Modelling the Dynamics of Campylobacteriosis Using Nonstandard Finite Difference Approach with Optimal Control

Campylobacter genus is the bacteria responsible for campylobacteriosis infections, and it is the commonest cause of gastroenteritis in adults and infants. The disease is hyperendemic in children in most parts of developing countries. It is a zoonotic disease that can be contracted via direct contact, food, and water. In this paper, we formulated a deterministic model for Campylobacteriosis as a zoonotic disease with optimal control and to determine the best control measure. The nonstandard ﬁ nite di ﬀ erence scheme was used for the model analysis. The disease-free equilibrium of the scheme in its explicit form was determined, and it was shown to be both locally and globally asymptotically stable. The campylobacteriosis model was extended to optimal control using prevention of susceptible humans contracting the disease and treatment of infected humans and animals. The objective function was optimised, and it was established that combining prevention of susceptible humans and treatment of infected animals was the e ﬀ ective control measure in combating campylobacteriosis infections. An analysis of the e ﬀ ects of contact between susceptible and infected animals as well susceptible and infected humans was conducted. It showed an increase in infected animals and humans whenever the contact rate increases and decreases otherwise. Biologically, it implies that campylobacteriosis infections can be controlled by ensuring that interactions among susceptible humans, infected animals, and infected humans is reduced to the barest minimum.


Introduction
Campylobacter genus is the bacteria responsible for campylobacteriosis infections. This is solely the commonest cause of gastroenteritis in adults and mostly infants [1]. The Campylobacter bacteria have been confirmed as the leading cause of diarrhea in the United States of America. Campylobacteriosis is mostly hyperendemic in children in most parts of developing countries. Campylobacteriosis can be spread or contracted through the fecal-oral path. It is a zoonotic disease that can be contracted via direct contact, food, and water. The disease is zoonotic in nature and hence can be spread from animals to humans and also from humans to humans [2].
A campylobacteriosis-infected person is usually asymptomatic at the incubation period, that is, between one and three days of infection. Diarrhea, fever, and abdominal cramps are usually the commonest symptoms of the disease. Symptoms of campylobacteriosis can last for at least five to eight days of infections. Children in developing countries mostly show symptoms of campylobacteriosis infections while adults rarely show any symptoms of infections. But on the contrary, the infection is less common in the developed world [1].
Symptomatic persons can infect others directly and contaminate water and food during the infectious period of campylobacteriosis. The mode of infection of campylobacteriosis is mostly through food, water, and milk that has been infected by contaminated feces which has been poorly treated. However, water contamination is basically via water fowl feces, sewage, and farm animal manure. On the other hand, human contamination is via leaked septic tanks into groundwater supply that is poorly disinfected or not disinfected at all. The disease is mostly foodborne and waterborne illness but can also be spread through direct contact with infected humans or animals via the fecal-oral path of transmission. But human-to-human spread is usually uncommon [3].
Understanding the spread dynamics of campylobacteriosis at policy and implementation levels of public health is necessary to design effective optimal control and costeffective strategies at prevention levels. Deterministic models enhance the general understanding of the disease spread by the provision of a theoretical frame which underlines factors that accounts for the spread and control of diseases [4,5].
The concept of deterministic modelling involves the process of constructing, testing, and validating models. These models are real representations of natural phenomena of systems or hypothesis in a mathematical perspective [6,7].
Generally, the intended use of a deterministic model is paramount in guiding the development of the model since the model structure has to adequately address its objective. Hence, understanding the mechanism and causes of patterns present in an observed data is usually an objective that initiates a deterministic modelling process [8,9]. Moreover, epidemiological models explain dynamics of infections and determine the best optimal control strategies and the most cost effective among these strategies [10,11]. However, authors in [12][13][14] proposed and formulated models that attempts to explain this hidden and existing phenomena.

Model Formulation and Description
The model diagram in Figure 1 shows the transmission dynamics of campylobacteriosis in humans and animals. This diagram is significant as it gives an overview of the disease transmission pattern in humans and animals. We divided the model into two parts, the total human and animal populations. These populations at any time, t are also divided into six subcompartments with respect to their disease status in the system. The total human population, represented by N h ðtÞ, is divided into subpopulations of susceptible humans S h ðtÞ, infected humans I h ðtÞ, and recovered humans R h ðtÞ. Susceptible humans are recruited through immigration into the population at a rate Λ h . They are infected with campylobacteriosis through ingestion of contaminated water, foods, and direct contact with infected animals and humans at a rate ðI v + I h Þβ: Infected humans recover from campylobacteriosis at a rate γ. Campylobacteriosis-related death rate is given byδ h . Recovered individuals may lose immunity and return to the susceptible group at a rate σ h . Campylobacteriosis natural death rate for all human compartments is μ h : Susceptible animals S v are recruited through immigration at a rate Λ v . Animals can be infected with campylobacteriosis through ingestion of contaminated food, water, and contact with infected animals at a rate ðI v + I h Þλ. Susceptible and infected animal natural death rate is μ v . Infected animal death rate as a result of campylobacteriosis is δ v , and animals may recover at a rate α. Animals may lose immunity at a rate σ v .
Hence, total human population is Total animal population, N v ðtÞ, is divided into subpopu-lations of susceptible animals S v ðtÞ, infectious animals I v ðtÞ, and recovered animals R v ðtÞ. Hence, total animal population is System of equations obtained from the model in Figure 1 are as follows: where β * m = I h + I v .

