of chemotherapy on malaria disease with infection immigrants and [16] applied optimal control methods associated with preventing exogenous reinfection based on a exogenous reinfection tuberculosis model. Authors in [17] conducted a research on the identification and reservoirs of pathogens for effective control of sporadic disease and epidemics. Listeria monocytogenes is among the major zoonotic food borne pathogen that is responsible for approximately twenty-eight percent of most food-related deaths in the United States annually and a major cause of serious product recalls worldwide. The dairy farm has been observed as a potential point and reservoir for Listeria monocytogenes.

Models are widely used in the study of transmission dynamics of infectious diseases. In recent times, the application of mathematical models in the study of infectious diseases has increased tremendously. Hence the emergence of a branch called mathematical epidemiology. Frequent diagnostic tests, the availability of clinical data, and electronic surveillance have facilitated the applications of mathematical models to critical examining of scientific hypotheses and the design of real-life strategies of controlling diseases [18, 19].

Authors in [20] constructed a coinfection model of malaria and cholera diseases with optimal control but never considered sensitivity analysis and analysis of the force of infection. Sensitivity analysis determines the most sensitive parameters to the model and the analysis of the force of infections determines the effects of the contact rate on the disease transmissions.

2. Model Formulation

In this section, we divide the model into subcompartments (groups) as shown in Figure 1. The total human population is divided into subcompartments consisting of susceptible humans , individuals that are infected with Anthrax, individuals that are infected with Listeriosis , individuals that are infected with both Anthrax and Listeriosis , and those that have recovered from Anthrax, Listeriosis, and both Anthrax and Listeriosis, respectively, , , and . The total vector population is represented by ; this is divided into subcompartments that consist of susceptible animals and animals infected with Anthrax , where is population of carcasses of animals in the soil that may have diet of Anthrax. Carcasses of animals which may have not been properly disposed of have the tendency of generating pathogens. The total vector and human populations are represented as

where

Figure 1: Flowchart for the coinfection model.

The concentration of carcasses and ingestion rate are denoted as and , respectively. Listeriosis related death rates are and , respectively, and Anthrax related death rates are and , respectively. Waning immunity rates are given by , and , and are the recovery rates, respectively, and are the bi-infected persons who have recovered from Anthrax only. The natural death rates of human and vector populations are and , respectively, and the modification parameter is given by . The coinfected persons who have recovered from Listeriosis are denoted by . This implies that

The following differential equations were obtained from the flowchart diagram of the coinfection model in Figure 1:

3. Analysis of Listeriosis Only Model

In this section, only the Listeriosis model is considered in the analysis of the transmission dynamics.

3.1. Disease-Free Equilibrium

We obtain the disease-free equilibrium of the Listeriosis only model by setting the system of equations in (4) to zero. At disease-free equilibrium, there are no infections and recovery.

3.2. Basic Reproduction Number

In this section, the concept of the Next-Generation Matrix would be employed in computing the basic reproduction number. Using the theorem in Van den Driessche and Watmough [21] on the Listeriosis model in (4), the basic reproduction number of the Listeriosis only model, , is given by

3.3. Existence of the Disease-Free Equilibrium

3.4. Endemic Equilibrium

The endemic equilibrium points are computed by setting the system of differential equations in the Listeriosis only model (4) to zero. The endemic equilibrium points are as follows:

3.5. Existence of the Endemic Equilibrium

Lemma 1. The Listeriosis only model has a unique endemic equilibrium if and only if the basic reproduction number

Proof. The Listeriosis force of infection, , satisfies the polynomial;

where , and

By mathematical induction, and whenever the basic reproduction number is less than one This implies that In conclusion, the Listeriosis model has no endemic equilibrium and the basic reproductive number is less than one

The analysis illustrates the impossibility of backward bifurcation in the Listeriosis model, because there is no existence of endemic equilibrium whenever the basic reproduction number is less than one

4. Analysis of Anthrax Only Model

In this section, only the Anthrax model is considered in the analysis of the transmission dynamics.

