# A New Generalized Fractional-Order Derivative and Bifurcation Analysis of Cholera and Human Immunodeficiency Co-Infection Dynamic Transmission

Authors: Kumama Regassa Cheneke; Koya Purnachandra Rao; Gereme Kenassa Edesssa
Journal: International Journal of Mathematics and Mathematical Sciences
 (2022)

---

## Abstract

In this study, the co-infection of HIV and cholera model has been developed and analyzed. The new fractional-order derivative is applied and the behavior of the solution is interpreted. The order of new generalized fractional-order derivative implication is presented. A new method is incorporated to determine the forward bifurcation at a thresholdR0=1. The developed method is used to determine the stability of steady-state points. The full model and the submodels’ disease-free equilibrium are locally asymptotically stable if the corresponding reproduction number is less than one and unstable if the production number is greater than one. The only HIV model exhibits forward bifurcation at the threshold point, R0=1. The numerical simulations solutions obtained using a new generalized fractional-order derivative shows that the total human population size approaches the disease-free equilibrium if the order of the fractional derivative is higher. Also, the simulated results show that the memory effects toward the invading disease are less whenever the order of the fractional derivative is near 0 but higher whenever the order of the fractional derivative is near 1. Furthermore, V. cholerae concentration in the environment increases whenever the intrinsic growth rate increases. The numerical solutions are carried out using MATLAB software.


---

## Body

## 1. Introduction

The individual infected with Human immunodeficiency virus (HIV) has a weak immune system [1–5]. HIV is the virus that invades targeted human T-cells directly [6, 7]. The advanced stage HIV of causes the fatal disease called acquired immunodeficiency syndrome (AIDS) [5]. In 2018, globally the estimated number of people living with HIV was 37.9 million and estimated deaths were about 1.2 million. Moreover, only about 62% of HIV tested individuals were under ART [5]. Particularly, in Ethiopia, 2018, there were about 690 people living with HIV, about 23,000 were newly infected individuals, and 11000 people died because of infection [5]. The common modes of HIV transmission are unsafe sex, usage of HIV exposed materials, mother to child during pregnancy, and untested blood transfusion. Individuals who get infected with HIV/AIDS can be treated with antiretroviral therapy (ART) [5]. The individual who get cholera infection can be dead within an hour if not treated instantly [8]. The transmission ways of cholera infection is through eating food and drinking water exposed to Vibrio cholerae that causes the infection [9–14]. Cholera infected individuals can be treated with oral rehydration solution (ORS). HIV infected individuals has high susceptibility to cholera infection [4]. The individuals identified with cholera infection need to be tested for HIV infection.The mathematical model is the widely used science to describe the dynamic behavior of population toward infections. For instance, a new generalized fractional order is applied in the epidemiology [15, 16]. Moreover, mathematical models are widely applied in modeling different infections [16–30].In this study, we are motivated by a new generalized fractional-order derivative described and applied in the recent works in [15, 16]. Moreover, we have modified the works of [4], and a new generalized fractional-order derivative is applied to the developed HIV-cholera co-infection model.


## 2. Mathematical Model

In this study, we have developed the HIV and cholera co-infection model (Figure1). We have divided the population targeted for this study into ten exclusive compartments which consist of the following. (i) Susceptible individuals (S): these are noninfectious individuals who can be infected with possible exposure to the infection. (ii) Only cholera-infected individuals (C): these are individuals infected with only cholera infection. (iii) Recovered individuals (R): these are recovered individuals who are infected by only cholera infection. (iv) HIV-infected individuals (I): these are individuals who get infected with HIV and at pre-AIDS stage. (v) HIV-cholera co-infected individuals (D): these are individuals who are infected both with HIV and cholera infection. They are at pre-AIDS stage. (vi) Recovered HIV individuals (E): these are HIV infected and cholera recovered individuals. They are at pre-AIDS stage. (vii) AIDS individuals (A): these are individuals who get infected with only HIV and at AIDS stage. (viii) AIDS-cholera-infected individuals F: these are individuals at the AIDS stage and infected with cholera infection. (ix) Recovered AIDS individuals G: these are cholera-recovered AIDS individuals. (x) Vibrio cholerae concentration (B): these are concentration of Vibrio cholerae in the environment that causes cholera infection with possible feeding the contaminated food or water.Figure 1
Flow diagram of HIV and cholera transmission dynamics.Furthermore, the following assumptions are considered in the development of the HIV and cholera model:(i)
The population is assumed to be homogeneous.(ii)
Recruitment rate of individuals into susceptible compartment isτ.(iii)
The total population size is nonconstant.(iv)
The total population size of human population under the study is denoted byNHt and defined as(1)NHt=St+Ct+Rt+It+Dt+Et+At+Ft+Gt.(v)
All human populations die at natural death rate ofμ.(vi)
All individuals inC and D die due to cholera infection at the rate of φ and ψ, respectively.(vii)
All individuals inA, F, and G die at disease-induced death rate of δ,ς, and ζ, respectively.(viii)
The transferring rate ofC to R due to treatment is ω.(ix)
The transferring rate ofD to E due to treatment is κ.(x)
The transferring rate ofF to G due to treatment is ρ.(xi)
The rate of losing immunity forR, E, and G are γ,η, and ϕ, respectively.(xii)
HIV is transmitted at transmission rate isβh.(xiii)
Cholera infection transmission rate isβ.(xiv)
Logistic growth model is considered for concentration ofV. cholerae in the environment.(xv)
Intrinsic growth rate isr.(xvi)
Concentration ofV. cholerae in the environment that causes 50% cholera infection is κ.(xvii)
K is the carrying capacity of V. cholerae in the environment.(xviii)
V. cholerae die at the rate ν.The mathematical model of the HIV-cholera co-infection is given by(2)dSdt=τ−λ1+λ2S−λ3S+γR−μS,dCdt=λ1+λ2S−λ3C−ω+φ+μC,dRdt=ωC−λ3R−γ+μR,dIdt=λ3S−λ1+λ2I+ηE−α+μI,dDdt=λ1+λ2I+λ3C−ψ+ξ+κ+μD,dEdt=λ3R+κD−η+θ+μE,dAdt=αI−λ1+λ2A+ϕG−δ+μA,dFdt=λ1+λ2A+ξD−ρ+ς+μF,dGdt=ρF+θE−ϕ+ζ+μG,dBdt=rB1−BK−νB,with nonnegative initial conditions, whereλ1=βcB/k+B,λ2=βC+D+F,andλ3=βhI+D+E.