Model Analysis
where Hence, Based on the existence and uniqueness theorem, F is C 1 . Hence, ∃ a unique global solution of the initial value problem of (3) and this solution should be nonnegative whenever its initial conditions are nonnegative.

Nonstandard Finite Difference Scheme
This is basically a numerical scheme with step size Δt that is usually used in the approximation of solution Xðt k Þ of autonomous system of differential equations of the form subject to where F is C 1 usually of the form where The scheme Definition 1. The scheme (12) can be referred to as a nonstandard finite difference scheme when it at least satisfies the following conditions: (i) D Δt ðX k Þ = ðX k+1 − ψX k Þ/φðΔtÞ, where ψand φ are positive functions which depend on parameters of the differential equations, step size, ðΔtÞ and satisfy where g denotes an approximation of the nonlocal right hand side of the system Definition 2. The nonstandard finite difference scheme is called elementary stable, if, for any value of the step size, its only fixed points are those of the original differential system, the linear stability properties of each fixed points being the same for both the differential system and the discrete scheme.
Based on the definition of the nonstandard finite difference (NSFD) scheme and the rules governing its construction in [15][16][17][18], the NSFD scheme for the system of (3). is given by 3 Computational and Mathematical Methods in Medicine where, The scheme in its explicit form is given by

Disease-Free Equilibrium
Given initial conditions, The disease-free equilibrium of the system of equations in its explicit form can established by linearising the system in its explicit form. The Jacobian matrix of the system of equations is given by

Computational and Mathematical Methods in Medicine
The corresponding eigenvalues of the Jacobian matrix are obtained as should converge to the disease-free equilibrium for any positive initial conditions when conditions (i) and (ii) are satisfied for every value of ðΔtÞ. Linearising system (3) at the DFE, the eigenvalues of the corresponding Jacobian matrix are given by It shows that the DFE is locally asymptotically stable for every value of ðΔtÞ if the conditions (i) and (ii) of Definition 1 are satisfied.

Global Stability of the Disease-Free Equilibrium
Theorem 4. The disease-free equilibrium is globally asymptotically stable if the conditions stated in Theorem3are satisfied.
Proof. The sequence; should converge to the disease-free equilibrium converges to Considering the system of equation in (19), we prove that the disease-free equilibrium is globally asymptotically stable using the conditions in Definition 1.
Hence, the DFE is GAS since conditions (i) and (ii) are satisfied for every value of ðΔtÞ.

Optimal Control Analysis of the Model
In this section, we carried out an analysis of optimal control to determine the impact of all intervention of the control schemes. This is derived by incorporating the following controls into the model in Figure 1 and the introduction of an objective functional that seeks to minimise: ðu 1 , u 2 , u 3 Þ, where u 1 denotes prevention of S h , u 2 denotes treatment of I h , and u 3 denotes treatment of I v .
By introducing all controls, the system in equation (3) becomes In epidemiological models, the essence of optimal control analysis is to minimise the spread or number of infections and cost associated with treatment and prevention controls. The objective functional required to achieve this is formulated by subject to the system of equations in (3). Control efforts of model in (3) is by linear combination of u 2 i ðtÞ, ði = 1, 2Þ. It is assumed to be a quadratic in nature by the assumption that cost is generally nonlinear in nature. Thus, the aim is to minimise the number of infection and reduce the cost of treatment.

Pontryagin's Maximum Principle.
This principle provides the necessary conditions that an optimal must satisfy. It changes the system in equations (3) and (44) into minimisation problem pointwise Hamiltonian ðHÞ, with respect to ðu 1 , u 2 , u 3 Þ. where and M R v are referred to as the adjoint variables.
The adjoint (costate) variables are solutions of adjoint systems below: This satisfies the transversality condition by combining the Pontryagin's maximum principle and the existence of the optimal control. Theorem 5. The optimal control vector ðu * 1 ðtÞ, u * 2 ðtÞ, and u * 3 ð tÞÞ that maximises the objective function ðJÞ over ∪ is given by , ( u * 2 t ð Þ = max 0, min 1, , u * 3 t ð Þ = max 0, min 1, Proof. The existence of an optimal control is as a result the convexity of the integral of J with respect to u 1 , u 2 , and u 3 , the Lipschitz property of the state system with respect to the state variables, and a priori boundedness of the state solutions [21,22]. The system in (47) was obtained by differentiating the Hamiltonian function and evaluated at optimal control. However, by equating the derivatives of the Hamiltonian with respect to the controls to zero, the following are obtained: In conclusion, by standard control arguments involving bounds on controls, The system in (49) leads to system in (48). The optimal control uniqueness for small t f was obtained as a result of the Lipschitz structure of system of equations and the priori boundedness of the state solutions and adjoint functions. Existence of optimal control uniqueness is in line with uniqueness of optimal system that comprises equations (3), (47), (48), and (49) [22][23][24].