4.1. Disease-Free Equilibrium

The disease-free equilibrium of the Anthrax only model is obtained by setting the system of equations in model (11) to zero. At disease-free equilibrium, there are no infections and recovery.

4.2. Basic Reproduction Number

In this section, the concept of the Next-Generation Matrix would be employed in computing the basic reproduction number. Using the theorem in Van den Driessche and Watmough [21] on the Anthrax model in (11), the basic reproduction number of the Anthrax only model, , is given by

4.3. Stability of the Disease-Free Equilibrium

Using the next-generation operator concept in Van den Driessche and Watmough [21] on the systems of equations in model (11), the linear stability of the disease-free equilibrium, , can be ascertained. The disease-free equilibrium is locally asymptotically stable whenever the basic reproduction number is less than one And it is unstable whenever the basic reproduction number is greater than one The disease-free equilibrium is the state at which there are no infections in the system. At disease-free equilibrium, there are no infections in the system.

4.4. Endemic Equilibrium

The endemic equilibrium points are computed by setting the system of differential equations in the Anthrax only model (11) to zero. The endemic equilibrium points are as follows:

The endemic equilibrium of the Anthrax only model is given by

4.5. Existence of the Endemic Equilibrium

Lemma 2. The Anthrax only model has a unique endemic equilibrium whenever the basic reproduction number is greater than one .

Proof. Considering the endemic equilibrium points of the Anthrax only model,

The endemic equilibrium point satisfies the given polynomial

where

and

By mathematical induction, and whenever the basic reproduction number is less than one This implies that In conclusion, the Anthrax only model has no endemic any time the basic reproductive number is less than one

The analysis illustrates the impossibility of backward bifurcation in the Anthrax only model. Because there is no existence of endemic equilibrium whenever the basic reproduction number is less than one .

5. Anthrax-Listeriosis Coinfection Model

In this section, the dynamics of the Anthrax-Listeriosis coinfection model in (3) is considered in the analysis of the transmission dynamics.

5.1. Disease-Free Equilibrium

The disease-free equilibrium of the Anthrax-Listeriosis model is obtained by setting the system of equations of model (3) to zero. At disease-free equilibrium, there are no infections and recovery.

The disease-free equilibrium is given by

5.2. Basic Reproduction Number

The concept of the next-generation operator method in Van den Driessche and Watmough [21] was employed on the system of differential equations in model (3) to compute the basic reproduction number of the Anthrax-Listeriosis coinfection model. The Anthrax-Listeriosis coinfection model has a reproduction number given by

where and are the basic reproduction numbers of Anthrax and Listeriosis, respectively.

and

Theorem 3. The disease-free equilibrium is locally asymptotically stable whenever the basic reproduction number is less than one and unstable otherwise.

5.3. Impact of Listeriosis on Anthrax

In this section, the impact of Listeriosis on Anthrax and vice versa is analysed. This is done by expressing the reproduction number of one in terms of the other by expressing the basic reproduction number of Listeriosis on Anthrax, that is, expressing in terms of

from .

Solving for in the above,

where

Also, letting

this implies

By substituting into the basic reproduction number of Listeriosis ,

where the basic reproduction number of Listeriosis only model is given in the relation

Now, taking the partial derivative of with respect to in (32) gives

If , the derivative , is strictly positive. Two scenarios can be deduced from the derivative , depending on the values of the parameters:

(1)If , it implies that and the epidemiological implication is that Anthrax has no significance effect on the transmission dynamics of Listeriosis.(2)If , it implies that , and the epidemiological implication is that an increase in Anthrax cases would result in an increase Listeriosis cases in the environment. That is Anthrax enhances Listeriosis infections in the environment.

