# Fractional Derivative and Optimal Control Analysis of Cholera Epidemic Model

Authors: Kumama Regassa Cheneke; Koya Purnachandra Rao; Geremew Kenassa Edessa
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, a cholera model with fractional derivative and optimal control analysis is presented. Numerical simulation analysis shows that increasing the order of fractional derivatives contributes to updating the memory of the population to control the effects of cholera infection through available controlling techniques. On the other hand, the optimal analysis gives an indication of applying controlling infection with available treatment and prevention techniques. It provides a better mechanism to prevent the happening of cholera infection. Moreover, cost-effectiveness evaluation of cholera contamination intervention with feasible three or four combos of manipulate measures hygiene, vaccination, remedy of infectives, and chlorination indicates that hygiene, vaccination, and chlorination are the desired higher mixture to govern in addition propagation of cholera contamination. Numerical simulations are performed with the MATLAB platform and numerical solutions and results are discussed.


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## Body

## 1. Introduction

Due to sudden outbreak of cholera infection and its huge impact on economic resources, optimal control strategies toward infection is highly recommended[1–14]. On the other hand, fractional derivative is one of the highly applicable differential calculus to describe both biological and physical phenomena [1, 15–19]. Different mathematical models of cholera infection are developed starting from the discovery of infection. Moreover, the fractional derivatives are also widely applied in the dynamics of cholera infection. However, optimal controls and fractional derivatives are not applied to cholera model with logistic growth for V. cholerae. Thus, to fulfill the observed gap, we are motivated to develop a cholera model with optimal control and fractional derivative analysis. In this chapter, a modified SIR-B mathematical model of the cholera epidemic model is analyzed to show the impact of optimal control and memory effect on dynamics of cholera infection. Organization of the chapter: in Section 2, materials and methods are presented. In Section 3, numerical simulations are presented.. In Section 4, results and discussions are presented. In Section 5 the conclusion is presented.


## 2. Materials and Methods

In this chapter, cholera epidemic is analyzed with fractional derivative and optimal control strategies. Both analytical and numerical methods are involved to describe the dynamics of the infection. Moreover, new-generation method is used to compute basic reproduction number and numerical simulations are performed using MATLAB.


### 2.1. Deterministic Model Formulation

In this study, we have developed the deterministic mathematical model of cholera with optimal control strategies. We have divided the total population as (i) susceptible (S). These are individuals free of the infection but can be infected with the infection if exposed to the infectious persons or environment contaminated with cholera causing bacterium. (ii) Infected (I). These are individuals that get infected with cholera and can transmit infection to others with possible contacts that cause infection. (iii) Recovered (R). These are individuals recovered from the cholera infection with treatment. The recovered individuals who loss immunity become susceptible. (iv) Concentration of Vibrio cholerae bacterium B (t). This covers an environment contaminated with Vibrio cholerae that cause cholera epidemic. Furthermore, the following assumptions are incorporated in the formulation of the model.(i)
The total population is considered to be nonconstant(ii)
All humans die due to cholera at the disease induced death rate ofφ(iii)
All waste products containVibrio cholerae assumed to be managed and there is no contribution of infected person to the reserved fresh water(iv)
All human population assumed to die at the natural death rateµ(v)
All new infected human individuals are recruited to the susceptible at the recruitment rateλ(vi)
Treatment is performed in the reserved fresh water(vii)
Cholera infected individuals recovered at recovery rate ofη(viii)
Cholera infection recovered individuals loss immunity at the rateγ(ix)
Cholera infection is transmitted from human to human at the rateβ(x)
V. cholerae is ingested from the environment at the ingestion rate βc(xi)
The saturated incidence rate is used in the transmission of cholera infection from environment to human(xii)
The concentration ofV. cholerae in water that causes 50 percent chance of cholera infection is denoted by κ(xiii)
The carrying capacity ofV. cholerae is K(xiv)
V. cholerae die at the rate of v(xv)
Cholera infection transmission preventive control effort isu1(xvi)
Immunity loss control effort isu2(xvii)
V. cholerae clearing control effort is u3In general, the dynamics of human population subject to the cholera infection and the stated assumptions are described by the subsequent schematic diagram.Based on Figure1 and model assumptions the mathematical model of cholera infection with optimal control measures is given by,(1)dSdt=λ−1−u1βcSBk+B+βSI+γR−u2ωS–μS,dIdt=1−u1βcSBk+B+βSI−η+u4+φ+μI,dRdt=η+u4I−γ+μR+u2ωS,anddBdt=1−u3rB1−BK−νB,with non-negative initial conditions: S0≥0,I0≥0,R0≥0,B0≥0,0≤ui≤1,andi=1,2,3,4.Figure 1
The flow diagram of SIR-B model of cholera transmission dynamics.


### 2.2. Fractional Order Derivative Problem

Fractional derivatives are one of the most widely applied differential calculus in science and engineering to describe natural and biological phenomena [10, 18]. Particularly, in this study, Caputo fractional derivative of order α is applied to describe the dynamics of cholera infection and described by the following equations:(2)D0CtαS=λ−βcSBk+B+βSI+γR−μS,D0CtαI=βcSBk+B+βSI−η+φ+μI,D0CtαR=ηI−γ+μR,andD0CtαB=rB1−BK−νB,with non-negative initial conditions subject to S0≥0,I0≥0,R0≥0,andB0≥0.


## 2.1. Deterministic Model Formulation

In this study, we have developed the deterministic mathematical model of cholera with optimal control strategies. We have divided the total population as (i) susceptible (S). These are individuals free of the infection but can be infected with the infection if exposed to the infectious persons or environment contaminated with cholera causing bacterium. (ii) Infected (I). These are individuals that get infected with cholera and can transmit infection to others with possible contacts that cause infection. (iii) Recovered (R). These are individuals recovered from the cholera infection with treatment. The recovered individuals who loss immunity become susceptible. (iv) Concentration of Vibrio cholerae bacterium B (t). This covers an environment contaminated with Vibrio cholerae that cause cholera epidemic. Furthermore, the following assumptions are incorporated in the formulation of the model.(i)
The total population is considered to be nonconstant(ii)
All humans die due to cholera at the disease induced death rate ofφ(iii)
All waste products containVibrio cholerae assumed to be managed and there is no contribution of infected person to the reserved fresh water(iv)
All human population assumed to die at the natural death rateµ(v)
All new infected human individuals are recruited to the susceptible at the recruitment rateλ(vi)
Treatment is performed in the reserved fresh water(vii)
Cholera infected individuals recovered at recovery rate ofη(viii)
Cholera infection recovered individuals loss immunity at the rateγ(ix)
Cholera infection is transmitted from human to human at the rateβ(x)
V. cholerae is ingested from the environment at the ingestion rate βc(xi)
The saturated incidence rate is used in the transmission of cholera infection from environment to human(xii)
The concentration ofV. cholerae in water that causes 50 percent chance of cholera infection is denoted by κ(xiii)
The carrying capacity ofV. cholerae is K(xiv)
V. cholerae die at the rate of v(xv)
Cholera infection transmission preventive control effort isu1(xvi)
Immunity loss control effort isu2(xvii)
V. cholerae clearing control effort is u3In general, the dynamics of human population subject to the cholera infection and the stated assumptions are described by the subsequent schematic diagram.Based on Figure1 and model assumptions the mathematical model of cholera infection with optimal control measures is given by,(1)dSdt=λ−1−u1βcSBk+B+βSI+γR−u2ωS–μS,dIdt=1−u1βcSBk+B+βSI−η+u4+φ+μI,dRdt=η+u4I−γ+μR+u2ωS,anddBdt=1−u3rB1−BK−νB,with non-negative initial conditions: S0≥0,I0≥0,R0≥0,B0≥0,0≤ui≤1,andi=1,2,3,4.Figure 1
The flow diagram of SIR-B model of cholera transmission dynamics.


