BIFURCATION, SENSITIVITY AND OPTIMAL CONTROL ANALYSIS OF MODELLING ANTHRAX-LISTERIOSIS CO-DYNAMICS

In this paper, a deterministic model for the co-dynamics of Anthrax and Listerios diseases was formulated to investigate the qualitative and quantitative relationship of both diseases by incorporating prevention and treatment controls. The basic reproduction number, stability, existence and equilibria of each disease was investigated separately. The Anthrax-Listeriosis co-dynamics model was analysed and it the idea of backward bifurcation existed. The impact of Anthrax infection on the transmission of Listeriosis was determined. The AnthraxListeriosis co-dynamics model was extended and included time dependent control variables. Pontryagin’s Maximum Principle was used to obtain the optimal control strategies needed for eradication of Anthrax-Listeriosis infections. We performed the numerical simulation of the co-dynamics model in order to give the quantitative implications of the results. It was established that Anthrax infection can be attributed to increased risk of Listeriosis but Listeriosis infection is not associate with the risk of Anthrax. Effective control of Anthrax means incorporating both the intervention srategies of Anthrax and Listeriosis.


INTRODUCTION
Researchers in [1,2] attempted to determine effectiveness of vaccination policies using SIR model. From the theoretical results of their study under constant vaccination, the dynamics of the disease model is similar to dynamics without vaccination. Several studies have used the methods of optimal control theory in the formulation of the models [3]. However, some of these studies focused on the effects of vaccination on the spread and transmission of the diseases as in the case of the authors in [4].
Moreover, authors in [5] studied a disease transmission model by considering the impact of a protective vaccine and came out with the optimal vaccine coverage threshold required for disease eradication. Also, in [6], optimal control was used to study a nonlinear SIR epidemic model with vaccination strategy. Some modelling techniques have been employed and established the role of optimal control using SIR epidemic model [7,8]. [9], formulated an SIR epidemic model by considering vaccination as control measure.
Authors in [10] formulated a model for the transmission of Listeriosis in animal and human populations but never considered optimal control strategies in combating the disease. [11], also applied optimal control to investigate the impact of chemo-therapy on malaria disease with infection immigrants and [12] applied optimal control methods associated with preventing exogenous reinfection based on a exogenous reinfection tuberculosis model. [13] researched on the identification and reservours 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 reservour for listeria monocytogenes.
Listeria monocytogenes is the third major and common pathogen responsible for bacterial meningitis among neonates in North America. Factors that are responsible and can increase the risk of Listeriosis include acquired and induced immune suppression linked with HIV infection, hematologic malignancies,cirrhosis, diabetes, hemochromatosis and renal failure with hemodialysis [14].
Deterministic models are often used to study the dynamics of diseases in epidemiology. In recent times application of models in the study of disease transmission has increased. The availability of clinical data and electronic surveillance has facilitated the applications of mathematical models to critical examining of scientific hypotheses and the design of strategies of combating diseases [15,16,17].

MODEL FORMULATION
In this section, we divide the population into compartments. Total human and vector populations were represented by N h and N v respectively. Total human and vector populations expressed as; (1) N h = S h + I a + I l + I al + R a + R l + R al .
(2)  The variables used in the frmulation of the model are described in Table 1.
Waning immunity rates are given by; ω, k and ψ. Where α, δ and σ are the recovery rates respectively and τ (1 − σ ) are the co-infected persons who have recovered from Anthrax only.
The co-infected infected persons who have recovered Listeriosis is denoted by This implies that;

ANALYSIS OF LISTERIOSIS ONLY MODEL
In this section, Listeriosis model only is considered.
. We obtain the DFE of the Listeriosis only model by using system of equations in (4).

Basic reproduction number.
In this section, we employ the concept of the Next Generation Matrix in computing ℜ 0l . Using the theorem in [23] on the Listeriosis model only in (4). (ℜ 0l ) , is given by: 3.3. Existence of the disease free equilibrium. The (DFE) of the Listeriosis model only was obtained using system of equations in (4). This was obtained as; ℜ 0l of the Listeriosis only model was established as; Using the next geration operator in [23,24] , the linear stability can be established on the system of equations in (4). The disease-free equilibrium, (ξ 0l ) is locally asymptotically stable whenever (ℜ 0l < 1) and unstable whenever (ℜ 0l > 1) .

Endemic equilibrium (EE).
The EE points are computed using system of equations in (4). The EE points are as follows: Proof. Listeriosis force of infection; π = C p v K +C p , satisfies the polynomial;
In conclusion, the Listeriosis model has no EE any time ℜ 0l < 1.
The analysis illustrates the impossibility of backward bifurcation in the Listeriosis only model. Since there is no existence of EE whenever ℜ 0l < 1.