However, by expressing the basic reproduction number of Anthrax on Listeriosis, that is expressing in terms of ,

where

By letting

it implies that

Therefore,

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

If the partial derivative of with respect to is greater than zero, , the biological implication is that an increase in the number of cases of Listeriosis would result in an increase in the number of cases of Anthrax in the environment. Moreover, the impact of Anthrax treatment on Listeriosis can also be analysed by taking the partial derivative of with respect to ,

Clearly, is a decreasing function of ; the epidemiological implication is that the treatment of Listeriosis would have an impact on the transmission dynamics of Anthrax.

5.4. Analysis of Backward Bifurcation

In this section, the phenomenon of backward bifurcation is carried out by employing the center manifold theory on the system of differential equations in model (3). Bifurcation analysis was carried out by employing the center manifold theory in Castillo-Chavez and Song [22]. Considering the human transmission rate and as the bifurcation parameters, it implies that and if and only if

and

By considering the following change of variables,

This would give the total population as

By applying vector notation

The Anthrax-Listeriosis coinfection model can be expressed as

The following system of differential equations is obtained:

Backward bifurcation is carried out by employing the center manifold theory on the system of differential equations in model (3). This concept involves the computation of the Jacobian of the system of differential equations in (49) at the disease-free equilibrium . The Jacobian matrix at disease-free equilibrium is given by

where

Clearly, the Jacobian matrix at disease-free equilibrium has a case of simple zero eigenvalue as well as other eigenvalues with negative real parts. This is an indication that the center manifold theorem is applicable. By applying the center manifold theorem in Castillo-Chavez and Song [22], the left and right eigenvectors of the Jacobian matrix are computed first. Letting the left and right eigenvector represented by

respectively, the following were obtained:

And

Moreover, by further simplifications, it can be shown that

It can be deduced that the coefficient would always be positive. Backward bifurcation will take place in the system of differential equations in (3) if the coefficient is positive. In conclusion, it implies that the disease-free equilibrium is not globally stable.

Figure 2 shows the simulation of the coinfection model indicating the phenomenon of backward bifurcation as evidence to the model analysis. This phenomenon usually exists in cases where the disease-free equilibrium and the endemic equilibrium coexist. Epidemically, the implication is that the concept of whenever the basic reproduction number is less than unity, the ability to control the disease is no longer sufficient. Figure 2 confirms the analytical results which shows that endemic equilibrium exists when the basic reproduction number is greater than unity.

Figure 2: Simulation of the coinfection model showing the existence of backward bifurcation.

6. Sensitivity Analysis of the Coinfection Model

In this section, we performed the sensitivity analysis of the basic reproduction number of the coinfection model to each of the parameter values. This is to determine the significance or contribution of each parameter on the basic reproduction number. The sensitivity index of the basic reproduction number to a parameter is given by the relation

Sensitivity analysis of the basic reproduction number of Anthrax and Listeriosis to each of the parameter values was computed separately, since the basic reproduction number of the coinfection model is usually

6.1. Sensitivity Indices of

In this section, we derive the sensitivity of , to each of the parameters. Table 1 shows the detailed sensitivity indices of the basic reproduction number of Anthrax to each of the parameter values. From the values in Table 1, it can be observed that the most sensitive parameters are human transmission rate, vector transmission rate, human recruitment rate, and vector recruitment rate. Since the basic reproduction number is less than one, increasing the human recruitment rate by would increase the basic reproduction number of Anthrax by . However, decreasing the human recruitment rate by would decrease the basic reproduction number of Anthrax by . Moreover, decreasing human and vector transmission rates by would decrease the basic reproduction number of Anthrax by and , respectively. However, increasing human and vector transmission rates by would increase the basic reproduction number of Anthrax by and , respectively. The sensitivity analysis determines the contribution of each parameter to the basic reproduction number. This is an improvement of the work done by authors in [2, 3].

Table 1: Sensitivity indices of to each of the parameter values.