## 2.2. Fractional Order Derivative Problem

Fractional derivatives are one of the most widely applied differential calculus in science and engineering to describe natural and biological phenomena [10, 18]. Particularly, in this study, Caputo fractional derivative of order α is applied to describe the dynamics of cholera infection and described by the following equations:(2)D0CtαS=λ−βcSBk+B+βSI+γR−μS,D0CtαI=βcSBk+B+βSI−η+φ+μI,D0CtαR=ηI−γ+μR,andD0CtαB=rB1−BK−νB,with non-negative initial conditions subject to S0≥0,I0≥0,R0≥0,andB0≥0.


## 3. Mathematical Analysis of Fractional Derivative Model

In this section, we verify the mathematical and biological acceptence of formulatedmeaningful of cholera model with fractional order derivative. Hence, we show positivity, boundedness, existence, anduniquenessof solution. Moreover, we present the basic analytical analysis of the fractional order derivative model.


### 3.1. Positivity Property

Theorem 1.
The solutions of cholera model with fractional derivative are positive for all positive initial conditions.Proof.
To show positivity of solutions we follow the procedures done in [18]. Consider the trajectory of solution along S-axis so that I0=0,R0=0,B0=0, and S0>0. Then, the first equation of model (2) reduces to the form as follows:(3)D0CtαS=λ−μS,with non-negative initial condition subject to S0≥0.
Moreover, by applying Laplace transforms the solution of equation (3) the following equation is obtained:(4)St=S0Eα−μt+λμ1−Eα−μt≥0.
SinceI0≥0,R0≥0, and R0≥0 similar to the preceding computation we can conclude that It≥0,Rt≥0, and Bt≥0 for all time along the axis. Thus, the solutions of a model with fractional derivative are non-negative for all time t.


### 3.2. Boundedness Property

Theorem 2.
The solution of dynamical system (2) is bounded for time t in the solution region Ω⊂ℜ+3×ℜ+ such that,(5)Ω=S,I,R,B∈ℜ+4:S0+I0+R0≤λμ,B≤K.Proof.
To show boundedness we apply technique applied in [15, 20, 21]. Hence, first we consider the fractional derivative of human population dynamics. Moreover, adding the corresponding left and right terms in cholera model, that belong to human population, it gives(6)D0CtαN=λ−μN−δN.
Then, equation (6) reduces to(7)D0CtαN≤λ−μN,where, N=S+I+R.
Similar to the works done in [15], the solution of the preceding inequality (7) is given by(8)N≤λμ−λμ−N0Eα−μt.
Since Mittag Leffler function and initial conditions are non-negative and bounded, then the preceding inequality (8) reduces to(9)N≤λμ.
Therefore, the solutions of model (2) is bounded for all time t.


### 3.3. Existence and Uniqueness

Theorem 3.
The solution of the cholera model with fractional derivative exist and is unique.Proof.
The right expressions of cholera model (2) with fractional derivative form a function f such that(10)f=λ−βcSBk+B+βSI+γR−μSβcSBk+B+βSI−η+φ+μIηI−γ+μRrB1−BK−νB=f1f2f3f4.
Moreover, let us compute the following partial derivatives,(11)∂f1∂S=−βcBk+B+βI−μ,∂f1∂I=βS,∂f1∂R=γ,∂f1∂B=−βckSk+B2.
We have shown that state variables are bounded. Moreover, functionf1 and its partial derivatives are continuous and bounded. Similarly, f2,f3,andf4 and their corresponding partial derivatives are continuous and bounded. Hence, there exists a unique solution of the model. Therefore, the constructed model both mathematically and biologically sounds meaningful.


### 3.4. Disease-Free Equilibrium (DFE)

A disease-free equilibriumE0C=S0,I0,R0,B0 of cholera model is a point where cholera infection in the population. Hence, setting D0CtαS=0,D0CtαI=0,D0CtαR=0,D0CtαB=0,I=0,andB=0 in the cholera model (2), the computed disease-equilibrium is given by(12)E0C=λμ,0,0,0.


### 3.5. Endemic Equilibrium (EE)

The endemic equilibrium pointE1=S∗,I∗,R∗,B∗ of cholera model is the point where the infection persists in the population or where rate of change of fractional derivatives vanish and it is obtained by solving the following equations:(13)λ−βcSBk+B+βSI+γR−μS=0,βcSBk+B+βSI−η+ϕ+μI=0,ηI−γ+μR=0,andrB1−BK−νB=0.Hence, the solution of preceding equation is given by(14)E1=S∗,I∗,R∗,B∗,Where I∗=λ−μS∗/η+ϕ+μ,B∗=K1−ν/r=B, and S∗=S, which is the solution of the equation(15)μ2R0λS2−BcBK+B+μR0+1S+λ=0.Theorem 4.
A disease-free equilibrium is both locally and globally stable ifR0≤1 and unstable if R0>1.Proof.
Similar to the works done in [22].Theorem 5.
An endemic equilibrium is both locally and globally stable ifR0>1 and unstable if R0≤1.Proof.
The prove follows from the generalization of stability of equilibrium for a model with forward bifurcation as stated in [17].


## 3.1. Positivity Property

Theorem 1.
The solutions of cholera model with fractional derivative are positive for all positive initial conditions.Proof.
To show positivity of solutions we follow the procedures done in [18]. Consider the trajectory of solution along S-axis so that I0=0,R0=0,B0=0, and S0>0. Then, the first equation of model (2) reduces to the form as follows:(3)D0CtαS=λ−μS,with non-negative initial condition subject to S0≥0.
Moreover, by applying Laplace transforms the solution of equation (3) the following equation is obtained:(4)St=S0Eα−μt+λμ1−Eα−μt≥0.
SinceI0≥0,R0≥0, and R0≥0 similar to the preceding computation we can conclude that It≥0,Rt≥0, and Bt≥0 for all time along the axis. Thus, the solutions of a model with fractional derivative are non-negative for all time t.


## 3.2. Boundedness Property

Theorem 2.
The solution of dynamical system (2) is bounded for time t in the solution region Ω⊂ℜ+3×ℜ+ such that,(5)Ω=S,I,R,B∈ℜ+4:S0+I0+R0≤λμ,B≤K.Proof.
To show boundedness we apply technique applied in [15, 20, 21]. Hence, first we consider the fractional derivative of human population dynamics. Moreover, adding the corresponding left and right terms in cholera model, that belong to human population, it gives(6)D0CtαN=λ−μN−δN.
Then, equation (6) reduces to(7)D0CtαN≤λ−μN,where, N=S+I+R.
Similar to the works done in [15], the solution of the preceding inequality (7) is given by(8)N≤λμ−λμ−N0Eα−μt.
Since Mittag Leffler function and initial conditions are non-negative and bounded, then the preceding inequality (8) reduces to(9)N≤λμ.
Therefore, the solutions of model (2) is bounded for all time t.


