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

Authors: Shaibu Osman; Houenafa Alain Togbenon; Dominic Otoo
Journal: Computational and Mathematical Methods in Medicine
 (2020)

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## Abstract

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 finite difference 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 effective control measure in combating campylobacteriosis infections. An analysis of the effects 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.


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## Body

## 1. 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 cost-effective 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 [12–14] proposed and formulated models that attempts to explain this hidden and existing phenomena.


## 2. Model Formulation and Description

The model diagram in Figure1 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.Figure 1
Model flow diagram showing the transmission dynamics of the disease.Table 1
Shows the parameter values used in the model simulation.
ParameterValueReferencesβ0.03Assumedλ0.004[4]Λv0.005AssumedΛh0.002Assumedμv0.0002[4]δv0.003Assumedα0.05[4]μh0.0001Assumedδh0.001Assumedσh0.004Assumedσv0.007[4]....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 Nht, is divided into subpopulations of susceptible humans Sht, infected humans Iht, and recovered humans Rht. Susceptible humans are recruited through immigration into the population at a rate Λh. They are infected with campylobacteriosis through ingestion of contaminated water, foods, and direct contact with infected animals and humans at a rate Iv+Ihβ. Infected humans recover from campylobacteriosis at a rate γ. Campylobacteriosis-related death rate is given byδh. Recovered individuals may lose immunity and return to the susceptible group at a rate σh. Campylobacteriosis natural death rate for all human compartments is μh. Susceptible animals Sv are recruited through immigration at a rate Λv. Animals can be infected with campylobacteriosis through ingestion of contaminated food, water, and contact with infected animals at a rate Iv+Ihλ. Susceptible and infected animal natural death rate is μv. Infected animal death rate as a result of campylobacteriosis is δv, and animals may recover at a rate α. Animals may lose immunity at a rate σv.Hence, total human population is(1)Nht=Sht+Iht+Rht.Total animal population,Nvt, is divided into subpopulations of susceptible animals Svt, infectious animals Ivt, and recovered animals Rvt.Hence, total animal population is(2)Nvt=Svt+Ivt+Rvt.System of equations obtained from the model in Figure1 are as follows:(3)dShdt=Λh+σhRh−βm∗βSh−μhSh,dIhdt=βm∗βSh−γIh−μh+δhIh,dRhdt=γIh−σh+μhRh,dSvdt=Λv−βm∗λSv−μvSv+σvRv,dIvdt=βm∗λSv−αIv−μv+δvIv,dRvdt=αIv−σv+μvRv,where βm∗=Ih+Iv.


## 3. Model Analysis

### 3.1. Positivity and Boundedness of Solutions

The solution of the system in (3) is a function of the form(4)X:t∈J⊂ℝ→Xt=ShtIhtRhtSvtIvtRvt⊂ℝ6.Considering,(5)F:X∈ℝ6→FX∈ℝ6,where(6)FX=dShdt=Λh+σhRh−βm∗βSh−μhSh,dIhdt=βm∗βSh−γIh−μh+δhIh,dRhdt=γIh−σh+μhRh,dSvdt=Λv−βm∗λSv−μvSv+σvRv,dIvdt=βm∗λSv−αIv−μv+δvIv,dRvdt=αIv−σv+μvRv.Hence,(7)dXdt=FX,X0=X0=Sh0,Ih0,Rh0,Sv0,Iv0,Rv0T.Based on the existence and uniqueness theorem,F is C1. Hence, ∃ a unique global solution of the initial value problem of (3) and this solution should be nonnegative whenever its initial conditions are nonnegative.


## 3.1. Positivity and Boundedness of Solutions

The solution of the system in (3) is a function of the form(4)X:t∈J⊂ℝ→Xt=ShtIhtRhtSvtIvtRvt⊂ℝ6.Considering,(5)F:X∈ℝ6→FX∈ℝ6,where(6)FX=dShdt=Λh+σhRh−βm∗βSh−μhSh,dIhdt=βm∗βSh−γIh−μh+δhIh,dRhdt=γIh−σh+μhRh,dSvdt=Λv−βm∗λSv−μvSv+σvRv,dIvdt=βm∗λSv−αIv−μv+δvIv,dRvdt=αIv−σv+μvRv.Hence,(7)dXdt=FX,X0=X0=Sh0,Ih0,Rh0,Sv0,Iv0,Rv0T.Based on the existence and uniqueness theorem,F is C1. Hence, ∃ a unique global solution of the initial value problem of (3) and this solution should be nonnegative whenever its initial conditions are nonnegative.