## 3. Mathematical Analysis of the Submodels

### 3.1. Only HIV Model

In the absence of cholera, infection model (2) reduces to the form:(3)dSdt=τ−βhIS−μS,dIdt=βhIS−α+μI,dAdt=αI−δ+μA,with nonnegative initial conditions.


#### 3.1.1. Invariant Region

Theorem 1.
The proper subsetΩ of ℜ+3 is a solution region of only the HIV model such that(4)Ω=S,I,A∈ℜ+3:Nh=S+I+A≤τμ.Proof.
Adding left and right expressions of model (3), we obtain(5)dNhdt=τ−μNh−δA≤τ−μNh.
Applying integration on the preceding inequality, we obtain(6)Nh≤τμ−τμ−Nh0e−μt.
As time gets larger and larger, the preceding expression is bounded above byτ/μ. Therefore, the solutions of only the HIV model are bounded in the invariant Ω such that(7)Ω=S,I,A∈ℜ+3:Nh=S+I+A≤τμ.


#### 3.1.2. Positivity Property

Theorem 2.
The solutions of only HIV model are nonnegative for all timet.Proof.
Considering the first equation of model (3), we obtain(8)dSdt=τ−βhIS−μS.
Ignoring the positive termτ, the forgoing equation reduces to the inequality of the form:(9)dSdt≥−βhIS−μS.
Applying integration and solving for variableS, the foregoing inequality reduces to(10)St≥S0e−μt−βh∫0tIξSξdξ.
In the preceding inequality,S0 and e−μt−βh∫0tIξSξdξ are nonnegative for all time t. Therefore, St is nonnegative for all time t. Similarly, other state variables are nonnegative. Therefore, all solutions are nonnegative.


#### 3.1.3. Existence and Uniqueness of Solutions

Proof.
The expressions on the right of equality in the only HIV model are continuously differentiable everywhere. Therefore, the solutions of the only HIV model exist and are unique by Cauchy–Lipchitz theorem, stated in [11].


#### 3.1.4. Disease-Free Equilibrium

The disease-free equilibriumE0h is a steady state where there is no disease in the population. To obtain disease-free equilibrium, we set the rate of change of variables in model (3) equals to zero along with I=A=0. That is,(11)τ−μS=0.Hence, solving the equation, we obtain(12)E0h=τμ,0,0.


#### 3.1.5. Endemic Equilibrium

An endemic equilibriumEh=S∗,I∗,A∗ is a steady state at which the disease persists in the population. To obtain the endemic equilibrium, we set(13)τ−βhIS−μS=0,βhIS−α+μI=0,αI−δ+μA=0.Solving the foregoing equation, we obtain(14)Eh=α+μβh,μR0−1βh,αμR0−1βhδ+μ.where R0=τβh/μα+μ.


#### 3.1.6. Reproduction Number

The reproduction numberR0h is computed from the only HIV model using the next-generation method. Moreover, from thw only HIV model, the rate of change of infected variables can be written as(15)ddtIA=βhS000IA−α+μ0−αδ+μIA=F−VIA.TakingF=βhS000 and V=α+μ0−αδ+μ, the next-generation matrix FV−1 is given by(16)FV−1=βhS000α+μ0−αδ+μ−1=βhSα+μ000.The eigenvalues of the foregoing next-generation matrix are given by(17)λ1=βhSα+μ,λ2=0.Therefore, the reproduction numberR0h=ρFV−1, spectral radius of next-generation matrix, is given by(18)R0h=βhS0α+μ.where S0=τ/μ.


#### 3.1.7. Bifurcation and Stability Analysis

Center manifold theory is widely used to determine the occurrence of forward bifurcation and backward bifurcation at equilibrium point. For instance, bifurcation analysis is performed in the works done in [28–35]. Moreover, based on application of center manifold theory, we have stated an efficient new generalized method to determine a kind of bifurcation that occurs at R0=1 and stability behavior of equilibriums of the model without computing bifurcation parameters required in center manifold theory on bifurcation analysis.Theorem 3.
(new generalized method for bifurcation and stability analysis).(i)
Bifurcation diagram is an efficient diagram to determine the occurrence of backward and forward bifurcations atR0=1. Hence, simulate endemic equilibrium versus reproduction number,(ii)
An endemic equilibrium of the epidemic model undergoes forward bifurcation atR0=1 if stability behavior of model equilibriums interchanged at R0=1. Moreover, endemic equilibrium of a model is stable only if R0>1.(iii)
An endemic equilibrium of the epidemic model undergoes backward bifurcation atR0=1 if stability behavior of a model equilibria share the same stability behavior if R0<1.(iv)
In backward bifurcation, the endemic equilibrium exhibits both stable and unstable behavior forR0 which is less than bifurcation parameter yielding point.(v)
An epidemic model with endemic equilibrium that undergoes forward bifurcation atR0=1 has locally and globally asymptotically stable endemic equilibrium if R0>1.(vi)
A model with endemic equilibrium that behaves forward bifurcation atR0=1 has locally and globally asymptotically stable disease-free equilibrium if R0≤1.In this section, we simulate endemic equilibrium versus reproduction number to identify the kind of bifurcation that occurs at the pointR0=1 and to determine the stability behavior of equilibriums if R0<1 and R0>1.Theorem 4.
The only HIV model exhibits forward bifurcation atR0=1 provided that only disease-free equilibrium is stable if R0<1 and endemic equilibrium is only positively stable if R0>1.Proof.
It follows from Figure2.