## 3.3. Existence and Uniqueness

Theorem 3.
The solution of the cholera model with fractional derivative exist and is unique.Proof.
The right expressions of cholera model (2) with fractional derivative form a function f such that(10)f=λ−βcSBk+B+βSI+γR−μSβcSBk+B+βSI−η+φ+μIηI−γ+μRrB1−BK−νB=f1f2f3f4.
Moreover, let us compute the following partial derivatives,(11)∂f1∂S=−βcBk+B+βI−μ,∂f1∂I=βS,∂f1∂R=γ,∂f1∂B=−βckSk+B2.
We have shown that state variables are bounded. Moreover, functionf1 and its partial derivatives are continuous and bounded. Similarly, f2,f3,andf4 and their corresponding partial derivatives are continuous and bounded. Hence, there exists a unique solution of the model. Therefore, the constructed model both mathematically and biologically sounds meaningful.


## 3.4. Disease-Free Equilibrium (DFE)

A disease-free equilibriumE0C=S0,I0,R0,B0 of cholera model is a point where cholera infection in the population. Hence, setting D0CtαS=0,D0CtαI=0,D0CtαR=0,D0CtαB=0,I=0,andB=0 in the cholera model (2), the computed disease-equilibrium is given by(12)E0C=λμ,0,0,0.


## 3.5. Endemic Equilibrium (EE)

The endemic equilibrium pointE1=S∗,I∗,R∗,B∗ of cholera model is the point where the infection persists in the population or where rate of change of fractional derivatives vanish and it is obtained by solving the following equations:(13)λ−βcSBk+B+βSI+γR−μS=0,βcSBk+B+βSI−η+ϕ+μI=0,ηI−γ+μR=0,andrB1−BK−νB=0.Hence, the solution of preceding equation is given by(14)E1=S∗,I∗,R∗,B∗,Where I∗=λ−μS∗/η+ϕ+μ,B∗=K1−ν/r=B, and S∗=S, which is the solution of the equation(15)μ2R0λS2−BcBK+B+μR0+1S+λ=0.Theorem 4.
A disease-free equilibrium is both locally and globally stable ifR0≤1 and unstable if R0>1.Proof.
Similar to the works done in [22].Theorem 5.
An endemic equilibrium is both locally and globally stable ifR0>1 and unstable if R0≤1.Proof.
The prove follows from the generalization of stability of equilibrium for a model with forward bifurcation as stated in [17].


## 4. Optimal Control Problem Analysis

In this section, we have constructed the model of cholera infection with optimal control as follows:(16)dSdt=λ−1−u1βcSBk+B+βSI+γR−u2ωS–μS,dIdt=1−u1βcSBk+B+βSI−η+u4+φ+μI,dRdt=η+u4I−γ+μR+u2ωS,anddBdt=1−u3rB1−BK−νB,with non-negative initial conditions: S0≥0,I0≥0,R0≥0,B0≥0,0≤ui≤1,andi=1,2,3,4.


### 4.1. Construction of Objective Function

AlthoughL1 type cost function is more robust than L2 type cost function, most papers done on optimal control problems adopt L2 type cost function aim for a simply calculations [21]. In this subsection we apply L2 type cost function to define the objective functional J that describes the optimal level of effort required to control cholera infection. Moreover, the aim of objective functional is to minimize the number of infected humans It, the concentrations of Vibrio cholerae in the environment Bt, and the cost of providing and applying the controls u1,u2,u3, and u4. Here, similar to the works done in [23], we choose the quadratic cost on controls. Hence, we have the following equation:(17)J=minu1,u2,u3,u4∫0tfc1I+c2B+12w1u12+w2u22+w3u32+w4u42dx,Where ci,i=1,2 and wi,i=1,2,3,4 are positive constants. Moreover, given the objective functional (17). We find an optimal controls u1∗,u2∗,u3∗, and u4∗ such that(18)Ju1∗,u2∗,u3∗,u4∗=minJu1,u2,u3,u4|u1,u2,u3,u4∈U,where,(19)U=u1,u2,u3,u4:0≤ui<1,0≤t≤tf,i=1,2,3,4,is the measurable set. In the next section, the existence of the optimal control will be discussed.


### 4.2. Existence of Optimal Solution

In this section, to show the existence of optimal solutions, in line with [24], we rewrite optimal control problem so that it can be expressed as coefficient matrix consists of either state variables or control functions. The optimal control problem can be written as follows:(20)Fx,u=Cx+u2,3,4x+u1,3Zx+Hx,where,(21)x=SIRB,C=−μ0γ00−η+φ+μ000η−γ+μ0000r−ν,u2,3,4=−u2ω0000−u400u2ωu400000−u3r,u1,3=u1000−u10000000000u3,Zx=βcSBk+B+βSI−βcSBk+B+βSI0rB2K,Hx=λ−βcSBk+B+βSIβcSBk+B+βSI0−rB2K.Theorem 6.
IfFx,u−Fx∗,u≤Kx−x∗, then the developed control system is uniformly Lipschitz continuous. Where, K=C+u2,3,4+u1,3+1<∞.Proof.
ConsiderFx,u−Fx∗,u = Cx+u2,3,4x+u1,3Zx+Hx−Cx∗+u2,3,4,5x∗+u1,3Zx∗+Hx∗ ≤ C+u2,3,4x−x∗+u1,3Zx−Zx∗+Hx−Hx∗ = C+u2,3,4x−x∗+u1,3x−x∗ ≤ C+u2,3,4+u1,3+1x−x∗≤Kx−x∗. Here, K=C+u2,3,4+u1,3+1<∞. Therefore, the function F is uniformly Lipschitz continuous. Thus, the solutions of control system exist and unique.