## 4. Nonstandard Finite Difference Scheme

This is basically a numerical scheme with step sizeΔt that is usually used in the approximation of solution Xtk of autonomous system of differential equations of the form(8)dXdt=FX,subject to(9)X0=X0,where F is C1 usually of the form(10)DΔtXk=FΔtXk,where(11)DΔtXk≈dXtkdt,Xk≈Xtk,FΔtXk≈FXk,tk≈t0+kΔt.The scheme(12)DΔtXk=FΔtXk.Definition 1.
The scheme (12) can be referred to as a nonstandard finite difference scheme when it at least satisfies the following conditions:(i)
DΔtXk=Xk+1−ψXk/φΔt, where ψand φ are positive functions which depend on parameters of the differential equations, step size, Δt and satisfy(13)ψΔt=1+OΔt,φΔt=Δt+OΔt2(ii)
FΔtXk=gXk,Xk+1,Δt, where g denotes an approximation of the nonlocal right hand side of the systemDefinition 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 [15–18], the NSFD scheme for the system of (3).is given by(14)Shn+1−Shnφ1Δt=Λh−βIhn+IvnShn+1−μhShn+1+σhRhn+1,Ihn+1−Ihnφ2Δt=βIhn+IvnShn+1−γIhn+1−μh+δhIhn+1,Rhn+1−Rhnφ3Δt=γIhn+1−σh+μhRhn+1,Svn+1−Svnφ4Δt=Λv−λIhn+IvnSvn+1−μvSvn+1+σvRvn+1,Ivn+1−Ivnφ5Δt=λIhn+IvnSvn+1−αIvn+1−μv+δvIvn+1,Rvn+1−Rvnφ6Δt=αIvn+1−σv+μvRvn+1,where(15)φjΔt,kj∗=1−e−kj∗Δtkj∗,kj∗=maxγi,j=1,2,3,⋯6,i=1,2,3,⋯6,with(16)γi=∂f/∂xix=x¯,fx¯=0,where;(17)φ1Δt=1−e−μhΔtμh,−βΛhμh,σh,0,−βΛhμh,0,φ2Δt=1−e−∣βΛh/μh−γ−μh−δh∣Δt∣βΛh/μh−γ−μh−δh∣,−βΛhμh,0,0,−βΛhμh,0,φ3Δt=1−e−σh+μhΔtσh+μh,γ,−σh−μh,0,0,0,φ4Δt=1−e−μvΔtμv,0,−λΛvμv,−μv,−λΛvμv,σv,φ5Δt=1−e−∣λΛv/μv−α−μv−δv∣Δt∣λΛv/μv−α−μv−δv∣,0,λΛvμv,0,0,λΛvμv−α−μv−δv,φ6Δt=1−e−σv+μvΔtσv+μv,0,0,0,α,σv−μv,where,(18)φ1Δt=1−e−μhΔtμh,φ2Δt=1−e−∣βΛh/μh−γ−μh−δh∣Δt∣βΛh/μh−γ−μh−δh∣,φ3Δt=1−e−σh+μhΔtσh+μh,φ4Δt=1−e−μvΔtμv,φ5Δt=1−e−∣λΛv/μv−α−μv−δv∣Δt∣λΛv/μv−α−μv−δv∣,φ6Δt=1−e−σv+μvΔtσv+μv.The scheme in its explicit form is given by(19)Shn+1=Λh+σhRhn+1φ1Δt+Shn1+βIhn+βIvn+μhφ1Δt,Ihn+1=βIhnShn+1+βIvnShn+1φ2Δt+Ihn1+γ+μh+δhφ2Δt,Rhn+1=γIhn+1φ3Δt+Rhn1+σh+μhφ3Δt,Svn+1=Λv+σvRvn+1φ4Δt+Svn1+λIhn+λIvn+μvφ4Δt,Ivn+1=λIhnSvn+1+λIvnSvn+1φ5Δt+Ivn1+α+μv+δvφ5Δt,Rvn+1=αIvn+1φ6Δt+Rvn1+σv+μvφ6Δt.