#### 3.1.8. Local Stability of HIV-Free Equilibrium

Theorem 5.
The HIV-free equilibrium of the only HIV model is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
The Jacobian matrixJ constructed from the only HIV model at disease-free equilibrium is given by(19)J=−μ−βhS000βhS0−α+μ00α−δ+μ.
The eigenvalues of the foregoing matrix are computed as(20)λ1=−μ,λ2=α+μR0h−1,λ3=−δ+μ.
Here, all computed eigenvalues are negative. Therefore, by stability theory of differential equations, the HIV-free equilibrium of the only HIV model is locally asymptotically stable if and only ifR0h<1 and unstable if R0h>1.


#### 3.1.9. Global Stability of HIV-Free Equilibrium

Theorem 6.
The HIV-free equilibrium of the only HIV model is globally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
It follows from Theorem3.


#### 3.1.10. Local Stability of Endemic Equilibrium

Theorem 7.
The HIV-free equilibrium of the only HIV model is locally asymptotically stable ifR0>1 and unstable if R0<1.Proof.
It follows from Theorem3.


#### 3.1.11. Local Stability of Endemic Equilibrium

Theorem 8.
The HIV-free equilibrium of the only HIV model is locally asymptotically stable ifR0>1 and unstable if R0<1.Proof.
It follows from Theorem3.


#### 3.1.12. Global Stability of Endemic Equilibrium

Theorem 9.
The endemic equilibrium of the only HIV model is globally asymptotically stable ifR0>1 and unstable if R0<1.Proof.
It follows from Theorem3.


### 3.2. Only Cholera Model

In the absence of HIV/AIDS in the population, the only cholera model is obtained from model (2) as(21)dSdt=τ−βcBk+B+βCS+γR−μS.dCdt=βcBk+B+βCS−ω+φ+μC,dRdt=ωC−γ+μ,dBdt=rB1−BK−νB,under nonnegative initial conditions.


#### 3.2.1. Invariant Region

Theorem 10.
The solutions of the only cholera model are invariant in the regionΩ2 proper subset of ℜ+4 such that Ω2=S,C,R,B:Nc=S+C+R<τ/μ,B≤K.Proof.
Adding the first three equations of the only cholera model, we obtain(22)dNcdt=τ−μNc−φC≤τ−μNc.
Solving the above inequality over time interval0,t, we obtain(23)Nc≤τμ−τμ−Nc0e−μt.
As timet⟶∞, the total human population in the only cholera model Nc⟶τ/μ for all time t. Also, from the last equation of the only cholera model, we have(24)dBdt≤rB1−BK.
Solving the preceding inequality, we obtain(25)1B+1K−BdB≤rKdt,lnBtB0−lnK−BtK−B0≤rKdt,Bt+BtB0K−B0ert/K≤KB0K−B0eK/rt,Bt≤KB0K−B0e−rt/K+B0.
As timet⟶∞, Bt⟶K. Thus, Bt is bounded above by carrying capacity of environment K. Hence, it completes the proof.


#### 3.2.2. Positivity Property

Theorem 11.
Solutions of the only cholera model are nonnegative for all timet.Proof.
The first equation of only cholera model (21) is given by(26)dSdt=τ−βcBSk+B−βSC+γR−μS.
Logically, ignoring the positive termτ+γR in the foregoing equation, we obtain an inequality of the form:(27)dSdt≥−βcBSk+B−βSC−μS.
Applying the integration and solving, we obtain(28)St≥S0e−μt−∫0tβcBξ/k+Bξ+βCξdξ.
SinceS0 and the exponential expression e−μt−∫0tβcBξ/k+Bξ+βCξdξ is positive for all time t, the variable St is nonnegative for all time t.


#### 3.2.3. Existence and Uniqueness of Solutions

Theorem 12.
Solutions of only cholera model (21) exist and are unique.Proof.
The expressions on the right of only cholera model (21) are continuously differentiable and bounded. Therefore, the solutions of only cholera model (21) exist and are unique by Cauchy–Lipchitz theorem.


#### 3.2.4. Cholera-Free Equilibrium

The cholera-free equilibrium of model (21) is computed by setting C=B=0 so that model (21) reduced to the form(29)τ−μS=0,−γ+μR=0.Solving the preceding equation, we obtain(30)S=τμ,R=0.Therefore, cholera-free equilibrium is given by(31)E0c=τμ,0,0,0.


#### 3.2.5. Endemic Equilibrium

The endemic equilibriumE1=S∗,C∗,R∗,B∗ of the only cholera model is computed by solving the equations:(32)τ−βcBSk+B−βSC+γR−μS=0,βcBSk+B+βSC−ω+φ+μC=0,ωC−γ+μR=0,rB1−BK−νB=0.The computed endemic equilibriumE1=S∗C∗R∗B∗ of the only cholera model is given by(33)E1=S∗,C∗,ωγ+μC∗,B∗.where I∗=τ−μS∗/ω+φ+μ,B∗=K1−ν/r=B, and S∗=S is the solution of the equation:(34)μ2R0cτS2−βcBK+B+μR0c+1S+τ=0.


