Modeling Anthrax with Optimal Control and Cost Effectiveness Analysis

Anthrax is an infection caused by bacteria and it affects both human and animal populations. The disease can be categorized under zoonotic diseases and humans can contract infections through contact with infected animals, ingest contaminated dairy and animal products. In this paper, we developed a mathematical model for anthrax transmission dynamics in both human and animal populations with optimal control. The qualitative solution of the model behaviour was analyzed by determining hv R , equilibrium points and sensitivity analysis. A vaccination class was incorporated into the model with waning immunity. Local and global stability of the model’s equilibria was found to be locally asymptotically stable whenever 1 hv R < and unstable otherwise. Analysis of parameter contribution was conducted to determine the contribution of each to hv R . It was revealed that reducing animal and human interaction rate, would decrease hv R . We extended the model to optimal control in order to find the best control strategy in reducing anthrax infections. It showed that the effective strategy in combating the anthrax epidemics is vaccination of animals and prevention of humans.


Introduction
Anthrax is an infectious disease that is caused by the bacteria Bacillus anthracis. contract the anthrax disease if they interact with infected animals or consumed contaminated dairy and animal products. Anthrax is a zoonotic disease found naturally in the soil and it affects both animal and human populations worldwide. Individuals can contract the disease from direct contact with animals or from contaminated animal products [1] [2].
Mathematical models are capable of describing a natural phenomenon and the transmission dynamics of infectious diseases. These models play a key role in combating infectious diseases in epidemiology. These models can explicitly explain the transmission mechanism and dynamics of disease [3] [4]. Complex models for transmission dynamics of diseases such as periodic orbits, Hoff bifurcations and multiple equilibrium have been proposed and worked on for some time now. They give a concise qualitative illustration of the disease dynamics and better analysis and implications for disease prediction of [5] [6]. [1] investigated the effects of constant vaccination on anthrax model but never considered effects of optimal control. From the theoretical results of their study under constant vaccination, the transmission of the disease model is similar to dynamics without vaccination. Moreover, [7] developed a model by considering vaccine impact and concluded by establishing the optimal vaccine coverage threshold required for disease eradication. However, [8] used optimal control in the study of a nonlinear SIR epidemic model with a vaccination strategy. [9] considered the application of optimal control to investigate the impact of chemo-therapy on malaria disease with infection immigrants and [10] applied optimal control methods associated with preventing exogenous reinfection based on a exogenous reinfection tuberculosis model. Authors in [11], investigated the essential role of three basic controls: personal protection, treatment, and mosquito reduction strategies in combating an infectious disease.
However, [12] formulated a general epidemic model of a vector-borne disease consisting two vertebrate host species and one insect vector species. The qualitative analysis of the study revealed that model exhibited a multiple endemic equilibrium. The spread of anthrax has an impact on the life's of people and cost of treatment is of concern in every human endeavour. Many studies in literature have been performed to establish the role of vaccination and treatment on transmission of diseases [13]. Some studies reveal the complex nature of the anthrax disease. The transmission dynamics of the disease varies depending on the prevailing conditions of the country. There has not been any mathematical model developed to explain the transmission dynamics of anthrax with optimal control and cost effectiveness in Ghana. We therefore developed a model to investigate the dynamics of anthrax with optimal control and cost effectiveness analysis using the anthrax data between 2005 to 2016 as shown in Figure 1.