## 5. Disease-Free Equilibrium

Given initial conditions,(20)Sht=0,Iht=0,Rht=0,St=0,Ivt=0,Rvt=0.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(21)11+μhφ1ΔtP1σhφ1Δt1+μhφ1Δt0P600P200P700P311+σh+μhφ3Δt0000P4011+μvφ4ΔtP8σvφΔt1+μvφ4Δt0P500P900000P1011+σv+μvφ6Δt,where(22)P1=−βΛh/μhφ1Δt1+μhφ1Δt,P6=−βΛh/μhφ1Δt1+μhφ1Δt,P2=1+βΛh/μhφ2Δt1+γ+μh+δhφ2Δt,P7=βΛh/μhφ2Δt1+γ+μh+δhφ2Δt,P3=γφ3Δt1+σh+μhφ3Δt,P8=−λΛv/μvφ4Δt1+μvφ4Δt,P4=−λΛv/μvφ4Δt1+μvφ4Δt,P9=λΛv/μvφ5Δt1+α+μv+δvφ5Δt,P5=λΛv/μvφ5Δt1+α+μv+δvφ5Δt,P10=αφ6Δt1+σv+μvφ6ΔtThe corresponding eigenvalues of the Jacobian matrix are obtained as(23)λ1=11+μhφ1Δt,λ2=1+βΛh/μhφ2Δt1+γ+μh+δhφ2Δt,λ3=11+σh+μhφ3Δt,λ4=11+μvφ4Δt,λ5=1+λΛv/μvφ5Δt1+α+μv+δvφ5Δt,λ6=11+σv+μvφ6Δt.


### 5.1. Local Stability of the Disease-Free Equilibrium

Theorem 3.
The DFE is locally asymptotically stable for every value ofΔtif the following conditions are satisfied:(i)
βΛh/μhγ+μh+δh<1(ii)
λΛv/μvα+μv+δv<1Proof.
The sequence(24)Shn,Ihn,Rhn,Svn,Ivn,Rvn,should converge to the disease-free equilibrium(25)DFE=Λhμh,0,0,Λvμv,0,0,for any positive initial conditions when conditions (i) and (ii) are satisfied for every value of Δt.
Linearising system (3) at the DFE, the eigenvalues of the corresponding Jacobian matrix are given by(26)λ1=11+μhφ1Δt,λ2=1+βΛh/μhφ2Δt1+γ+μh+δhφ2Δt,λ3=11+σh+μhφ3Δt,λ4=11+μvφ4Δt,λ5=1+λΛv/μvφ5Δt1+α+μv+δvφ5Δt,λ6=11+σv+μvφ6Δt.
It shows that the DFE is locally asymptotically stable for every value ofΔt if the conditions (i) and (ii) of Definition 1 are satisfied.
Forλ1,(27)λ1=11+μhφ1Δt,
λ1<1 since 1+μhφ1Δt>1.
Forλ2,(28)λ2=1+βΛh/μhφ2Δt1+γ+μh+δhφ2Δt,
λ2<1 if and only if 1+βΛh/μhφ2Δt<1+γ+μh+δhφ2Δt.
Forλ3,(29)λ3=11+σh+μhφ3Δt,
λ3<1 since 1+σh+μhφ3Δt>1
Forλ4,(30)λ4=11+μvφ4Δt,
λ4<1 since 1+μvφ4Δt>1.
Forλ5,(31)λ5=1+λΛv/μvφ5Δt1+α+μv+δvφ5Δt,
λ5≤1 on condition that 1+λΛv/μvφ5Δt<1+α+μv+δvφ5Δt.
Forλ6,(32)λ6=11+σv+μvφ6Δt,
λ6<1 since 1+σv+μvφ6Δt>1.