#### 3.2.6. Reproduction Number

The reproduction number is computed from model (21) using the next-generation method; by denoting the new infected individuals arrives at the class by f and remaining terms by v, we obtain(35)f=βcBSk+B+βSCrB1−BK,v=ω+φ+μCνB.The constructed Jacobian matrices are(36)F=βSβcSk0r,V=ω+φ+μ00ν,FV−1=βτμω+φ+μβcτμkν0rν.Basically, the eigenvalues of the preceding next-generation matrix areλ1=βτ/μω+φ+μ,λ2=r/ν. Hence,(37)R0c=maxrν,βτμω+φ+μ.


#### 3.2.7. Local Stability of Disease-Free Equilibrium

Theorem 13.
Cholerae-free equilibrium of the only cholera model is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
Similar to the works carried outin [15].


## 3.1. Only HIV Model

In the absence of cholera, infection model (2) reduces to the form:(3)dSdt=τ−βhIS−μS,dIdt=βhIS−α+μI,dAdt=αI−δ+μA,with nonnegative initial conditions.


### 3.1.1. Invariant Region

Theorem 1.
The proper subsetΩ of ℜ+3 is a solution region of only the HIV model such that(4)Ω=S,I,A∈ℜ+3:Nh=S+I+A≤τμ.Proof.
Adding left and right expressions of model (3), we obtain(5)dNhdt=τ−μNh−δA≤τ−μNh.
Applying integration on the preceding inequality, we obtain(6)Nh≤τμ−τμ−Nh0e−μt.
As time gets larger and larger, the preceding expression is bounded above byτ/μ. Therefore, the solutions of only the HIV model are bounded in the invariant Ω such that(7)Ω=S,I,A∈ℜ+3:Nh=S+I+A≤τμ.


### 3.1.2. Positivity Property

Theorem 2.
The solutions of only HIV model are nonnegative for all timet.Proof.
Considering the first equation of model (3), we obtain(8)dSdt=τ−βhIS−μS.
Ignoring the positive termτ, the forgoing equation reduces to the inequality of the form:(9)dSdt≥−βhIS−μS.
Applying integration and solving for variableS, the foregoing inequality reduces to(10)St≥S0e−μt−βh∫0tIξSξdξ.
In the preceding inequality,S0 and e−μt−βh∫0tIξSξdξ are nonnegative for all time t. Therefore, St is nonnegative for all time t. Similarly, other state variables are nonnegative. Therefore, all solutions are nonnegative.


### 3.1.3. Existence and Uniqueness of Solutions

Proof.
The expressions on the right of equality in the only HIV model are continuously differentiable everywhere. Therefore, the solutions of the only HIV model exist and are unique by Cauchy–Lipchitz theorem, stated in [11].


### 3.1.4. Disease-Free Equilibrium

The disease-free equilibriumE0h is a steady state where there is no disease in the population. To obtain disease-free equilibrium, we set the rate of change of variables in model (3) equals to zero along with I=A=0. That is,(11)τ−μS=0.Hence, solving the equation, we obtain(12)E0h=τμ,0,0.


### 3.1.5. Endemic Equilibrium

An endemic equilibriumEh=S∗,I∗,A∗ is a steady state at which the disease persists in the population. To obtain the endemic equilibrium, we set(13)τ−βhIS−μS=0,βhIS−α+μI=0,αI−δ+μA=0.Solving the foregoing equation, we obtain(14)Eh=α+μβh,μR0−1βh,αμR0−1βhδ+μ.where R0=τβh/μα+μ.


### 3.1.6. Reproduction Number

The reproduction numberR0h is computed from the only HIV model using the next-generation method. Moreover, from thw only HIV model, the rate of change of infected variables can be written as(15)ddtIA=βhS000IA−α+μ0−αδ+μIA=F−VIA.TakingF=βhS000 and V=α+μ0−αδ+μ, the next-generation matrix FV−1 is given by(16)FV−1=βhS000α+μ0−αδ+μ−1=βhSα+μ000.The eigenvalues of the foregoing next-generation matrix are given by(17)λ1=βhSα+μ,λ2=0.Therefore, the reproduction numberR0h=ρFV−1, spectral radius of next-generation matrix, is given by(18)R0h=βhS0α+μ.where S0=τ/μ.


### 3.1.7. Bifurcation and Stability Analysis

Center manifold theory is widely used to determine the occurrence of forward bifurcation and backward bifurcation at equilibrium point. For instance, bifurcation analysis is performed in the works done in [28–35]. Moreover, based on application of center manifold theory, we have stated an efficient new generalized method to determine a kind of bifurcation that occurs at R0=1 and stability behavior of equilibriums of the model without computing bifurcation parameters required in center manifold theory on bifurcation analysis.Theorem 3.
(new generalized method for bifurcation and stability analysis).(i)
Bifurcation diagram is an efficient diagram to determine the occurrence of backward and forward bifurcations atR0=1. Hence, simulate endemic equilibrium versus reproduction number,(ii)
An endemic equilibrium of the epidemic model undergoes forward bifurcation atR0=1 if stability behavior of model equilibriums interchanged at R0=1. Moreover, endemic equilibrium of a model is stable only if R0>1.(iii)
An endemic equilibrium of the epidemic model undergoes backward bifurcation atR0=1 if stability behavior of a model equilibria share the same stability behavior if R0<1.(iv)
In backward bifurcation, the endemic equilibrium exhibits both stable and unstable behavior forR0 which is less than bifurcation parameter yielding point.(v)
An epidemic model with endemic equilibrium that undergoes forward bifurcation atR0=1 has locally and globally asymptotically stable endemic equilibrium if R0>1.(vi)
A model with endemic equilibrium that behaves forward bifurcation atR0=1 has locally and globally asymptotically stable disease-free equilibrium if R0≤1.In this section, we simulate endemic equilibrium versus reproduction number to identify the kind of bifurcation that occurs at the pointR0=1 and to determine the stability behavior of equilibriums if R0<1 and R0>1.Theorem 4.
The only HIV model exhibits forward bifurcation atR0=1 provided that only disease-free equilibrium is stable if R0<1 and endemic equilibrium is only positively stable if R0>1.Proof.
It follows from Figure2.