### 4.3. Construction of Hamiltonian Function

The necessary condition that an optimal control must satisfy comes from Pontryagin’s Maximum Principle [11–13, 20, 22, 25]. The principle converts equations of objective functional into a problem of minimizing point wise a Hamiltonian H, with respect to the controls u1,u2,u3, and u4. Let λ1,λ2,λ3 and λ4 be the adjoint variables with respect to state variables S,I,R, and B, respectively. Then Hamiltonian function can be constructed as follows:(22)H=c1I+c2B+12w1u12+w2u22+w3u32+w4u42+λ1λ−1−u1βcSBk+B+βSI+γR−u2ωS–μS+λ21−u1βcSBk+B+βSI−η+u4+φ+μI+λ3η+u4I−γ+μR+u2ωS+λ41−u3rB1−BK−νB.With conditions:(1)
dλ1/dt=−∂H/∂S,dλ2/dt=−∂H/∂I,dλ3/dt=−∂H/∂R,dλ4/dt=−∂H/∂B(2)
∂H/∂u1=0,∂H/∂2=0,∂H/∂u3=0,∂H/∂u4=0Theorem 7.
SupposeS∗,I∗,R∗ and B∗ are optimal state solutions with associated optimal control variables u1∗,u2∗,u3∗ and u4∗ for the optimal control problem with conditions. Then there exists adjoint variables λ1,λ2,λ3 and λ4 satisfy the following equations:(23)dλ1dt=1−u1βcBk+B+βIλ1−λ2+u2ωλ1−λ3+μλ1,dλ2dt=−c1+1−u1βSλ1−λ2+η+u4λ2−λ3+φ+μλ2,dλ3dt=γλ3−λ1+μλ3,anddλ4dt=−c2+1−u1βckSk+B2λ1−λ2−1−u3r1−2BKλ4+νλ4.And with transversality conditions,(24)λ1tf=λ2tf=λ3tf=λ4tf=0.And optimality conditions,(25)u1∗=βcSB/k+B+βSIλ2−λ1w1,u2∗=ωSλ1−λ3w2,u3∗=rB1−B/Kλ4w3,andu4∗=Iλ2−λ3w4.Proof.
We apply Pontryagin’s principles to obtain the adjoint equations and the optimal controls. Then, differentiating the constructed Hamiltonian equation (22) with respect to S (t), I (t), R (t), and B (t), and applying adjoint variable conditions, we obtain the following equations:(26)dλ1dt=−∂H∂S,dλ2dt=−∂H∂I,dλ3dt=−∂H∂R,anddλ4dt=−∂H∂B.
Then,(27)dλ1dt=1−u1βcBk+B+βIλ1−λ2+u2ωλ1−λ3+μλ1,dλ2dt=−c1+1−u1βSλ1−λ2+η+u4λ2−λ3+φ+μλ2,dλ3dt=γλ3−λ1+μλ3,anddλ4dt=−c2+1−u1βckSk+B2λ1−λ2−1−u3r1−2BKλ4+νλ4.
Also, optimal control conditions gives the following equations:(28)∂H∂u1=0,∂H∂2=0,∂H∂u3=0,and∂H∂u4=0.
Thus,∂H/∂u1=w1u1+βcSB/k+B+βSIλ1−λ2=0 at u1∗, implies u1∗=βcSB/k+B+βSIλ2−λ1/w1. ∂H/∂u2=w2u2+ωSλ3−λ1=0 at u2∗, implies u2∗=ωSλ1−λ3/w2. ∂H/∂u3=w3u3−rB1−B/Kλ4=0 at u3∗, implies u3∗=rB1−B/Kλ4/w3. ∂H/∂u4=w4u4+Iλ3−λ2=0 at u4∗, implies u4∗=Iλ2−λ3/w4.
In compact form,(29)u1∗=maxminβcSB/k+B+βSIλ2−λ1w1,1,0,u2∗=maxminωSλ1−λ3w2,1,0,u3∗=maxminrB1−B/Kλ4w3,1,0,andu4∗=maxminIλ2−λ3w4,1,0.
Therefore, we obtain the optimal controls and states by solving the optimal system which includes the state system with initial conditions, the adjoint system, and the characterization of the optimal controls. Thus, substituting optimal controlsu1∗t,u2∗t,u3∗t, and u4∗t in the control model equations, we arrived at the optimality system.(30)dSdt=λ−1−u1βcSBk+B+βSI+γR−u2ωS–μS,dIdt=1−u1βcSBk+B+βSI−η+u4+φ+μI,dRdt=η+u4I−γ+μR+u2ωS,dBdt=1−u3rB1−BK−νB,dλ1dt=1−u1βcBk+B+βIλ1−λ2+u2ωλ1−λ3+μλ1,dλ2dt=−c1+1−u1βSλ1−λ2+η+u4λ2−λ3+φ+μλ2,dλ3dt=γλ3−λ1+μλ3,dλ4dt=−c2+1−u1βckSk+B2λ1−λ2−1−u3r1−2BKλ4+νλ4.
With initial conditions:S0≥0,I0≥0,R0≥0,B0≥0, transversality conditions: λ1tf=λ2tf=λ3tf=λ4tf=0, and control conditions: 0≤u1≤1,0≤u2≤1,0≤u3≤1, and 0≤u4≤1. Moreover, the controls ui are in function of time t. Also, due to the boundedness of the state and adjoint variables, the optimality system has unique solution for small time tf.


## 4.1. Construction of Objective Function

AlthoughL1 type cost function is more robust than L2 type cost function, most papers done on optimal control problems adopt L2 type cost function aim for a simply calculations [21]. In this subsection we apply L2 type cost function to define the objective functional J that describes the optimal level of effort required to control cholera infection. Moreover, the aim of objective functional is to minimize the number of infected humans It, the concentrations of Vibrio cholerae in the environment Bt, and the cost of providing and applying the controls u1,u2,u3, and u4. Here, similar to the works done in [23], we choose the quadratic cost on controls. Hence, we have the following equation:(17)J=minu1,u2,u3,u4∫0tfc1I+c2B+12w1u12+w2u22+w3u32+w4u42dx,Where ci,i=1,2 and wi,i=1,2,3,4 are positive constants. Moreover, given the objective functional (17). We find an optimal controls u1∗,u2∗,u3∗, and u4∗ such that(18)Ju1∗,u2∗,u3∗,u4∗=minJu1,u2,u3,u4|u1,u2,u3,u4∈U,where,(19)U=u1,u2,u3,u4:0≤ui<1,0≤t≤tf,i=1,2,3,4,is the measurable set. In the next section, the existence of the optimal control will be discussed.


## 4.2. Existence of Optimal Solution

In this section, to show the existence of optimal solutions, in line with [24], we rewrite optimal control problem so that it can be expressed as coefficient matrix consists of either state variables or control functions. The optimal control problem can be written as follows:(20)Fx,u=Cx+u2,3,4x+u1,3Zx+Hx,where,(21)x=SIRB,C=−μ0γ00−η+φ+μ000η−γ+μ0000r−ν,u2,3,4=−u2ω0000−u400u2ωu400000−u3r,u1,3=u1000−u10000000000u3,Zx=βcSBk+B+βSI−βcSBk+B+βSI0rB2K,Hx=λ−βcSBk+B+βSIβcSBk+B+βSI0−rB2K.Theorem 6.
IfFx,u−Fx∗,u≤Kx−x∗, then the developed control system is uniformly Lipschitz continuous. Where, K=C+u2,3,4+u1,3+1<∞.Proof.
ConsiderFx,u−Fx∗,u = Cx+u2,3,4x+u1,3Zx+Hx−Cx∗+u2,3,4,5x∗+u1,3Zx∗+Hx∗ ≤ C+u2,3,4x−x∗+u1,3Zx−Zx∗+Hx−Hx∗ = C+u2,3,4x−x∗+u1,3x−x∗ ≤ C+u2,3,4+u1,3+1x−x∗≤Kx−x∗. Here, K=C+u2,3,4+u1,3+1<∞. Therefore, the function F is uniformly Lipschitz continuous. Thus, the solutions of control system exist and unique.