### 5.2. 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;(33)Shn,Ihn,Rhn,Svn,Ivn,Rvn,should converge to the disease-free equilibrium(34)Λhμh,0,0,Λvμv,0,0,for any positive initial condition whenever conditions (i) and (ii) of Definition 1 are satisfied for every value of Δt.
From Definition1, the DFE is LAS for every value of Δt whenever conditions (i) and (ii) of Definition 1 hold.
Suppose that forn>0,
Shn,Ihn,Rhn,Svn,Ivn,Rvn converges to Λh/μh,0,0,Λv/μv,0,0.
Then, it can be shown that(35)Shn+1,Ihn+1,Rhn+1,Svn+1,Ivn+1,Rvn+1,converges to(36)Λhμh,0,0,Λvμv,0,0.Considering the system of equation in (19), we prove that the disease-free equilibrium is globally asymptotically stable using the conditions in Definition 1.ForIhn+1,(37)Ihn+1=βIhnShn+1+βIvnShn+1φ2Δt+Ihn1+γ+μh+δhφ2Δt,then,Ihn+1→0,as n→∞.ForRhn+1,(38)Rhn+1=γIhn+1φ3Δt+Rhn1+σh+μhφ3Δt,then,Rhn+1→0 as n→∞.ForShn+1,(39)Shn+1=Λh+σhRhn+1φ1Δt+Shn1+βIhn+βIvn+μhφ1Δt,then,Shn+1→Λh/μh as n→∞.ForIvn+1,(40)Ivn+1=λIhnSvn+1+λIvnSvn+1φ5Δt+Ivn1+α+μv+δvφ5Δt,then,Ivn+1→0 as n→∞.ForRvn+1,(41)Rvn+1=αIvn+1φ6Δt+Rvn1+σv+μvφ6Δt,then,Rvn+1→0 as n→∞.ForSvn+1,(42)Svn+1=Λv+σvRvn+1φ4Δt+Svn1+λIhn+λIvn+μvφ4Δt,then,Svn+1→Λv/μv as n→∞.Hence, the DFE is GAS since conditions (i) and (ii) are satisfied for every value ofΔt.


## 5.1. Local Stability of the Disease-Free Equilibrium

Theorem 3.
The DFE is locally asymptotically stable for every value ofΔtif the following conditions are satisfied:(i)
βΛh/μhγ+μh+δh<1(ii)
λΛv/μvα+μv+δv<1Proof.
The sequence(24)Shn,Ihn,Rhn,Svn,Ivn,Rvn,should converge to the disease-free equilibrium(25)DFE=Λhμh,0,0,Λvμv,0,0,for any positive initial conditions when conditions (i) and (ii) are satisfied for every value of Δt.
Linearising system (3) at the DFE, the eigenvalues of the corresponding Jacobian matrix are given by(26)λ1=11+μhφ1Δt,λ2=1+βΛh/μhφ2Δt1+γ+μh+δhφ2Δt,λ3=11+σh+μhφ3Δt,λ4=11+μvφ4Δt,λ5=1+λΛv/μvφ5Δt1+α+μv+δvφ5Δt,λ6=11+σv+μvφ6Δt.
It shows that the DFE is locally asymptotically stable for every value ofΔt if the conditions (i) and (ii) of Definition 1 are satisfied.
Forλ1,(27)λ1=11+μhφ1Δt,
λ1<1 since 1+μhφ1Δt>1.
Forλ2,(28)λ2=1+βΛh/μhφ2Δt1+γ+μh+δhφ2Δt,
λ2<1 if and only if 1+βΛh/μhφ2Δt<1+γ+μh+δhφ2Δt.
Forλ3,(29)λ3=11+σh+μhφ3Δt,
λ3<1 since 1+σh+μhφ3Δt>1
Forλ4,(30)λ4=11+μvφ4Δt,
λ4<1 since 1+μvφ4Δt>1.
Forλ5,(31)λ5=1+λΛv/μvφ5Δt1+α+μv+δvφ5Δt,
λ5≤1 on condition that 1+λΛv/μvφ5Δt<1+α+μv+δvφ5Δt.
Forλ6,(32)λ6=11+σv+μvφ6Δt,
λ6<1 since 1+σv+μvφ6Δt>1.