### 3.1.8. Local Stability of HIV-Free Equilibrium

Theorem 5.
The HIV-free equilibrium of the only HIV model is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
The Jacobian matrixJ constructed from the only HIV model at disease-free equilibrium is given by(19)J=−μ−βhS000βhS0−α+μ00α−δ+μ.
The eigenvalues of the foregoing matrix are computed as(20)λ1=−μ,λ2=α+μR0h−1,λ3=−δ+μ.
Here, all computed eigenvalues are negative. Therefore, by stability theory of differential equations, the HIV-free equilibrium of the only HIV model is locally asymptotically stable if and only ifR0h<1 and unstable if R0h>1.


### 3.1.9. Global Stability of HIV-Free Equilibrium

Theorem 6.
The HIV-free equilibrium of the only HIV model is globally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
It follows from Theorem3.


### 3.1.10. Local Stability of Endemic Equilibrium

Theorem 7.
The HIV-free equilibrium of the only HIV model is locally asymptotically stable ifR0>1 and unstable if R0<1.Proof.
It follows from Theorem3.


### 3.1.11. Local Stability of Endemic Equilibrium

Theorem 8.
The HIV-free equilibrium of the only HIV model is locally asymptotically stable ifR0>1 and unstable if R0<1.Proof.
It follows from Theorem3.


### 3.1.12. Global Stability of Endemic Equilibrium

Theorem 9.
The endemic equilibrium of the only HIV model is globally asymptotically stable ifR0>1 and unstable if R0<1.Proof.
It follows from Theorem3.


## 3.1.1. Invariant Region

Theorem 1.
The proper subsetΩ of ℜ+3 is a solution region of only the HIV model such that(4)Ω=S,I,A∈ℜ+3:Nh=S+I+A≤τμ.Proof.
Adding left and right expressions of model (3), we obtain(5)dNhdt=τ−μNh−δA≤τ−μNh.
Applying integration on the preceding inequality, we obtain(6)Nh≤τμ−τμ−Nh0e−μt.
As time gets larger and larger, the preceding expression is bounded above byτ/μ. Therefore, the solutions of only the HIV model are bounded in the invariant Ω such that(7)Ω=S,I,A∈ℜ+3:Nh=S+I+A≤τμ.


## 3.1.2. Positivity Property

Theorem 2.
The solutions of only HIV model are nonnegative for all timet.Proof.
Considering the first equation of model (3), we obtain(8)dSdt=τ−βhIS−μS.
Ignoring the positive termτ, the forgoing equation reduces to the inequality of the form:(9)dSdt≥−βhIS−μS.
Applying integration and solving for variableS, the foregoing inequality reduces to(10)St≥S0e−μt−βh∫0tIξSξdξ.
In the preceding inequality,S0 and e−μt−βh∫0tIξSξdξ are nonnegative for all time t. Therefore, St is nonnegative for all time t. Similarly, other state variables are nonnegative. Therefore, all solutions are nonnegative.


## 3.1.3. Existence and Uniqueness of Solutions

Proof.
The expressions on the right of equality in the only HIV model are continuously differentiable everywhere. Therefore, the solutions of the only HIV model exist and are unique by Cauchy–Lipchitz theorem, stated in [11].


## 3.1.4. Disease-Free Equilibrium

The disease-free equilibriumE0h is a steady state where there is no disease in the population. To obtain disease-free equilibrium, we set the rate of change of variables in model (3) equals to zero along with I=A=0. That is,(11)τ−μS=0.Hence, solving the equation, we obtain(12)E0h=τμ,0,0.


## 3.1.5. Endemic Equilibrium

An endemic equilibriumEh=S∗,I∗,A∗ is a steady state at which the disease persists in the population. To obtain the endemic equilibrium, we set(13)τ−βhIS−μS=0,βhIS−α+μI=0,αI−δ+μA=0.Solving the foregoing equation, we obtain(14)Eh=α+μβh,μR0−1βh,αμR0−1βhδ+μ.where R0=τβh/μα+μ.


## 3.1.6. Reproduction Number

The reproduction numberR0h is computed from the only HIV model using the next-generation method. Moreover, from thw only HIV model, the rate of change of infected variables can be written as(15)ddtIA=βhS000IA−α+μ0−αδ+μIA=F−VIA.TakingF=βhS000 and V=α+μ0−αδ+μ, the next-generation matrix FV−1 is given by(16)FV−1=βhS000α+μ0−αδ+μ−1=βhSα+μ000.The eigenvalues of the foregoing next-generation matrix are given by(17)λ1=βhSα+μ,λ2=0.Therefore, the reproduction numberR0h=ρFV−1, spectral radius of next-generation matrix, is given by(18)R0h=βhS0α+μ.where S0=τ/μ.