## 4.3. Construction of Hamiltonian Function

The necessary condition that an optimal control must satisfy comes from Pontryagin’s Maximum Principle [11–13, 20, 22, 25]. The principle converts equations of objective functional into a problem of minimizing point wise a Hamiltonian H, with respect to the controls u1,u2,u3, and u4. Let λ1,λ2,λ3 and λ4 be the adjoint variables with respect to state variables S,I,R, and B, respectively. Then Hamiltonian function can be constructed as follows:(22)H=c1I+c2B+12w1u12+w2u22+w3u32+w4u42+λ1λ−1−u1βcSBk+B+βSI+γR−u2ωS–μS+λ21−u1βcSBk+B+βSI−η+u4+φ+μI+λ3η+u4I−γ+μR+u2ωS+λ41−u3rB1−BK−νB.With conditions:(1)
dλ1/dt=−∂H/∂S,dλ2/dt=−∂H/∂I,dλ3/dt=−∂H/∂R,dλ4/dt=−∂H/∂B(2)
∂H/∂u1=0,∂H/∂2=0,∂H/∂u3=0,∂H/∂u4=0Theorem 7.
SupposeS∗,I∗,R∗ and B∗ are optimal state solutions with associated optimal control variables u1∗,u2∗,u3∗ and u4∗ for the optimal control problem with conditions. Then there exists adjoint variables λ1,λ2,λ3 and λ4 satisfy the following equations:(23)dλ1dt=1−u1βcBk+B+βIλ1−λ2+u2ωλ1−λ3+μλ1,dλ2dt=−c1+1−u1βSλ1−λ2+η+u4λ2−λ3+φ+μλ2,dλ3dt=γλ3−λ1+μλ3,anddλ4dt=−c2+1−u1βckSk+B2λ1−λ2−1−u3r1−2BKλ4+νλ4.And with transversality conditions,(24)λ1tf=λ2tf=λ3tf=λ4tf=0.And optimality conditions,(25)u1∗=βcSB/k+B+βSIλ2−λ1w1,u2∗=ωSλ1−λ3w2,u3∗=rB1−B/Kλ4w3,andu4∗=Iλ2−λ3w4.Proof.
We apply Pontryagin’s principles to obtain the adjoint equations and the optimal controls. Then, differentiating the constructed Hamiltonian equation (22) with respect to S (t), I (t), R (t), and B (t), and applying adjoint variable conditions, we obtain the following equations:(26)dλ1dt=−∂H∂S,dλ2dt=−∂H∂I,dλ3dt=−∂H∂R,anddλ4dt=−∂H∂B.
Then,(27)dλ1dt=1−u1βcBk+B+βIλ1−λ2+u2ωλ1−λ3+μλ1,dλ2dt=−c1+1−u1βSλ1−λ2+η+u4λ2−λ3+φ+μλ2,dλ3dt=γλ3−λ1+μλ3,anddλ4dt=−c2+1−u1βckSk+B2λ1−λ2−1−u3r1−2BKλ4+νλ4.
Also, optimal control conditions gives the following equations:(28)∂H∂u1=0,∂H∂2=0,∂H∂u3=0,and∂H∂u4=0.
Thus,∂H/∂u1=w1u1+βcSB/k+B+βSIλ1−λ2=0 at u1∗, implies u1∗=βcSB/k+B+βSIλ2−λ1/w1. ∂H/∂u2=w2u2+ωSλ3−λ1=0 at u2∗, implies u2∗=ωSλ1−λ3/w2. ∂H/∂u3=w3u3−rB1−B/Kλ4=0 at u3∗, implies u3∗=rB1−B/Kλ4/w3. ∂H/∂u4=w4u4+Iλ3−λ2=0 at u4∗, implies u4∗=Iλ2−λ3/w4.
In compact form,(29)u1∗=maxminβcSB/k+B+βSIλ2−λ1w1,1,0,u2∗=maxminωSλ1−λ3w2,1,0,u3∗=maxminrB1−B/Kλ4w3,1,0,andu4∗=maxminIλ2−λ3w4,1,0.
Therefore, we obtain the optimal controls and states by solving the optimal system which includes the state system with initial conditions, the adjoint system, and the characterization of the optimal controls. Thus, substituting optimal controlsu1∗t,u2∗t,u3∗t, and u4∗t in the control model equations, we arrived at the optimality system.(30)dSdt=λ−1−u1βcSBk+B+βSI+γR−u2ωS–μS,dIdt=1−u1βcSBk+B+βSI−η+u4+φ+μI,dRdt=η+u4I−γ+μR+u2ωS,dBdt=1−u3rB1−BK−νB,dλ1dt=1−u1βcBk+B+βIλ1−λ2+u2ωλ1−λ3+μλ1,dλ2dt=−c1+1−u1βSλ1−λ2+η+u4λ2−λ3+φ+μλ2,dλ3dt=γλ3−λ1+μλ3,dλ4dt=−c2+1−u1βckSk+B2λ1−λ2−1−u3r1−2BKλ4+νλ4.
With initial conditions:S0≥0,I0≥0,R0≥0,B0≥0, transversality conditions: λ1tf=λ2tf=λ3tf=λ4tf=0, and control conditions: 0≤u1≤1,0≤u2≤1,0≤u3≤1, and 0≤u4≤1. Moreover, the controls ui are in function of time t. Also, due to the boundedness of the state and adjoint variables, the optimality system has unique solution for small time tf.