## 5.2. 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;(33)Shn,Ihn,Rhn,Svn,Ivn,Rvn,should converge to the disease-free equilibrium(34)Λhμh,0,0,Λvμv,0,0,for any positive initial condition whenever conditions (i) and (ii) of Definition 1 are satisfied for every value of Δt.
From Definition1, the DFE is LAS for every value of Δt whenever conditions (i) and (ii) of Definition 1 hold.
Suppose that forn>0,
Shn,Ihn,Rhn,Svn,Ivn,Rvn converges to Λh/μh,0,0,Λv/μv,0,0.
Then, it can be shown that(35)Shn+1,Ihn+1,Rhn+1,Svn+1,Ivn+1,Rvn+1,converges to(36)Λhμh,0,0,Λvμv,0,0.Considering the system of equation in (19), we prove that the disease-free equilibrium is globally asymptotically stable using the conditions in Definition 1.ForIhn+1,(37)Ihn+1=βIhnShn+1+βIvnShn+1φ2Δt+Ihn1+γ+μh+δhφ2Δt,then,Ihn+1→0,as n→∞.ForRhn+1,(38)Rhn+1=γIhn+1φ3Δt+Rhn1+σh+μhφ3Δt,then,Rhn+1→0 as n→∞.ForShn+1,(39)Shn+1=Λh+σhRhn+1φ1Δt+Shn1+βIhn+βIvn+μhφ1Δt,then,Shn+1→Λh/μh as n→∞.ForIvn+1,(40)Ivn+1=λIhnSvn+1+λIvnSvn+1φ5Δt+Ivn1+α+μv+δvφ5Δt,then,Ivn+1→0 as n→∞.ForRvn+1,(41)Rvn+1=αIvn+1φ6Δt+Rvn1+σv+μvφ6Δt,then,Rvn+1→0 as n→∞.ForSvn+1,(42)Svn+1=Λv+σvRvn+1φ4Δt+Svn1+λIhn+λIvn+μvφ4Δt,then,Svn+1→Λv/μv as n→∞.Hence, the DFE is GAS since conditions (i) and (ii) are satisfied for every value ofΔt.


## 6. 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 Figure1 and the introduction of an objective functional that seeks to minimise: u1,u2,u3, where u1 denotes prevention of Sh, u2 denotes treatment of Ih, and u3 denotes treatment of Iv.By introducing all controls, the system in equation (3) becomes(43)dShdt=Λh+σh−1−u1βIv+IhSh−μhSh,dIhdt=1−u1βIv+IhSh−u2+γIh−δh+μhIh,dRhdt=u2+γIh−σh+μhRh,dSvdt=Λv−1−u1λIv+IhSv−μvSv+σvRv,dIvdt=1−u1λIv+IhSv−αIv−δv+μvIv,dRvdt=αIv−σv+μvRv.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(44)J=minu1,u2,u3∫0tfB1Iv+B2Ih+B3u12+B4u22+B5u32dt,subject to the system of equations in (3).Control efforts of model in (3) is by linear combination ofui2t, i=1,2. 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.In (44), B1, B2, B3, B4, and B5 denote weight constants to facilitate balance terms in the integral to avoid dominance of one another. B1Ih,B2Iv are costs associated with Ih and Iv, respectively. B3u12, is cost associated with prevention of Sh. B4u22, is cost of treatment of Ih, and B5u32 is cost associated with treatment of Iv. tf, is the period of intervention. Hence, B1Ih,B2Iv denotes a linear function for cost associated with infections and B3u12,B4u22,B5u32, denotes a quadratic function for the cost associated with controls [19, 20].The objective is to find the optimal functionsu1∗t,u2∗t,u3∗t such that(45)Ju1∗t,u2∗t,u3∗t=minu1,u2,u3∈∪Ju1,u2,u3,where∪=u:u,0≤uit≤1,tε0,tf,i=1,2,3 denotes the control set.