ANALYSIS OF ANTHRAX ONLY MODEL
In this section, Anthrax only model is considered in the analysis of disease transmission.
. We obtain the DFE of the Anthrax only model using system of equations in (14).
4.2. Basic reproduction number (ℜ 0a ) .. In this section, We employed the concept of the Next Generation Matrix in computing (ℜ 0a ). Using the theorem in [23] on the Anthrax model in equation (14) , ℜ 0a of the Anthrax only model is given by: 4.3. Stability of the disease-free equilibrium. Using the next generation operator concept in [23] on the systems of equation in model (14) , the linear stability of (ξ 0a ), can be established.
The DFE is locally asymptotically stable whenever ℜ 0a < 1 and unstable otherwise.
4.4. Endemic equilibrium. The EE points are computed using system of equations in (14).
The EE points are as follows: The EE of the Anthrax only model is given by; Hence;

Existence of the endemic equilibrium.
Lemma 2. The Anthrax only model has a unique EE whenever ℜ 0a > 1. Considering the EE points of the Anthrax only model; The EE point satisfies the given polynomial; Where; and the Anthrax only model has no endemic any time ℜ 0a < 1.
The analysis shows the impossibility of backward bifurcation in the Anthrax only model.
Because there is no existence of EE whenever ℜ 0a < 1.

ANTHRAX-LISTERIOSIS CO-INFECTION MODEL
We consider the dynamics of the Anthrax-Listeriosis co-infection of system of equations in equation (3).

Disease free equilibrium (DFE).
The DFE of the Anthrax-Listeriosis model is obtained using system of equations in (3). Hence; Basic reproduction number. The concept of the next generation operator method in [23] was employed on the system of equations in (3) to compute ℜ al of the Anthrax-Listeriosis co-infection model. The ℜ al given by; Where, ℜ a and ℜ l are the reproduction numvers of Anthrax and Listeriosis respectively.

Impact of Listeriosis on Anthrax.
In this section, We analysed the impact of Listeriosis on Anthrax and the vice versa. This was done by expressing the reproduction number of one in terms of the other. By expressing the basic reproduction number of Listeriosis on Anthrax, that By substituting µ h into ℜ l ; where R 0l is given the relation; Now, taking the partial derivative of ℜ l with respect to ℜ a ; , is strictly positive. Two scenarios can be deduced from the derivative ∂ ℜ l ∂ ℜ a , depending on the values of the parameters; and the biological implications is that Anthrax has no significance effect on the spread of Listeriosis. ( , and the biological implications 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 where; Now, taking the partial derivative of ℜ a with respect to ℜ l in equation (36), gives; If the partial derivative of ℜ a with respect to ℜ l is greater than zero, ∂ ℜ a ∂ ℜ l > 0 , the biological implication is that an inrease in the number of cases of Listeriosis would result in an increase in the number of cases of Anthrax in the environment. Moreover, the the impact of Anthrax treatment on Listeriosis can also be analysed by taking the partial derivative of ℜ a with respect to α, ∂ ℜ a ∂ α .
Clearly, ℜ a is a decreasing function of α, 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 was carried out by aplying the centre manifold theory on the sysyem of equations in (3) as outlined in [25]. Considering β h and v as bifurcation parameters, it implies that ℜ a = 1 and ℜ l = 1 if and only if, Considering the following change of variables; This would give the total population as; By applying vector notation; The Anthrax-Listeriosis co-infection model can be expressed as; The following system of equation is obtained; (42) Backward bifurcation was carried out by employing the centre manifold theory on the sysyem of equations in (3). This concept involves computation of the Jacobian of the system of equations in (42) at DFE. The Jacobian matrix at DFE is given by; , Clearly, the Jacobian matrix at DFE has a case of simple zero eigenvalue as well as other eigenvalues with negative real parts. This is a clear indication that the centre maniflod theorem is applicable. By applying the centre manifold theorem in [22,25], the left and right eigenvectors of the Jacobian matrix J(ξ 0 ) is computed first. Letting the left and right eigenvector reprensented by: y = y 1 , y 2 , y 3 , y 4 , y 5 , y 6 , y 7 , y 8 , y 9 , y 10 and w = w 1 , w 2 , w 3 , w 4 , w 5 , w 6 , w 7 , w 8 , w 9 , w 10 T respectively.
and y 1 = y 3 = y 5 = y 6 = y 7 = y 8 = y 9 = 0 , By further simplifications, it can be shown that; It can be deduced that the coefficient b would always be positive. Backward bifurcation will take place in the system of equations in (3) if the coefficient a is positive. In conclusion, the DFE is not globally stable.
This phenomenon only exists in situations where DFE and EE coexists. Epidemiological implication is that the idea that whenever ℜ 0 < 1, the disease can be controled is no longer a sufficient condition.