## 5. Numerical Simulation and Cost-effectiveness Analysis

### 5.1. Numerical Simulation

The parameters presented in Table1 are used in the simulation of solutions of SIR-B model and are either assumed logically or taken from the literature.Table 1
Descriptions of parameter value.
ParameterValue per dayλ15βc0.2143β0.000002ξ0.5ζ0.1μ0.0000548γ0.1η0.05φ0.15ν0.033κ106r0.004K109ω0.025γ0.003S (0)57500C (0)1700R (0)0B (0)275000w11w21w31w41c158c236In Figure2, the memory effect is illustrated by considering the level of impacts imposed. The more the memory effect considered, the more the possibility to inhibit the growth of V. cholerae in the environment. The high-order of fractional derivative value shows more impacts on the growth of V. cholerae.Figure 2
Simulations ofV. cholerae dynamics with different memory effect.In Figure3, the order of fractional derivative shows the impact of memory effect on cholera recovered individuals. The more the memory effect on the population, the more individuals become cholera recovered with immunity.Figure 3
Simulations of recovered individuals with different memory effect.In Figure4, the impact of memory effect on cholera recovered individuals is illustrated. Cholera infected individuals become less if the order of fractional derivative is more. The memory effect on individuals reduces the number of individuals who become infected with cholera infection.Figure 4
Simulations of infected individuals with different memory effect.Next, we investigate the impact of control strategies on the transmission dynamics of HIV in human population. Moreover, the applied control strategies are described as follows:Strategy 1: apply combination of vaccination, hygiene, and chlorination (treatment of water)Strategy 2: apply combination of hygiene, treatment of infective individuals, and chlorination (treatment of water)Strategy 3: apply combination of vaccination, hygiene, treatment of infective individuals, and chlorination (treatment of water)Strategy 4: apply combination of vaccination, hygiene, and treatment of infective individualsStrategy 5: apply combination of vaccination, treatment of infective individuals, and chlorination (treatment of water)The optimal control solution is obtained by solving the optimality system. The well-known fourth-order Runge–Kutta’s methods, both forward and backward methods are applied to solve the optimality system. The details of control strategy are described as follows:Strategy 1: Combination of Vaccination, Hygiene, and Chlorination. In this strategy the combination of the vaccinationu2, hygiene u1, and chlorination u3 are used to optimize the objective function while neglecting the treatment of infective individuals u4 control measure. In Figures 5(a), 5(c), and 5(d) we observed that the number of susceptible individuals, recovered individuals, and pathogens increase while in Figure 3(b) the number of infected individuals decrease due to this strategy over the 50 days of simulations. Therefore, the control strategy acts effectively in controlling infective individuals.Strategy 2: Combination of Hygiene, Treatment of Infective Individuals, and Chlorination In this strategy the combination of hygieneu1, chlorination u3 and treatment of infective individuals u4 are used to optimize the objective function while neglecting the vaccination u2 control measure. In Figures 6(b)–6(d) we observe that the number of infected individuals, recovered individuals, and V. cholerae concentration decrease over the 50 days of simulations. But, in Figure 6(a) the number of susceptible individuals increases over the 50 days. Thus, this strategy acts as reducing the number of individuals get infected.Strategy 3: Combination of Vaccination, Treatment of Infective Individuals, and Chlorination. In this strategy the combination of vaccinationu2, chlorination u3, and treatment of infective individuals u4 are used to optimize the objective function while neglecting the hygiene u1 control measure. In Figures 7(a) and 7(b) we observed that more number of susceptible individuals get decrease and number of infected individuals decrease in the presence of control intervention. In Figures 7(c) and 7(d), in the presence of control, the number of recovered individuals and V. cholerae concentration increase strictly over the days of simulations. Therefore, neglecting hygiene acts as fueling the concentration of V. cholerae in the environment.Strategy 4: Combination of Vaccination, Hygiene, and Treatment of Infective Individuals. In this strategy the combination of vaccinationu2, hygiene u1, and treatment of infective individuals u4 are used to optimize the objective function while neglecting the chlorination u3 control measure. In Figure 8(a), we observe that applying this strategy increase the number of susceptible individuals. But, in Figures 8(b)–8(d), in the presence of control, the number of infective individuals recovered, and V. cholerae concentration decrease over the days of simulations. Therefore, the control strategy acts as controlling infection and reducing the concentration of V. cholerae.Strategy 5: Combination of All Four Controls Vaccination, Treatment of Infective Individuals, and Chlorination. In this strategy the combination of vaccinationu2, hygiene u1 and treatment of infective individuals u4 and chlorination (u3) control measures are applied to optimize the objective functional. In Figure 9(a), we observe that number of susceptible individuals increase the presence of control. Also, the number of recovered individuals increase in the presence of control whereas decrease in the absence of control. In Figures 9(b)–9(d) in the presence of control, the number of infective individuals and V. cholerae concentration decrease over the 50 days of simulations. Therefore, the control strategy acts effectively in controlling infective individuals and pathogens (V. cholerae).Figure 5
Simulations of the cholera model showing the effect of optimal control strategy with vaccination, hygiene, and chlorination control measures.
(a)(b)(c)(d)Figure 6
Simulations of the cholera model showing the effect of optimal control strategy with hygiene, treatment of infective individuals, and chlorination control measures.
(a)(b)(c)(d)Figure 7
Numerical simulations of combination of vaccination, treatment of infective individuals, and chlorination.
(a)(b)(c)(d)Figure 8
Numerical simulations of combination of vaccination, treatment of infective individuals, and hygiene.
(a)(b)(c)(d)Figure 9
Simulations of effects of applying all control measures on cholera infection dynamics.
(a)(b)(c)(d)In Figure10, the control profiles are simulated. It is shown that u1 and u3 are effective from beginning to the end of period infection. Also, u4 is effective for the first five days and decrease for the remaining days of infection. Finally, u2 is less effective than other controls in the days of infection.Figure 10
Simulations of control measures effects on controlling status.In Figure11, the applied combined control cost is simulated. The simulations results show that the cost of applying all controls or the cost of ignoring either u1 or u2 costs more over the first ten days of infection. But, ignoring either u1 and u3 costs less after ten days of infection.Figure 11
Simulations of cost of applied control measures.In Figure12 the dynamics of size of infected individuals versus susceptible individuals are simulated. The number of cholera infected individuals increase to the maximum and onwards get decrease while the susceptible individuals decrease.Figure 12
Simulation of the phase portrait ofS(t) versus I(t).In Figure13 the dynamics of human individual size is simulated in 3-D. Moreover, the more the infected individuals, the more the recovered individuals.Figure 13
Simulation of human individuals on 3-dimensional space.


### 5.2. Cost-effectiveness Analysis

To understand the cost-effectiveness of the applied control measures we apply the Incremental Cost-Effectiveness Ratio (ICER).(31)ICER=Difference of costs investedDifference of number of individuals in health status.In ICER, when comparing two competing intervention strategies incrementally, one intervention should be compared with the next-less effective alternative. The ICER numerator includes the differences in intervention costs, averted disease costs, costs of prevented cases and averted productivity losses if applicable. While, ICER’s denominator is the difference in health outcomes (e.g. total number of infection averted, number of susceptibility cases prevented).The ICER given in Table2 are computed as follows:(32)ICER1=149.41464.4312×105=0.00003,(33)ICER2=142.4517−149.41464.6623×105−4.4312×105=−0.0003,(34)ICER3=61.6417−42.45174.9521×105−4.6623×105=−0.0028,(35)ICER4=111.6205−61.64174.9523×105−4.9521×105=2.4989,(36)ICER5=111.6420−111.620574.9524×105−4.9523×105=0.002124.Table 2
Rank of strategies based on the number of the infection averted.
StrategiesTotal avertedTotal costs ($)ICERNo strategy00Strategy 1 (u1 = 0)4.4312 x 105149.41460.00003Strategy 2 (u4 = 0)4.6623 x 105142.4517−0.0003Strategy 3 (u3 = 0)4.9521 x 10561.6417−0.0028Strategy 4 (u2 = 0)4.9523 x 105111.62052.4989Strategy 5 (no zero4.9524 x 105111.64200.002124Comparing the computations, the ICER (4) value is greater than other ICER values. Hence, we discard strategy 4 and repeat the preceding procedures of computing ICER values for the remaining strategy.The ICER given in Table3 are computed as follows:(37)ICER1=149.41464.4312×105=0.00003,(38)ICER2=142.4517−149.41464.6623×105−4.4312×105=−0.0003,(39)ICER3=61.6417−142.45174.9521×105−4.6623×105=−0.0028,(40)ICER5=111.6420−61.64174.9524×105−4.9521×105=1.66667.Table 3
Rank of strategies based on the number of the infection averted.
StrategiesTotal avertedTotal costs ($)ICERNo strategy00Strategy 1 (u1 = 0)4.4312 x 105149.41460.00003Strategy 2 (u4 = 0)4.6623 x 105142.4517−0.0003Strategy 3 (u3 = 0)4.9521 x 10561.6417−0.0028Strategy 5 (no zero4.9524 x 105111.64201.66667Comparing the computations, the ICER (5) value is greater than others ICER values of strategies. Hence, we discard strategy 5 and repeat the preceding procedures of computing ICER values for the remaining strategy.The ICER given in Table4 are computed as follows:(41)ICER1=149.41464.4312×105=0.00003,(42)ICER2=142.4517−149.41464.6623×105−4.4312×105=−0.0003,(43)ICER3=61.6417−142.45174.9521×105−4.6623×105=−0.0028.Table 4
Rank of strategies based on the number of the infection averted.
StrategiesTotal avertedTotal costs ($)No strategy00Strategy 1 (u1 = 0)4.4312 x 1050.00003Strategy 2 (u4 = 0)4.6623 x 105−0.0003Strategy 3 (u3 = 0)4.9521 x 105−0.0028Comparing the computations, the ICER (1) value is greater than others ICER values of strategies. Hence, we discard strategy 1 and repeat the preceding procedures of computing ICER values for the remaining strategy.The ICER is computed as follows:(45)ICER2=142.4517−149.41464.6623×105−4.4312×105=−0.0003,(46)ICER3=61.6417−142.45174.9521×105−4.6623×105=4.0405.Comparing the computations on Table5, the ICER (3) value is greater than other ICER (2). Hence, we discard strategy 3. Therefore, strategy 2 is the better strategy for controlling of cholera transmission and prevention. Hence, prevention with hygiene, water treatment, and vaccination is preferred among four and three controlling measures in the presence of cholera treatment.Table 5
Rank of strategies based on the number of the infection averted.
StrategiesTotal avertedTotal costs ($)ICERNo strategy00Strategy 2 (u4 = 0)4.6623 x 105142.4517−0.0003Strategy 3 (u3 = 0)4.9521 x 10561.64174.0405