### 6.1. 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 H, with respect to u1,u2,u3.(46)H=B1Iv+B2Ih+B3u12+B4u22+B5u32+MIh1−u1βIv+IhSh−u2+γIh−δh+μhIh+MRhu2+γIh−σh+μhRh+MSvΛv−1−u1λIv+IhSv−μvSv+σvRv+MIv1−u1λIv+IhSv−u3+αIv−δv+μvIvMRvu3+αIv−σv+μvRv,whereMSh, MIh, MRh, MSv, MIv, and MRv are referred to as the adjoint variables.The adjoint (costate) variables are solutions of adjoint systems below:(47)dMShdt=1−u1Iv+IhβMSh−MIh+μhMSh,dMIhdt=−B2+1−u1βShMSh−MIh+u2+γMIh−MRh+μh+δhMIh+1−u1λSvMSv−MIv+bλVvMVv−MIv,dMRhdt=−σhMSh+σh+μhMRh,dMSvdt=1−u1λIv+IhMSv−MIv+μvMSv,dMIvdt=−B1+1−u1βShMSh−MIh+1−u1λSvMSv−MIv+μv+δvMIv+u4+αMIv−MRv,dMRvdt=−σhMSv+σv+μvMRv.This satisfies the transversality condition(48)MShtf=MIhtf=MRhtf=MSvtf=MIvtf=MRvtf=0,by combining the Pontryagin’s maximum principle and the existence of the optimal control.Theorem 5.
The optimal control vectoru1∗t,u2∗t,andu3∗tthat maximises the objective function J over ∪ is given by(49)u1∗t=max0,min1,βMIh−MShIv+IhSh∗2B3+λMIv−MSvIv+IhSv∗2B3,u2∗t=max0,min1,MIh−MRhIh∗2B4,u3∗t=max0,min1,MIv−MRvIv∗2B5,
where
MSh, MIh, MRh, MSv, MIv, and MRvare the solutions of equation (47) and (48).Proof.
The existence of an optimal control is as a result the convexity of the integral ofJ with respect to u1, u2, and u3, 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:(50)u1=u~1≔βMIh−MShIv+IhSh∗2B3+λMIv−MSvIv+IhSv∗2B3u2=u~2≔MIh−MRhIh∗2B4,u3=u~3≔MIv−MRvIv∗2B5.In conclusion, by standard control arguments involving bounds on controls,(51)u1∗=0,ifu~1≤0,u~1if0<u~1<1,1ifu~1≥1,u2∗=0,ifu~2≤0,u~2if0<u~2<1,1ifu~2≥1,u3∗=0,ifu~3≤0,u~3if0<u~3<1,1ifu~3≥1.The system in (49) leads to system in (48). The optimal control uniqueness for small tf 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) [22–24].


## 6.1. 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 H, with respect to u1,u2,u3.(46)H=B1Iv+B2Ih+B3u12+B4u22+B5u32+MIh1−u1βIv+IhSh−u2+γIh−δh+μhIh+MRhu2+γIh−σh+μhRh+MSvΛv−1−u1λIv+IhSv−μvSv+σvRv+MIv1−u1λIv+IhSv−u3+αIv−δv+μvIvMRvu3+αIv−σv+μvRv,whereMSh, MIh, MRh, MSv, MIv, and MRv are referred to as the adjoint variables.The adjoint (costate) variables are solutions of adjoint systems below:(47)dMShdt=1−u1Iv+IhβMSh−MIh+μhMSh,dMIhdt=−B2+1−u1βShMSh−MIh+u2+γMIh−MRh+μh+δhMIh+1−u1λSvMSv−MIv+bλVvMVv−MIv,dMRhdt=−σhMSh+σh+μhMRh,dMSvdt=1−u1λIv+IhMSv−MIv+μvMSv,dMIvdt=−B1+1−u1βShMSh−MIh+1−u1λSvMSv−MIv+μv+δvMIv+u4+αMIv−MRv,dMRvdt=−σhMSv+σv+μvMRv.This satisfies the transversality condition(48)MShtf=MIhtf=MRhtf=MSvtf=MIvtf=MRvtf=0,by combining the Pontryagin’s maximum principle and the existence of the optimal control.Theorem 5.
The optimal control vectoru1∗t,u2∗t,andu3∗tthat maximises the objective function J over ∪ is given by(49)u1∗t=max0,min1,βMIh−MShIv+IhSh∗2B3+λMIv−MSvIv+IhSv∗2B3,u2∗t=max0,min1,MIh−MRhIh∗2B4,u3∗t=max0,min1,MIv−MRvIv∗2B5,
where
MSh, MIh, MRh, MSv, MIv, and MRvare the solutions of equation (47) and (48).Proof.
The existence of an optimal control is as a result the convexity of the integral ofJ with respect to u1, u2, and u3, 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:(50)u1=u~1≔βMIh−MShIv+IhSh∗2B3+λMIv−MSvIv+IhSv∗2B3u2=u~2≔MIh−MRhIh∗2B4,u3=u~3≔MIv−MRvIv∗2B5.In conclusion, by standard control arguments involving bounds on controls,(51)u1∗=0,ifu~1≤0,u~1if0<u~1<1,1ifu~1≥1,u2∗=0,ifu~2≤0,u~2if0<u~2<1,1ifu~2≥1,u3∗=0,ifu~3≤0,u~3if0<u~3<1,1ifu~3≥1.The system in (49) leads to system in (48). The optimal control uniqueness for small tf 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) [22–24].


