A Mathematical Model for Coinfection of Listeriosis and Anthrax Diseases

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 coinfectionmodel 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.


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][11][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.

Model Formulation
In this section, we divide the model into subcompartments (groups) as shown in Figure 1.The total human population ( ℎ ) is divided into subcompartments consisting of susceptible humans ( ℎ ), 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  V ; this is divided into subcompartments that consist of susceptible animals ( V ) and animals infected with Anthrax ( V ), 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  =   V/( +   ).
The concentration of carcasses and ingestion rate are denoted as  and V, 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 (1 − ) are the bi-infected persons who have recovered from Anthrax only.The natural death rates of human and vector populations are  ℎ and  V , respectively, and the modification parameter is given by .The coinfected persons who have recovered from Listeriosis are denoted by (1 − )(1 − ).This implies that ( The following differential equations were obtained from the flowchart diagram of the coinfection model in Figure 1:

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.Proof.The Listeriosis force of infection, ( =   V/( +   )), satisfies the polynomial; where  = Ω ℎ ( +  ℎ ) +   (( +  ℎ ) +  ℎ ( +  ℎ + )), and International Journal of Mathematics and Mathematical Sciences By mathematical induction,  > 0 and  > 0 whenever the basic reproduction number is less than one (R 0 < 1).This implies that  * = −/ ≤ 0. In conclusion, the Listeriosis model has no endemic equilibrium and the basic reproductive number is less than one (R 0 < 1).
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 (R 0 < 1).

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.

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, (R 0 ), is given by Watmough [21] on the systems of equations in model (11), the linear stability of the disease-free equilibrium, ( 0 ), can be ascertained.The disease-free equilibrium is locally asymptotically stable whenever the basic reproduction number is less than one (R 0 < 1).And it is unstable whenever the basic reproduction number is greater than one (R 0 > 1).
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.

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 )

Existence of the Endemic Equilibrium
Lemma 2. The Anthrax only model has a unique endemic equilibrium whenever the basic reproduction number (R 0 ) is greater than one (R 0 > 1).
Proof.Considering the endemic equilibrium points of the Anthrax only model, The endemic equilibrium point satisfies the given polynomial where and By mathematical induction,  1 > 0 and  1 > 0 whenever the basic reproduction number is less than one (R 0 < 1).This implies that  *  = − 1 / 1 ≤ 0. In conclusion, the Anthrax only model has no endemic any time the basic reproductive number is less than one (R 0 < 1).
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 (R 0 < 1).

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.

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 and Theorem 3. The disease-free equilibrium ( 0 ) is locally asymptotically stable whenever the basic reproduction number is less than one (R  < 1) and unstable otherwise.

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 Solving for  ℎ in the above, where and Also, letting this implies By substituting  ℎ into the basic reproduction number of Listeriosis (R  ), where the basic reproduction number of Listeriosis only model ( 0 ) is given in the relation Now, taking the partial derivative of R  with respect to R  in (32) gives If ( + ) ≥ ( +  + ), the derivative (R  /R  ), is strictly positive.Two scenarios can be deduced from the derivative (R  /R  ), depending on the values of the parameters: and (1) If R  /R  = 0, it implies that (+) = (++) and the epidemiological implication is that Anthrax has no significance effect on the transmission dynamics of Listeriosis.
(2) If R  /R  > 0, 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 R  in terms of R  , where By letting it implies that Therefore, Now, taking the partial derivative of R  with respect to R  in equation (40) gives If the partial derivative of R  with respect to R  is greater than zero, (R  /R  > 0), 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 R  with respect to , (R  /).
Clearly, R  is a decreasing function of ; the epidemiological implication is that the treatment of Listeriosis would have an impact on the transmission dynamics of Anthrax.

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 ( ℎ ) and V as the bifurcation parameters, it implies that R  = 1 and R  = 1 if and only if and (48) 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 ( 0 ).The Jacobian matrix at diseasefree equilibrium is given by where and 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 ( 0 ) 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 (55) 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.

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 (R 0 ) to a parameter  is given by the relation Sensitivity analysis of the basic reproduction number of Anthrax R 0 and Listeriosis R 0 to each of the parameter values was computed separately, since the basic reproduction number of the coinfection model is usually

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.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].

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.

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 Bacteria ingestion rate 0.5 [20] 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.

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 diseasefree 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].coinfected population as the value of the human contact rate increases.

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

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

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

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

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

)
[21]Basic Reproduction Number.The concept of the nextgeneration 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 (R  ) given by R  = max {R  , R  } (25) where R  and R  are the basic reproduction numbers of Anthrax and Listeriosis, respectively.

Table 1 :
[2,3]tivity indices of R 0 to each of the parameter values.Sensitivity Indices of R 0 .In this section, we derive the sensitivity of R 0 , to each of the parameters.Table1shows the detailed sensitivity indices of the basic reproduction number of Anthrax (R 0 ) to each of the parameter values.From the values in Table1, 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 10% would increase the basic reproduction number of Anthrax by 12.164%.However, decreasing the human recruitment rate by 10% would decrease the basic reproduction number of Anthrax by 12.164%.Moreover, decreasing human and vector transmission rates by 10% would decrease the basic reproduction number of Anthrax by 1.216% and 0.243%, respectively.However, increasing human and vector transmission rates by 10% would increase the basic reproduction number of Anthrax by 1.216% and 0.243%, 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].6.2.Sensitivity Indices of R 0. In this section, we derive the sensitivity of R 0 to each of the parameters.The detailed sensitivity indices of the basic reproduction number of Listeriosis (R 0 ) to each of the parameter values are shown in Table2.We observe from the values in Table2that 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 10% would cause a decrease in the basic reproduction number of Listeriosis by 0.201487%.However, increasing the human recruitment rate by 10% would cause an increase in the basic reproduction number of Listeriosis by 0.201487%.Moreover, decreasing Listeriosis contribution to environment and bacteria ingestion rate by 10% would cause a decrease in the basic reproduction number of Listeriosis.Increasing Listeriosis contribution to environment and bacteria ingestion rate by 10% would cause an increase in the basic reproduction number of Listeriosis.
7.1.Simulation of Model Showing the Effects of Increasing Force of Infection on Infectious Anthrax and Listeriosis PopulationsOnly.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  ℎ = 0.01, ℎ = 0.02, ℎ = 0.03, and  ℎ = 0.04.Figure3shows 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

Table 2 :
Sensitivity indices of R 0 to each of the parameter values.

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