## 5.1. Numerical Simulation

The parameters presented in Table1 are used in the simulation of solutions of SIR-B model and are either assumed logically or taken from the literature.Table 1
Descriptions of parameter value.
ParameterValue per dayλ15βc0.2143β0.000002ξ0.5ζ0.1μ0.0000548γ0.1η0.05φ0.15ν0.033κ106r0.004K109ω0.025γ0.003S (0)57500C (0)1700R (0)0B (0)275000w11w21w31w41c158c236In Figure2, the memory effect is illustrated by considering the level of impacts imposed. The more the memory effect considered, the more the possibility to inhibit the growth of V. cholerae in the environment. The high-order of fractional derivative value shows more impacts on the growth of V. cholerae.Figure 2
Simulations ofV. cholerae dynamics with different memory effect.In Figure3, the order of fractional derivative shows the impact of memory effect on cholera recovered individuals. The more the memory effect on the population, the more individuals become cholera recovered with immunity.Figure 3
Simulations of recovered individuals with different memory effect.In Figure4, the impact of memory effect on cholera recovered individuals is illustrated. Cholera infected individuals become less if the order of fractional derivative is more. The memory effect on individuals reduces the number of individuals who become infected with cholera infection.Figure 4
Simulations of infected individuals with different memory effect.Next, we investigate the impact of control strategies on the transmission dynamics of HIV in human population. Moreover, the applied control strategies are described as follows:Strategy 1: apply combination of vaccination, hygiene, and chlorination (treatment of water)Strategy 2: apply combination of hygiene, treatment of infective individuals, and chlorination (treatment of water)Strategy 3: apply combination of vaccination, hygiene, treatment of infective individuals, and chlorination (treatment of water)Strategy 4: apply combination of vaccination, hygiene, and treatment of infective individualsStrategy 5: apply combination of vaccination, treatment of infective individuals, and chlorination (treatment of water)The optimal control solution is obtained by solving the optimality system. The well-known fourth-order Runge–Kutta’s methods, both forward and backward methods are applied to solve the optimality system. The details of control strategy are described as follows:Strategy 1: Combination of Vaccination, Hygiene, and Chlorination. In this strategy the combination of the vaccinationu2, hygiene u1, and chlorination u3 are used to optimize the objective function while neglecting the treatment of infective individuals u4 control measure. In Figures 5(a), 5(c), and 5(d) we observed that the number of susceptible individuals, recovered individuals, and pathogens increase while in Figure 3(b) the number of infected individuals decrease due to this strategy over the 50 days of simulations. Therefore, the control strategy acts effectively in controlling infective individuals.Strategy 2: Combination of Hygiene, Treatment of Infective Individuals, and Chlorination In this strategy the combination of hygieneu1, chlorination u3 and treatment of infective individuals u4 are used to optimize the objective function while neglecting the vaccination u2 control measure. In Figures 6(b)–6(d) we observe that the number of infected individuals, recovered individuals, and V. cholerae concentration decrease over the 50 days of simulations. But, in Figure 6(a) the number of susceptible individuals increases over the 50 days. Thus, this strategy acts as reducing the number of individuals get infected.Strategy 3: Combination of Vaccination, Treatment of Infective Individuals, and Chlorination. In this strategy the combination of vaccinationu2, chlorination u3, and treatment of infective individuals u4 are used to optimize the objective function while neglecting the hygiene u1 control measure. In Figures 7(a) and 7(b) we observed that more number of susceptible individuals get decrease and number of infected individuals decrease in the presence of control intervention. In Figures 7(c) and 7(d), in the presence of control, the number of recovered individuals and V. cholerae concentration increase strictly over the days of simulations. Therefore, neglecting hygiene acts as fueling the concentration of V. cholerae in the environment.Strategy 4: Combination of Vaccination, Hygiene, and Treatment of Infective Individuals. In this strategy the combination of vaccinationu2, hygiene u1, and treatment of infective individuals u4 are used to optimize the objective function while neglecting the chlorination u3 control measure. In Figure 8(a), we observe that applying this strategy increase the number of susceptible individuals. But, in Figures 8(b)–8(d), in the presence of control, the number of infective individuals recovered, and V. cholerae concentration decrease over the days of simulations. Therefore, the control strategy acts as controlling infection and reducing the concentration of V. cholerae.Strategy 5: Combination of All Four Controls Vaccination, Treatment of Infective Individuals, and Chlorination. In this strategy the combination of vaccinationu2, hygiene u1 and treatment of infective individuals u4 and chlorination (u3) control measures are applied to optimize the objective functional. In Figure 9(a), we observe that number of susceptible individuals increase the presence of control. Also, the number of recovered individuals increase in the presence of control whereas decrease in the absence of control. In Figures 9(b)–9(d) in the presence of control, the number of infective individuals and V. cholerae concentration decrease over the 50 days of simulations. Therefore, the control strategy acts effectively in controlling infective individuals and pathogens (V. cholerae).Figure 5
Simulations of the cholera model showing the effect of optimal control strategy with vaccination, hygiene, and chlorination control measures.
(a)(b)(c)(d)Figure 6
Simulations of the cholera model showing the effect of optimal control strategy with hygiene, treatment of infective individuals, and chlorination control measures.
(a)(b)(c)(d)Figure 7
Numerical simulations of combination of vaccination, treatment of infective individuals, and chlorination.
(a)(b)(c)(d)Figure 8
Numerical simulations of combination of vaccination, treatment of infective individuals, and hygiene.
(a)(b)(c)(d)Figure 9
Simulations of effects of applying all control measures on cholera infection dynamics.
(a)(b)(c)(d)In Figure10, the control profiles are simulated. It is shown that u1 and u3 are effective from beginning to the end of period infection. Also, u4 is effective for the first five days and decrease for the remaining days of infection. Finally, u2 is less effective than other controls in the days of infection.Figure 10
Simulations of control measures effects on controlling status.In Figure11, the applied combined control cost is simulated. The simulations results show that the cost of applying all controls or the cost of ignoring either u1 or u2 costs more over the first ten days of infection. But, ignoring either u1 and u3 costs less after ten days of infection.Figure 11
Simulations of cost of applied control measures.In Figure12 the dynamics of size of infected individuals versus susceptible individuals are simulated. The number of cholera infected individuals increase to the maximum and onwards get decrease while the susceptible individuals decrease.Figure 12
Simulation of the phase portrait ofS(t) versus I(t).In Figure13 the dynamics of human individual size is simulated in 3-D. Moreover, the more the infected individuals, the more the recovered individuals.Figure 13
Simulation of human individuals on 3-dimensional space.