## 7. 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 att=0 and t=tf. Our goal is to solve for tf=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 iterations 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 Rateβ on Infected Humans

Figure2 shows the analysis of contact rate β on infected humans. As the contact rate β increases, there seem to be an increase in the number of infections. As the contact rate β decreases, there is a corresponding decrease in the number of infected humans. This confirms the effects of contact rate β on infected humans. Hence, infections can be curbed by ensuring that the value of contact rate β reduces to the bearest minimum.Figure 2
Effects of contact rate on infected humans.


### 7.2. Analysis of Contact Rateλ on Infected Animals

Figure3 shows the analysis of contact rate λ on infected 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 λ on infected animals. Hence, infections can be curbed by ensuring that the value of contact rate λ reduces to the bearest minimum.Figure 3
Effects of contact rate on infected animals.


### 7.3. Strategy 1: Optimal Prevention ofSh and Treatment of Iv

We optimised the objective function using prevention ofSh and treatment of Iv as control measures. This was done by setting the treatment of infected humans, u2, 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.Figure 4
Population dynamics of infected animals and humans with and without optimal control.Figure 5
Population dynamics of recovered animals and humans with and without optimal control.Biologically, the implication is that campylobacteriosis infections can be controlled effectively by prevention of humans and treatment of infected animals.


### 7.4. Strategy 2: Optimal Prevention ofSh and Treatment of Ih

We optimised the objective function using prevention ofSh and treatment of Ih as control measures. This was done by setting the treatment of infected humans, u3, 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.Figure 6
Population dynamics of infected animals and humans with and without optimal control.Figure 7
Population dynamics of recovered animals and humans with and without optimal control.Biologically, the implication is that campylobacteriosis infections can be curbed effectively by prevention of humans and treatment of infected animals.


## 7.1. Analysis of Contact Rateβ on Infected Humans

Figure2 shows the analysis of contact rate β on infected humans. As the contact rate β increases, there seem to be an increase in the number of infections. As the contact rate β decreases, there is a corresponding decrease in the number of infected humans. This confirms the effects of contact rate β on infected humans. Hence, infections can be curbed by ensuring that the value of contact rate β reduces to the bearest minimum.Figure 2
Effects of contact rate on infected humans.


## 7.2. Analysis of Contact Rateλ on Infected Animals

Figure3 shows the analysis of contact rate λ on infected 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 λ on infected animals. Hence, infections can be curbed by ensuring that the value of contact rate λ reduces to the bearest minimum.Figure 3
Effects of contact rate on infected animals.


## 7.3. Strategy 1: Optimal Prevention ofSh and Treatment of Iv

We optimised the objective function using prevention ofSh and treatment of Iv as control measures. This was done by setting the treatment of infected humans, u2, 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.Figure 4
Population dynamics of infected animals and humans with and without optimal control.Figure 5
Population dynamics of recovered animals and humans with and without optimal control.Biologically, the implication is that campylobacteriosis infections can be controlled effectively by prevention of humans and treatment of infected animals.


## 7.4. Strategy 2: Optimal Prevention ofSh and Treatment of Ih

We optimised the objective function using prevention ofSh and treatment of Ih as control measures. This was done by setting the treatment of infected humans, u3, 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.Figure 6
Population dynamics of infected animals and humans with and without optimal control.Figure 7
Population dynamics of recovered animals and humans with and without optimal control.Biologically, the implication is that campylobacteriosis infections can be curbed effectively by prevention of humans and treatment of infected animals.


## 8. 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, disease-free 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.



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*Source: 8843299-2020-11-19.xml*