Parameter
Description
Sensitivity Index

Human recruitment rate

Vector recruitment rate

Human transmission rate

Vector transmission rate

Anthrax recovery rate

Human natural death rate

Vector natural death rate

Anthrax related death rate

Modification parameter

6.2. Sensitivity Indices of

In this section, we derive the sensitivity of to each of the parameters. The detailed sensitivity indices of the basic reproduction number of Listeriosis to each of the parameter values are shown in Table 2. We observe from the values in Table 2 that the most sensitive parameters are bacteria ingestion rate, Listeriosis related death, human recruitment rate, and Listeriosis contribution to environment. Decreasing the human recruitment rate by would cause a decrease in the basic reproduction number of Listeriosis by . However, increasing the human recruitment rate by would cause an increase in the basic reproduction number of Listeriosis by . Moreover, decreasing Listeriosis contribution to environment and bacteria ingestion rate by would cause a decrease in the basic reproduction number of Listeriosis. Increasing Listeriosis contribution to environment and bacteria ingestion rate by would cause an increase in the basic reproduction number of Listeriosis.

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

Parameter
Description
Sensitivity Index

Human recruitment rate

Co-infected human recovery rate

Human natural death rate

Listeriosis death rate among co-infected

Modification parameter

Bacteria ingestion rate

Listeriosis contribution to environment

Concentration of carcasses

Listeriosis recovery rate

Carcasses mortality rate

Listeriosis related death

7. Numerical Methods and Results

In this section, we carried out the numerical simulations of the coinfection model to illustrate the results of the qualitative analysis of the model which has already been performed. The variable and parameter values in Table 3 were used in the simulation of the coinfection model in (3). For the purposes of illustrations, we assumed some of the parameter values. Table 3 shows the detailed description of parameters and values that were used in the simulations of model (3). We used a Range-Kutta fourth-order scheme in the numerical solutions of the system of differential equations in model (3) by using matlab program.

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

Human transmission rate
0.01
[23]

Vector transmission rate
0.05
assumed

Anthrax waning immunity
0.02
assumed

Vector natural death rate
0.0004
[23]

Human recruitment rate
0.001
assumed

Vector recruitment rate
0.005
[23]

Anthrax recovery rate
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

Human natural death rate
0.2
[23]

Listeriosis waning immunity
0.001
assumed

Modification parameter
0.45
assumed

Co-infected who recover from Anthrax only
0.025
assumed

Concentration of carcasses
10000
[20]

Bacteria ingestion rate
0.5
[20]

7.1. Simulation of Model Showing the Effects of Increasing Force of Infection on Infectious Anthrax and Listeriosis Populations Only

In this section, we simulate the system of differential equations in model (3) by varying the human contact rate to see its effects on infected Anthrax population, infected Listeriosis population, and Anthrax-Listeriosis coinfected population. This was done by setting the values of human contact rate as ,,, and . Figure 3 shows an increase in the infected Anthrax population as the value of contact rate increases. Moreover, as the value of the human contact rate decreases, the number of Anthrax infected population decreases with time. However, an increase or decrease in the human contact rate increases or decreases the Listeriosis infected population with time as confirmed in Figure 4. The number of Anthrax-Listeriosis coinfected population shows a sharp reduction in the number of individuals infected with both diseases but the there is an increase in the number of infectious population as shown in Figure 5. An increase or decrease in the human contact rate shows an increase or decrease in the number of Anthrax-Listeriosis coinfected population as indicated in Figure 5. Analysis of force of infection gives a better understanding of the effects of the contact rate which was not considered by the work of authors in [2, 3, 20].

Figure 3: Simulation showing the effects of increasing the force of infection on Anthrax infected population only.

Figure 4: Simulation showing the effects of increasing the force of infection on Anthrax infected population only.

Figure 5: Simulation showing the effects of increasing the force of infection on Anthrax infected population only.

7.2. Simulation of Model Showing Infected Anthrax, Listeriosis, and Coinfected Populations

In this section, we simulate the model (3) to see the behaviour of Anthrax infected population, Listeriosis infected population, and Anthrax-Listeriosis coinfected population. Figure 6 shows an increase in the number of Anthrax infected individuals and a sharp increase in the number of Listeriosis infected individuals. Figure 7 shows a sharp reduction in the number of Anthrax-Listeriosis coinfected population from the beginning and it increases steadily at a point in time. Since the number of susceptible human populations increases in the system with time, there are higher chances of individuals being infected with Anthrax, Listeriosis, and Anthrax-Listeriosis coinfection. This is because the concept of mass action was one of the assumptions that was incorporated in our model.