## 3.1.7. Bifurcation and Stability Analysis

Center manifold theory is widely used to determine the occurrence of forward bifurcation and backward bifurcation at equilibrium point. For instance, bifurcation analysis is performed in the works done in [28–35]. Moreover, based on application of center manifold theory, we have stated an efficient new generalized method to determine a kind of bifurcation that occurs at R0=1 and stability behavior of equilibriums of the model without computing bifurcation parameters required in center manifold theory on bifurcation analysis.Theorem 3.
(new generalized method for bifurcation and stability analysis).(i)
Bifurcation diagram is an efficient diagram to determine the occurrence of backward and forward bifurcations atR0=1. Hence, simulate endemic equilibrium versus reproduction number,(ii)
An endemic equilibrium of the epidemic model undergoes forward bifurcation atR0=1 if stability behavior of model equilibriums interchanged at R0=1. Moreover, endemic equilibrium of a model is stable only if R0>1.(iii)
An endemic equilibrium of the epidemic model undergoes backward bifurcation atR0=1 if stability behavior of a model equilibria share the same stability behavior if R0<1.(iv)
In backward bifurcation, the endemic equilibrium exhibits both stable and unstable behavior forR0 which is less than bifurcation parameter yielding point.(v)
An epidemic model with endemic equilibrium that undergoes forward bifurcation atR0=1 has locally and globally asymptotically stable endemic equilibrium if R0>1.(vi)
A model with endemic equilibrium that behaves forward bifurcation atR0=1 has locally and globally asymptotically stable disease-free equilibrium if R0≤1.In this section, we simulate endemic equilibrium versus reproduction number to identify the kind of bifurcation that occurs at the pointR0=1 and to determine the stability behavior of equilibriums if R0<1 and R0>1.Theorem 4.
The only HIV model exhibits forward bifurcation atR0=1 provided that only disease-free equilibrium is stable if R0<1 and endemic equilibrium is only positively stable if R0>1.Proof.
It follows from Figure2.


## 3.1.8. Local Stability of HIV-Free Equilibrium

Theorem 5.
The HIV-free equilibrium of the only HIV model is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
The Jacobian matrixJ constructed from the only HIV model at disease-free equilibrium is given by(19)J=−μ−βhS000βhS0−α+μ00α−δ+μ.
The eigenvalues of the foregoing matrix are computed as(20)λ1=−μ,λ2=α+μR0h−1,λ3=−δ+μ.
Here, all computed eigenvalues are negative. Therefore, by stability theory of differential equations, the HIV-free equilibrium of the only HIV model is locally asymptotically stable if and only ifR0h<1 and unstable if R0h>1.


## 3.1.9. Global Stability of HIV-Free Equilibrium

Theorem 6.
The HIV-free equilibrium of the only HIV model is globally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
It follows from Theorem3.


## 3.1.10. Local Stability of Endemic Equilibrium

Theorem 7.
The HIV-free equilibrium of the only HIV model is locally asymptotically stable ifR0>1 and unstable if R0<1.Proof.
It follows from Theorem3.


## 3.1.11. Local Stability of Endemic Equilibrium

Theorem 8.
The HIV-free equilibrium of the only HIV model is locally asymptotically stable ifR0>1 and unstable if R0<1.Proof.
It follows from Theorem3.


## 3.1.12. Global Stability of Endemic Equilibrium

Theorem 9.
The endemic equilibrium of the only HIV model is globally asymptotically stable ifR0>1 and unstable if R0<1.Proof.
It follows from Theorem3.


## 3.2. Only Cholera Model

In the absence of HIV/AIDS in the population, the only cholera model is obtained from model (2) as(21)dSdt=τ−βcBk+B+βCS+γR−μS.dCdt=βcBk+B+βCS−ω+φ+μC,dRdt=ωC−γ+μ,dBdt=rB1−BK−νB,under nonnegative initial conditions.


### 3.2.1. Invariant Region

Theorem 10.
The solutions of the only cholera model are invariant in the regionΩ2 proper subset of ℜ+4 such that Ω2=S,C,R,B:Nc=S+C+R<τ/μ,B≤K.Proof.
Adding the first three equations of the only cholera model, we obtain(22)dNcdt=τ−μNc−φC≤τ−μNc.
Solving the above inequality over time interval0,t, we obtain(23)Nc≤τμ−τμ−Nc0e−μt.
As timet⟶∞, the total human population in the only cholera model Nc⟶τ/μ for all time t. Also, from the last equation of the only cholera model, we have(24)dBdt≤rB1−BK.
Solving the preceding inequality, we obtain(25)1B+1K−BdB≤rKdt,lnBtB0−lnK−BtK−B0≤rKdt,Bt+BtB0K−B0ert/K≤KB0K−B0eK/rt,Bt≤KB0K−B0e−rt/K+B0.
As timet⟶∞, Bt⟶K. Thus, Bt is bounded above by carrying capacity of environment K. Hence, it completes the proof.


### 3.2.2. Positivity Property

Theorem 11.
Solutions of the only cholera model are nonnegative for all timet.Proof.
The first equation of only cholera model (21) is given by(26)dSdt=τ−βcBSk+B−βSC+γR−μS.
Logically, ignoring the positive termτ+γR in the foregoing equation, we obtain an inequality of the form:(27)dSdt≥−βcBSk+B−βSC−μS.
Applying the integration and solving, we obtain(28)St≥S0e−μt−∫0tβcBξ/k+Bξ+βCξdξ.
SinceS0 and the exponential expression e−μt−∫0tβcBξ/k+Bξ+βCξdξ is positive for all time t, the variable St is nonnegative for all time t.