SENSITIVITY ANALYSIS OF THE CO-INFECTION MODEL
We perfomed sensitivity annalysis of ℜ al of the co-dynamics model. This is to determine the significance each parameter on ℜ al [26,27]. The sensitivity index ofℜ 0 to a parameter x is given by the relation: The sensitivity analysis of ℜ 0a and ℜ 0a were determined separately, since the basic reproduction number of the co-infection model is usually; (45) ℜ 0 = max{ ℜ 0a , ℜ 0l } 6.1. Sensitivity indices of ℜ 0a . In this section, we derive the sensitivity of ℜ 0a to each of the parameters. Detailed sensitivity indices of ℜ 0a are shown in Table 3. The values in  6.2. Sensitivity indices of ℜ 0l . In this section, we derive the sensitivity of ℜ 0l to each of the parameters. Detailed sensitivity indices of ℜ 0l are shown in Table 4. The values in Table 4 showed that the most sensitive parameters are the human recruitment rate, Listeriosis contribution

EXTENSION OF THE MODEL TO OPTIMAL CONTROL
In this section, we extended model analysis to optimal control. This was carried out to determine the impact of intervention schemes. The optimal control problem is derived by incorporating the following control functions into the Anthrax-Listeriosis co-infection model in Figure 1 and the introduction of an objective functional that seeks to minimise: (u 1 , u 2 , u 3 , u 4 , u 5 ).
The controls u 1 (t) and u 2 (t) denotes the efforts on preventing Anthrax and Listeriosis respectively.
In biological modelling, the objective of optimal control is to minimise spread or infections, cost of treatment and cost of prevention. The objective functional that can be used to achieve this is given by: subject to the system of equations in (46) . Wheret f is the final period of the intervention. This implies that (A 1 I a , A 2 I l , A 3 I al , A 4 I v ), represents a linear function for the cost associated with infections and A 5 u 2 1 , A 6 u 2 2 , A 7 u 2 3 , Au 2 4 , A 9 u 2 5 , represents a quadratic function for the cost associated with preventions and treatments.
The model control efforts is by linear combination of u 2 i (t), (i = 1, 2) . The quadratic in nature of the control efforts are as a result of the assumption that costs are generally non-linear in nature. Thus, our aim is to minimise the number of infectives and reduce cost of treatment.
Moreover, the characterisation of the optimal control is obtained by solving the partial derivative of the Hamiltonian function with respect to the control sets and equating the derivatives to zero.
where; u i = u * i , and i = 1, 2, 3, . . . , n. The following are obtained; (55) By re-arranging and simplification; (56) By employing the phenomenon of standard control arguments involving the bounds on the controls, it can be concluded that; where; (58)

NUMERICAL RESULTS
Numerical solutions of the optimal system are illlustrated using Runge-Kutta fourth order scheme. These were obtained by solving the state systems, adjoints equations and the transversality conditions. This is a two-point boundary-value problem with two boundary conditions at initial time, t = 0 and final time, t = t f . The objective is to solve this optimal problem at time, t f = 120 days. This period represents the time at which prevention and treatment strategies should end.
Following optimal control strategies were considered; prevention of Anthrax infection u 1 , prevention of Listeriosis infection u 2 , control efforts u 3 and u 4 on treatment of Anthrax and Listeriosis respectively. Control efforts u 5 on the treatment of Anthrax-Listeriosis co-infection.
The four most effective were selected.
The description of the variables and parameters used in the simlation of the co-dynamics model is shown in Table 2.

CONCLUSION
Anthrax-Listeriosis co-infection model was formulated and incorporated the following control strategies; prevention of persons, prevention of vectors, treatment of infected persons, treatment of infected vectors and treatment of Anthrax-Listeriosis co-infected persons. The Co-dynamics model was qualitatively and quantitatively analysed for understanding of the transmission mechanism of Anthrax and Listeriosis co-infection.
Observations: the disease free equilibrium of the Anthrax only model was locally stable whenever ℜ 0a was less one and a unique endemic equilibrium whenever ℜ 0a was greater than one. Moreover, we observe that the disease free equilibrium of the Listeriosis only model was locally stable whenever ℜ 0l was less one and a unique endemic equilibrium whenever ℜ 0l was greater than one.
Our model analysis also revealed that the disease free equilibrium of the Anthrax-Listeriosis co-infection model is locally stable whenever ℜ al was less one and unstable otherwise. Our model exhibeted the phenomenon of backward bifurcation. Epidemiological implications: the idea of Co-dynamics model been locally stable whenever ℜ al was less than one and unstable otherwise does not fully apply. Hence, the Anthrax-Listerios co-dynamics model showed a case of co-existence of the disease free equilibrium and endemic equilibrium whenever ℜ al was less than one.
We observed that the impact of Listeriosis on Anthrax infections showed that Anthrax infections can be linked with increased risk of Listeriosis but the reverse was not the case. Moreover, prevention and treatment of Anthrax without keeping Listeriosis under control was not the best strategy of combating either of these diseases. Prevention and treatment of Listeriosis can only be the effective way of combating Listeriosis if only Anthrax is kept under control. Anthrax infections can be linked to increased bacteria growth as shown in Figure 3 and Figure 5.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.