Model Formulation and Description
The model divides human and animal populations at any time, (t) into compartments with respect to their disease status as shown in Hence total animal population; The system of ordinary differential equations obtained from the model in

Positivity and Boundedness of Solutions
In epidemiological models, conditions under which the system should have non-negative solutions is paramount. The model would be biologically meaningful if all the solutions with positive initial conditions remain positive at every point in time.
then the solution of S t I t R t S t I t V t R t are also non-negative for all 0 t > . Human total population at any given time: In the absence of mortality due to Anthrax infections, the above equation becomes; Solving the equation and as t → ∞ , the population size, However, for total animal (vector) population at any given time: Solving the differential equation and as t → ∞ , the population size, Hence, feasible region for the dynamical system in (1) is given by: 3 4 4 .
where Π is positively invariant.
The DFE point is given by;

The Basic Reproductive Number
Using the next generation matrix approach in [4], the reproductive rate can be established. The reproductive rate combines the biology of infections with the social and behaviour of the factors influencing contact rate. The basic reproductive rate refers to the number of secondary cases one infectious individual will produce in a completely susceptible population [14] [15]. This is a threshold parameter that governs the spread of a disease. Considering only the infective compartments in the system of differential equations in (3): Let f be the number of new infection coming into the system and v be the number of infectives that are leaving the system either by death or birth.
The Jacobian matrix of f and v at DFE is obtained by F and V as follows: Computing the product of By computing the eigenvalues of This implies that the basic reproductive rate is given by; At DFE, it becomes; where; ( )

Global Stability of the Disease-Free Equilibrium
Proposition 3. The disease-free equilibrium (DFE) of model (3) Taking time derivative of ( ) P t along solutions of the differential equations I  I  t  t  t  S I I  I  I  S I I   b I I V  I  I   I  I Time derivative of P along the solutions of differential equations in (3) yields: The highest compact invariant set in , , It shows that 0 ξ is globally asymptotically stable in Ω . By LaSalle's invariant principle [16].

Endemic Equilibrium
Considering the system in 3, at equilibrium, * 0 for epidemiological control measures because an epidemic may persist at steady state even when ( ) 1 hv R < .

Global Stability of Endemic Equilibrium
In this section, the global behaviour of the system in Equation (3) is analysed.
Considering the non-linear Lyapunov function.

Sensitivity Analysis
Basically, the essence of sensitivity analysis is to determine the contribution of each parameter to the reproductive rate. This is help to identify the parameters with high impact on hv R . The basic reproductive rate is usually analysed to find out whether or not treatment of the infectives, mortality and vaccination could help in the control or eradication of the disease in the population [19] [20]. Definition 6. The normalised forward sensitivity index of a variable, y, which depends differentially on a parameter, x, defined as:

Sensitivity Indices of hv R
In epidemiological models, the value of hv R determines the ability of the infection to spread within the population. We will determine the reduction in infec- Considering all the parameters of the system in model (3), we derive the sensitivity of hv R to each of the parameters in the model.
The sensitivity indices of hv R with respect to each of the parameters of the system in model (3), are given in Table 2.
The detailed sensitivity analysis of ( )

Optimal Control of the Anthrax Model
The essence optimal control in disease modelling to to determine the best con-  A u is the cost associated with h S .
and is called the control set.
The solutions of adjoint are system; which satisfies the transversality condition;  *  *  *  1  2  3  4 , , , u t u t u t u t that maximizes the objective function (J) over  , given by;  Proof. Optimal control exists as a result the convexity of the integral of J w.r.t 1 2 3 , , u u u and 4 u , Lipschitz property of the state system with respect to the state variables and a priori boundedness of the state solutions [9]. The system in (31) was obtained by differentiating the Hamiltonian function evaluated at optimal control. By equating the derivatives of Hamiltonian with respect to the controls to zero, we obtained the following; By standard control arguments which involve bounds on the controls, it can be concluded that; The system in (34) leads to the system in (33) in Theorem (7). Uniqueness of op-timal of the system is guaranteed by imposing a condition on time interval [26]. □