### 3.2.3. Existence and Uniqueness of Solutions

Theorem 12.
Solutions of only cholera model (21) exist and are unique.Proof.
The expressions on the right of only cholera model (21) are continuously differentiable and bounded. Therefore, the solutions of only cholera model (21) exist and are unique by Cauchy–Lipchitz theorem.


### 3.2.4. Cholera-Free Equilibrium

The cholera-free equilibrium of model (21) is computed by setting C=B=0 so that model (21) reduced to the form(29)τ−μS=0,−γ+μR=0.Solving the preceding equation, we obtain(30)S=τμ,R=0.Therefore, cholera-free equilibrium is given by(31)E0c=τμ,0,0,0.


### 3.2.5. Endemic Equilibrium

The endemic equilibriumE1=S∗,C∗,R∗,B∗ of the only cholera model is computed by solving the equations:(32)τ−βcBSk+B−βSC+γR−μS=0,βcBSk+B+βSC−ω+φ+μC=0,ωC−γ+μR=0,rB1−BK−νB=0.The computed endemic equilibriumE1=S∗C∗R∗B∗ of the only cholera model is given by(33)E1=S∗,C∗,ωγ+μC∗,B∗.where I∗=τ−μS∗/ω+φ+μ,B∗=K1−ν/r=B, and S∗=S is the solution of the equation:(34)μ2R0cτS2−βcBK+B+μR0c+1S+τ=0.


### 3.2.6. Reproduction Number

The reproduction number is computed from model (21) using the next-generation method; by denoting the new infected individuals arrives at the class by f and remaining terms by v, we obtain(35)f=βcBSk+B+βSCrB1−BK,v=ω+φ+μCνB.The constructed Jacobian matrices are(36)F=βSβcSk0r,V=ω+φ+μ00ν,FV−1=βτμω+φ+μβcτμkν0rν.Basically, the eigenvalues of the preceding next-generation matrix areλ1=βτ/μω+φ+μ,λ2=r/ν. Hence,(37)R0c=maxrν,βτμω+φ+μ.


### 3.2.7. Local Stability of Disease-Free Equilibrium

Theorem 13.
Cholerae-free equilibrium of the only cholera model is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
Similar to the works carried outin [15].


## 3.2.1. Invariant Region

Theorem 10.
The solutions of the only cholera model are invariant in the regionΩ2 proper subset of ℜ+4 such that Ω2=S,C,R,B:Nc=S+C+R<τ/μ,B≤K.Proof.
Adding the first three equations of the only cholera model, we obtain(22)dNcdt=τ−μNc−φC≤τ−μNc.
Solving the above inequality over time interval0,t, we obtain(23)Nc≤τμ−τμ−Nc0e−μt.
As timet⟶∞, the total human population in the only cholera model Nc⟶τ/μ for all time t. Also, from the last equation of the only cholera model, we have(24)dBdt≤rB1−BK.
Solving the preceding inequality, we obtain(25)1B+1K−BdB≤rKdt,lnBtB0−lnK−BtK−B0≤rKdt,Bt+BtB0K−B0ert/K≤KB0K−B0eK/rt,Bt≤KB0K−B0e−rt/K+B0.
As timet⟶∞, Bt⟶K. Thus, Bt is bounded above by carrying capacity of environment K. Hence, it completes the proof.


## 3.2.2. Positivity Property

Theorem 11.
Solutions of the only cholera model are nonnegative for all timet.Proof.
The first equation of only cholera model (21) is given by(26)dSdt=τ−βcBSk+B−βSC+γR−μS.
Logically, ignoring the positive termτ+γR in the foregoing equation, we obtain an inequality of the form:(27)dSdt≥−βcBSk+B−βSC−μS.
Applying the integration and solving, we obtain(28)St≥S0e−μt−∫0tβcBξ/k+Bξ+βCξdξ.
SinceS0 and the exponential expression e−μt−∫0tβcBξ/k+Bξ+βCξdξ is positive for all time t, the variable St is nonnegative for all time t.


## 3.2.3. Existence and Uniqueness of Solutions

Theorem 12.
Solutions of only cholera model (21) exist and are unique.Proof.
The expressions on the right of only cholera model (21) are continuously differentiable and bounded. Therefore, the solutions of only cholera model (21) exist and are unique by Cauchy–Lipchitz theorem.


## 3.2.4. Cholera-Free Equilibrium

The cholera-free equilibrium of model (21) is computed by setting C=B=0 so that model (21) reduced to the form(29)τ−μS=0,−γ+μR=0.Solving the preceding equation, we obtain(30)S=τμ,R=0.Therefore, cholera-free equilibrium is given by(31)E0c=τμ,0,0,0.


## 3.2.5. Endemic Equilibrium

The endemic equilibriumE1=S∗,C∗,R∗,B∗ of the only cholera model is computed by solving the equations:(32)τ−βcBSk+B−βSC+γR−μS=0,βcBSk+B+βSC−ω+φ+μC=0,ωC−γ+μR=0,rB1−BK−νB=0.The computed endemic equilibriumE1=S∗C∗R∗B∗ of the only cholera model is given by(33)E1=S∗,C∗,ωγ+μC∗,B∗.where I∗=τ−μS∗/ω+φ+μ,B∗=K1−ν/r=B, and S∗=S is the solution of the equation:(34)μ2R0cτS2−βcBK+B+μR0c+1S+τ=0.


## 3.2.6. Reproduction Number

The reproduction number is computed from model (21) using the next-generation method; by denoting the new infected individuals arrives at the class by f and remaining terms by v, we obtain(35)f=βcBSk+B+βSCrB1−BK,v=ω+φ+μCνB.The constructed Jacobian matrices are(36)F=βSβcSk0r,V=ω+φ+μ00ν,FV−1=βτμω+φ+μβcτμkν0rν.Basically, the eigenvalues of the preceding next-generation matrix areλ1=βτ/μω+φ+μ,λ2=r/ν. Hence,(37)R0c=maxrν,βτμω+φ+μ.