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

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

7.3. Simulation of Model Showing Susceptible Human Bacteria Populations

In this section, we simulate model (3), to observe the behaviour of the susceptible human population and how the bacteria (carcasses) growth behaves with time in the epidemics. Figure 8 shows an increase in both the susceptible and bacteria growth. An increase in the number of susceptible from the beginning confirms the increase in the number of Anthrax infection and Listeriosis infection in Figure 6. The increase in the number of susceptible human populations could be attributed to our model being an open system.

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

8. Conclusion

In this paper, we analysed the transmission dynamics of Anthrax-Listeriosis coinfection model. The compartmental model was analysed qualitatively and quantitatively to fully understand the transmission mechanism of Anthrax-Listeriosis coinfection. 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 and a unique endemic equilibrium whenever the basic reproduction number is greater than one. The disease-free equilibrium of the Listeriosis model only is locally stable when the basic reproduction number is less than one and a unique endemic equilibrium whenever the basic reproduction number is greater than one. Our model analysis also reveals that the disease-free equilibrium of the Anthrax-Listeriosis coinfection model is locally stable whenever the basic reproduction number is less than one. The phenomenon of backward bifurcation was exhibited by our model. The biological implication is that the idea of the model been locally stable whenever the reproduction number is less than unity and unstable otherwise does not apply. This means that the Anthrax-Listeriosis coinfection model shows a case of coexistence of the disease-free equilibrium and the endemic equilibrium whenever the basic reproduction number is less than one.

We performed the sensitivity analysis of the basic reproductive number to each of the parameters to determine which parameter is more sensitive. The sensitivity indices of the basic reproduction number of Anthrax to each of the parameter values revealed that the most sensitive parameters are human transmission rate, vector transmission rate, human recruitment rate, and vector recruitment rate. Since the basic reproduction number is less than one, increasing the human recruitment rate would increase the basic reproduction number. This analysis is an improvement of the work done by [2, 3]. They considered the dynamics of Anthrax in animal population but never considered sensitivity analysis to determine the most sensitive parameter to the model.

The sensitivity indices of the basic reproduction number of Listeriosis to each of the parameter values shows that the most sensitive parameters are bacteria ingestion rate, Listeriosis related death, human recruitment rate, and Listeriosis contribution to environment.

We simulate the Anthrax-Listeriosis coinfection model by varying the human contact rate to see its effects on infected Anthrax population, infected Listeriosis population, and Anthrax-Listeriosis coinfected population. This analysis is an improvement of the work done by authors in [2, 3, 20]. Our simulation shows an increase in the infected Anthrax population, an increase the number Listeriosis infected population, and an increase in the number of Anthrax-Listeriosis coinfected population as the value of the human contact rate increases.

Data Availability

The data supporting this deterministic model are from previously published articles and they have been duly cited in this paper. Those parameter values taken from published articles are cited in Table 3 of this paper. These published articles are also cited at relevant places within the text as references.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

W. F. Schlech and D. Acheson, “Foodborne listeriosis,” Clinical Infectious Diseases, vol. 31, no. 3, pp. 770–775, 2000.

C. M. Saad-Roy, P. van den Driessche, and A.-A. Yakubu, “A mathematical model of anthrax transmission in animal populations,” Bulletin of Mathematical Biology, vol. 79, no. 2, pp. 303–324, 2017.

A. Friedman and A. A. Yakubu, “Anthrax epizootic and migration: persistence or extinction,” Mathematical Biosciences, vol. 241, no. 1, pp. 137–144, 2013.

Z. Lu, X. Chi, and L. Chen, “The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission,” Mathematical and Computer Modelling, vol. 36, no. 9-10, pp. 1039–1057, 2002.