Numerical Results
The numerical simulations of the effects of control strategies on disease dynamics is shown. This is done by solving the optimal system consisting of Equation (3), co-state Equation (31), transversality conditions (32) and characterisation (34). Optimal system were solved by applying an iterative scheme. A fourth order Range-Kutta scheme was applied to solve the state equations with a guess of controls over time. We also apply the current iterations solutions of state systems to solve adjoint equations by backward fourth order Range-Kutta scheme.
Finally, an update of controls by using a convex combination of previous controls and value from (33). This process was repeated and iterations stops if values of unknowns at the previous iterations are very close to the ones at the present iterations [27].
Most effective strategies were presented as follows: Combination of 1 u and u . The plots of the three most effective strategies were selected and presented as follows; Strategy 1: Optimal treatment of infectious animals and treatment of humans.
Using 4 u and 2 u , we optimised the objective functional, (J) by setting 1 u and 3 u to zero. Due to these control strategies applied, from Figure 3, it can be seen that the number of v I and h I have reduced substantially. This implies spread of anthrax can be curbed through effective treatment of infectious animals and the treatment of infectious humans. This strategy can best be reached by treatment all infectious animals and humans in the system. Figure 4 shows an exponential reduction in the number of recovered animals and humans the system.      Strategy 2: Optimal vaccination of animals and prevention of humans. Using 3 u and 1 u , we optimise J by setting the controls 2 u and 4 u to zero.
As a result of this control strategies used, it can be seen that the number of v I and h I have reduced considerably as shown in Figure 5. This means that anthrax spread can be tackled through 3 u and 1 u . This can be done though education of farmers on vaccination of animals against anthrax. Also, educating the public on the dangers associated with consumption of infected meat and products from animals infected with anthrax. Moreover, there are some reductions in the populations of h R and v R as shown in Figure 6. Strategy 3: Optimal vaccination of animals and treatment of infectious animals.
Using 3 u and 4 u , we optimise the objective functional by setting 1 u and 2 u to zero. As a result of this, it can be observed that there have been a reduction in the population of v I and h I as indicated in Figure 7. This means anthrax spread can handled by effective vaccination of susceptible animals and treatment of infectious animals. In order to achieve this, proper vaccination of animals and the treatment of all infectious animals should be conducted. Figure  8 shows a considerable reduction in both human and animal population.

Cost Effectiveness Analysis
In this section, a cost-benefit analysis is conducted to determine the costs associated with prevention of susceptible humans, treatment of infectious humans, vaccination of susceptible animals and treatment of infectious animals. In this paper, we consider the Infection Averted Ratio, (IAR) approach.

Intervention Averted Ratio (IAR)
The intervention averted ratio is defined as the ratio of the number of infection averted to the number of recovered. Number of infection averted is the diffeences between infectious population without optimal control and infectious population with optimal control. The IAR for each strategy was determined using the model's parameter values. Generally, the strategy with the highest ratio is taken as the best and most effective strategy to be considered.

Number of infection averted IAR
Number of recovered = The values in Table 3 shows the intervention averted ratios of the various strategies. Strategy 2 has the highest ratio and hence, it is the most effective strategy. The second most highest ratio is strategy 3. Hence, anthrax spread can be tackled through 1 u and 3 u . This can be done though education of farmers on vaccination of animals against anthrax. Also, educating the public on the dangers associated with consumption of infected meat and products from animals infected with anthrax. The diagram in Figure 9 shows IAR plots on the effects of control strategies.

Conclusions
The model's qualitative analysis exhibited an existence of multiple endemic equilibria. Its biological implications are that, proper anthrax disease can best be done if 1 hv R < . Moreover, sensitivity analysis of hv R showed that, an increase in animal recovery rate would decrease hv R . However, a decrease in animal recovery rate would increase hv R . Additionally, an increase in animal and human transmission rates would increase hv R and a decrease in animal and human transmission rates decreases hv R .
Analysis of optimal control and cost effectiveness of our model showed that the best and most effective strategy is vaccination of animals and prevention of susceptible humans in the system. Prevention control on humans and vaccination of animals should be considered as priority when fighting anthrax infections. There should be proper campaign on anthrax prevention and more animals should be vaccinated against the disease.

Data Availability Statement
Data supporting this model analysis are from previously published works and reported studies and all these have been cited in this paper. Some parameter values were assumed and others taken from published articles. These articles have also been cited accordingly at relevant places within the text as references.