## 5.2. Cost-effectiveness Analysis

To understand the cost-effectiveness of the applied control measures we apply the Incremental Cost-Effectiveness Ratio (ICER).(31)ICER=Difference of costs investedDifference of number of individuals in health status.In ICER, when comparing two competing intervention strategies incrementally, one intervention should be compared with the next-less effective alternative. The ICER numerator includes the differences in intervention costs, averted disease costs, costs of prevented cases and averted productivity losses if applicable. While, ICER’s denominator is the difference in health outcomes (e.g. total number of infection averted, number of susceptibility cases prevented).The ICER given in Table2 are computed as follows:(32)ICER1=149.41464.4312×105=0.00003,(33)ICER2=142.4517−149.41464.6623×105−4.4312×105=−0.0003,(34)ICER3=61.6417−42.45174.9521×105−4.6623×105=−0.0028,(35)ICER4=111.6205−61.64174.9523×105−4.9521×105=2.4989,(36)ICER5=111.6420−111.620574.9524×105−4.9523×105=0.002124.Table 2
Rank of strategies based on the number of the infection averted.
StrategiesTotal avertedTotal costs ($)ICERNo strategy00Strategy 1 (u1 = 0)4.4312 x 105149.41460.00003Strategy 2 (u4 = 0)4.6623 x 105142.4517−0.0003Strategy 3 (u3 = 0)4.9521 x 10561.6417−0.0028Strategy 4 (u2 = 0)4.9523 x 105111.62052.4989Strategy 5 (no zero4.9524 x 105111.64200.002124Comparing the computations, the ICER (4) value is greater than other ICER values. Hence, we discard strategy 4 and repeat the preceding procedures of computing ICER values for the remaining strategy.The ICER given in Table3 are computed as follows:(37)ICER1=149.41464.4312×105=0.00003,(38)ICER2=142.4517−149.41464.6623×105−4.4312×105=−0.0003,(39)ICER3=61.6417−142.45174.9521×105−4.6623×105=−0.0028,(40)ICER5=111.6420−61.64174.9524×105−4.9521×105=1.66667.Table 3
Rank of strategies based on the number of the infection averted.
StrategiesTotal avertedTotal costs ($)ICERNo strategy00Strategy 1 (u1 = 0)4.4312 x 105149.41460.00003Strategy 2 (u4 = 0)4.6623 x 105142.4517−0.0003Strategy 3 (u3 = 0)4.9521 x 10561.6417−0.0028Strategy 5 (no zero4.9524 x 105111.64201.66667Comparing the computations, the ICER (5) value is greater than others ICER values of strategies. Hence, we discard strategy 5 and repeat the preceding procedures of computing ICER values for the remaining strategy.The ICER given in Table4 are computed as follows:(41)ICER1=149.41464.4312×105=0.00003,(42)ICER2=142.4517−149.41464.6623×105−4.4312×105=−0.0003,(43)ICER3=61.6417−142.45174.9521×105−4.6623×105=−0.0028.Table 4
Rank of strategies based on the number of the infection averted.
StrategiesTotal avertedTotal costs ($)No strategy00Strategy 1 (u1 = 0)4.4312 x 1050.00003Strategy 2 (u4 = 0)4.6623 x 105−0.0003Strategy 3 (u3 = 0)4.9521 x 105−0.0028Comparing the computations, the ICER (1) value is greater than others ICER values of strategies. Hence, we discard strategy 1 and repeat the preceding procedures of computing ICER values for the remaining strategy.The ICER is computed as follows:(45)ICER2=142.4517−149.41464.6623×105−4.4312×105=−0.0003,(46)ICER3=61.6417−142.45174.9521×105−4.6623×105=4.0405.Comparing the computations on Table5, the ICER (3) value is greater than other ICER (2). Hence, we discard strategy 3. Therefore, strategy 2 is the better strategy for controlling of cholera transmission and prevention. Hence, prevention with hygiene, water treatment, and vaccination is preferred among four and three controlling measures in the presence of cholera treatment.Table 5
Rank of strategies based on the number of the infection averted.
StrategiesTotal avertedTotal costs ($)ICERNo strategy00Strategy 2 (u4 = 0)4.6623 x 105142.4517−0.0003Strategy 3 (u3 = 0)4.9521 x 10561.64174.0405


## 6. Results and Discussions

In this chapter, dynamics of cholera infection is described with analytical and numerical analysis. The fractional and optimal control effects are described. Moreover, the numerical simulation results obtained in Figure2 shows that increasing order of fractional derivative reduces the V. cholerae concentration in the environment. In Figure 3 the recovered population size increase as the order of fractional derivative increase whereas in Figure 4 the number of cholera infected individuals size decrease as the order of fractional derivative increase. Moreover, the numerical simulations results obtained from Figures 5–9 depicts the combined controls reduces the outbreak of infections in the population and increase the cholera-free human population. Furthermore, in Figures 10 and 11 the control profiles are simulated.Moreover, numerical simulation result indicates applying control strategythat includes control measures u1 and u3 from the beginning results in more effective techniques to control further spreading of cholera infection. Moreover, applying prevention with hygiene, water treatment, and vaccination is preferred among four and three controlling measures in the presence of cholera treatment. In Figures 12–14 the dynamics of individual size is simulated. The result shows that the numbers of susceptible individuals decrease in the presence of cholera infection. The fractional derivative analysis results is supported by results achieved in [15, 26]. In Figure 6, the impact of control strategies on the individuals is simulated along with associated coasts and applied control measures. The applied control reduces the number of infected individuals and V. cholerae concentration. Moreover, applying prevention controlling measures both on human and environment is effective from the starting to the end of the infection. The cost of applying all control is highly relative to others. However, the cost of including vaccination and treatment in applying three control measures is less compared to other control measures. The optimal control analysis is supported with results obtained in [21, 27–29].Figure 14
Simulation ofS(t), I(t), and B(t) dynamics.


## 7. Conclusions

In this study, we formulate a mathematical model of cholera infection and perform analysis with fractional derivative effect on memory of individuals and optimal control strategies impact on dynamics of infection. Moreover, the intervention with optimal control strategies block the occurrences meanwhile reduces theV. cholerae concentration in the environment. Generally, alarming memory of individuals to ward infection reduces the chance of infection occurrences and minimize the impacts to be faced with possibility of infection outbreak.



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