A. A. Lashari and G. Zaman, “Optimal control of a vector borne disease with horizontal transmission,” Nonlinear Analysis: Real World Applications, vol. 13, no. 1, pp. 203–212, 2012.

A. A. Lashari, “Optimal control of an SIR epidemic model with a saturated treatment,” Applied Mathematics & Information Sciences, vol. 10, no. 1, pp. 185–191, 2016.

S. Mushayabasa and C. P. Bhunu, “Is HIV infection associated with an increased risk for cholera? Insights from a mathematical model,” BioSystems, vol. 109, no. 2, pp. 203–213, 2012.

A. B. Gumel and S. M. Moghadas, “A qualitative study of a vaccination model with non-linear incidence,” Applied Mathematics and Computation, vol. 143, no. 2-3, pp. 409–419, 2003.

T. K. Kar and A. Batabyal, “Stability analysis and optimal control of an SIR epidemic model with vaccination,” BioSystems, vol. 104, no. 2-3, pp. 127–135, 2011.

O. D. Makinde, “Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy,” Applied Mathematics and Computation, vol. 184, no. 2, pp. 842–848, 2007.

T. T. Yusuf and F. Benyah, “Optimal control of vaccination and treatment for an SIR epidemiological model,” World Journal of Modelling and Simulation, vol. 8, no. 3, pp. 194–204, 2012.

G. Zaman, Y. H. Kang, and I. H. Jung, “Optimal treatment of an SIR epidemic model with time delay,” BioSystems, vol. 98, no. 1, pp. 43–50, 2009.

G. Zaman, Y. Han Kang, and I. H. Jung, “Stability analysis and optimal vaccination of an SIR epidemic model,” BioSystems, vol. 93, no. 3, pp. 240–249, 2008.

O. Shaibu, D. M. Oluwole, and M. T. David, “Stability analysis and modelling of listeriosis dynamics in human and animal populations,” The Global Journal of Pure and Applied Mathematics (GJPAM), vol. 14, no. 1, pp. 115–137, 2018.

O. D. Makinde and K. O. Okosun, “Impact of chemo-therapy on optimal control of malaria disease with infected immigrants,” BioSystems, vol. 104, no. 1, pp. 32–41, 2011.

K. Hattaf, M. Rachik, S. Saadi, Y. Tabit, and N. Yousfi, “Optimal control of tuberculosis with exogenous reinfection,” Applied Mathematical Sciences, vol. 3, no. 5-8, pp. 231–240, 2009.

M. K. Borucki, J. Reynolds, C. C. Gay et al., “Dairy farm reservoir of Listeria monocytogenes sporadic and epidemic strains,” Journal of Food Protection, vol. 67, no. 11, pp. 2496–2499, 2004.

H. W. Hethcote, “Qualitative analyses of communicable disease models,” Mathematical Biosciences, vol. 28, no. 3-4, pp. 335–356, 1976.

N. C. Grassly and C. Fraser, “Mathematical models of infectious disease transmission,” Nature Reviews Microbiology, vol. 6, no. 6, pp. 477–487, 2008.

K. O. Okosun and O. D. Makinde, “A co-infection model of malaria and cholera diseases with optimal control,” Mathematical Biosciences, vol. 258, pp. 19–32, 2014.

P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29–48, 2002.

C. Castillo-Chavez and B. Song, “Dynamical models of tuberculosis and their applications,” Mathematical Biosciences and Engineering, vol. 1, no. 2, pp. 361–404, 2004.

S. Mushayabasa, T. Marijani, and M. Masocha, “Dynamical analysis and control strategies in modeling anthrax,” Computational and Applied Mathematics, vol. 36, no. 3, pp. 1333–1348, 2017.

R. Brookmeyer, E. Johnson, and R. Bollinger, “Modeling the optimum duration of antibiotic prophylaxis in an anthrax outbreak,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 100, no. 17, pp. 10129–10132, 2003.