Numerical Analysis
In this section, we solved the optimal system by employing the Range-Kutta fourth-order scheme. We solved the state systems, adjoint equations, and the transversality conditions by considering it as a two-point boundary value problem with boundary conditions at t = 0 and t = t f . Our goal is to solve for t f = 90 days or three months. At this value, it is assumed that campylobacteriosis can easily spread. System of equations of the model in Figure 1 is solved numerically using the Range-Kutta fourth-order scheme with a guess on controls over a period of time. Moreover, we used curent iter-ations of the model equations in Figure 1, the costate equations, and transversality conditions by backward approach. Convex combinations of controls in previous iteratons and charaterisations of values from the system are then updated. The process is repeated continuously, and iteration stops if values of unknowns at previous iteration is as close as those at present iteration [25,26]. A number of combination of controls were considered and the best and most effective selected.
7.1. Analysis of Contact Rate ðβÞ on Infected Humans. Figure 2 shows the analysis of contact rate ðβÞ on infected humans. As the contact rate ðβÞ increases, there seem to be an increase in the number of infections. As the contact rate ðβÞ decreases, there is a corresponding decrease in the number of infected humans. This confirms the effects of contact rate ðβÞ on infected humans. Hence, infections can be curbed by ensuring that the value of contact rate ðβÞ reduces to the bearest minimum. Figure 3 shows the analysis of contact rate ðλÞ on infected      Computational and Mathematical Methods in Medicine animals. As the contact rate increases, there seem to be an increase in the number of infections. As the contact rate decreases, there is a corresponding decrease in the number of infected animals. This confirms the effects of contact rate ðλÞ on infected animals. Hence, infections can be curbed by ensuring that the value of contact rate ðλÞ reduces to the bearest minimum.

Strategy 1: Optimal Prevention of S h and Treatment of I v .
We optimised the objective function using prevention of S h and treatment of I v as control measures. This was done by setting the treatment of infected humans, u 2 , to zero. Figure 4 indicates a reduction in number of campylobacteriosis-infected animals and humans. Figure 5 indicates an increase in campylobacteriosis recovery in both animal and human populations. Biologically, the implication is that campylobacteriosis infections can be controlled effectively by prevention of humans and treatment of infected animals.

Strategy 2: Optimal Prevention of S h and Treatment of I h .
We optimised the objective function using prevention of S h and treatment of I h as control measures. This was done by setting the treatment of infected humans, u 3 , to zero. Figure 6 indicates a reduction in number of campylobacteriosis-infected animals and humans. Figure 7 indicates an increase in campylobacteriosis recovery in both animal and human populations.
Biologically, the implication is that campylobacteriosis infections can be curbed effectively by prevention of humans and treatment of infected animals.

Conclusion
A deterministic model that explains the spread dynamics of campylobacteriosis infection was formulated and analysed for its qualitative and quantitative solutions. The qualitative analysis of the model was carried out using the nonstandard finite difference scheme for boundedness of solution, diseasefree equilibrium, and its local and global stability. Campylobacteriosis disease-free equilibrium of the scheme in its explicit form was established. Analysis of the scheme established that the disease-free equilibrium was both locally and globally asymptotically stable.
The campylobacteriosis model was extended to optimal control using prevention of susceptible humans, treatment of infected humans, and treatment of infected animals. The objective functional was optimised, and it was established that combining prevention of susceptible humans and treatment of infected animals was the effective control measure in combating campylobacteriosis infections.
An analysis of the effects of contact rate between susceptible and infected animals as well as susceptible and infected humans was conducted. This showed an increase in infected animals and humans whenever the contact rate increases and decreases otherwise. Biologically, campylobacteriosis infections can be controlled by ensuring that interactions between susceptible humans, infected animals, and infected humans is reduced to the bearest minimum.

Data Availability
The parameter values supporting this deterministic model simulations were assumed and others taken from some published articles. All these were dully acknowledged and cited in

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Computational and Mathematical Methods in Medicine this paper. These published articles are also cited at relevant places within the text as references.

Conflicts of Interest
We declare that there are no conflict of interest regarding the publication of this article.