## 3.2.7. Local Stability of Disease-Free Equilibrium

Theorem 13.
Cholerae-free equilibrium of the only cholera model is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
Similar to the works carried outin [15].


## 4. Analysis of the Full Model

### 4.1. Invariant Region

Theorem 14.
The solutions of model (2) are invariant in the region Ω, proper subset of eight-dimensional space over the set of nonnegative real numbers such that(38)Ω=S,C,R,I,D,E,A,F,G,B:Nh=S+C+R+I+D+E+A+F+G≤τμ,B≤K.Proof.
The proof of this theorem is similar to the proof done in only cholera model (21), to show the invariant region.


### 4.2. Existence and Uniqueness of Solutions

Theorem 15.
The solutions of model (2) exist and are unique in the invariant region Ω⊂ℜ+8.Proof.
All expressions on the right-hand side of equality in model (2) are continuously differentiable. Therefore, by Cauchy–Lipchitz theorem stated, the solutions of model (2) exist and are unique.


### 4.3. Disease-Free Equilibrium

The disease-free equilibriumE0 of model (2) is computed as(39)E0=τμ,0,0,0,0,0,0,0.


### 4.4. Reproduction Number

The next-generation method is applied to compute the reproduction number from model (2). Moreover, we defined the following vectors from model (2) logically as(40)f=βcBk+B+βC+D+FSβhI+D+ESβcBk+BI+βC+D+FI+βhI+D+ECβhI+D+ER0βcBk+B+βC+D+FA0rB1−BK,v=ω+φ+μC−ηE+α+μIψ+ξ+κ+μD−κD+η+θ+μE−αI−ϕG+δ+μA−ξD+ρ+ς+μF−ρF−θE+ϕ+ζ+μGνB,F=βS0βS00βS0Sβck0βhSβhSβhS000000000000000000000000000000000000000000000000000r,V=a00000000α+μ0−η000000b0000000−κc00000−α00δ+μ0−ϕ000−ξ00d00000−θ0−ρe00000000ν,FV−1=βSa0βSb+βξSdb00βSd0βcτμkν−r0βhSfβhSb+βhSκcb+βhSκηcbfβhSc+βhSηcf0000000000000000000000000000000000000000000000000000,where a=ω+φ+μ,b=ψ+ξ+κ+μ,c=η+θ+μ,d=ρ+ς+μ,e=ϕ+ζ+μ,andf=α+μ. The foregoing next-generation matrix shows that the reproduction number produced by HIV infected individual is βhS/f. The reproduction number produced by cholera-infected individual and environment is βS/a. Therefore, the basic reproduction number R0 of the full model is given by(41)R0=maxR0h,R0c.


### 4.5. Local Stability of Disease-Free Equilibrium of the Full Model

Theorem 16.
The disease-free equilibrium of the co-infection model is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
To prove the theorem, we constructed a Jacobian matrix evaluated at disease-free equilibrium from the co-infection model as follows:(42)J=−μ−βγ−βh−β−βh−βh0−β0−βcκ0β−a00β00β0βcκ0ω−b0000000000βh−cβhβh+η00000000−d000000000κ−e0000000α00−f0ϕ00000ξ00−g0000000θ0ρ−h0000000000r−ν,a=ω+φ+μ,b=γ+μ,c=α+μ,d=ψ+ξ+κ+μ,e=η+θ+μ,f=δ+μ,g=ρ+ς+μ,h=ϕ+ζ+μ.
The computed eigenvalues of the preceding Jacobian matrix are(43)λ1=−μ,λ2=r−ν,λ3=aR0c−1,λ4=−b,λ5=−d,λ6=−e,λ7=−f,λ8=−g,λ9=−h,λ10=cR0h−1.
Here, all computed eigenvalues are negative ifr<ν,R0c<1,andR0h<1. Therefore, by stability theory, the disease-free equilibrium of the full model is locally asymptotically stable if R0<1 and unstable if R0>1.


## 4.1. Invariant Region

Theorem 14.
The solutions of model (2) are invariant in the region Ω, proper subset of eight-dimensional space over the set of nonnegative real numbers such that(38)Ω=S,C,R,I,D,E,A,F,G,B:Nh=S+C+R+I+D+E+A+F+G≤τμ,B≤K.Proof.
The proof of this theorem is similar to the proof done in only cholera model (21), to show the invariant region.


## 4.2. Existence and Uniqueness of Solutions

Theorem 15.
The solutions of model (2) exist and are unique in the invariant region Ω⊂ℜ+8.Proof.
All expressions on the right-hand side of equality in model (2) are continuously differentiable. Therefore, by Cauchy–Lipchitz theorem stated, the solutions of model (2) exist and are unique.


## 4.3. Disease-Free Equilibrium

The disease-free equilibriumE0 of model (2) is computed as(39)E0=τμ,0,0,0,0,0,0,0.


## 4.4. Reproduction Number

The next-generation method is applied to compute the reproduction number from model (2). Moreover, we defined the following vectors from model (2) logically as(40)f=βcBk+B+βC+D+FSβhI+D+ESβcBk+BI+βC+D+FI+βhI+D+ECβhI+D+ER0βcBk+B+βC+D+FA0rB1−BK,v=ω+φ+μC−ηE+α+μIψ+ξ+κ+μD−κD+η+θ+μE−αI−ϕG+δ+μA−ξD+ρ+ς+μF−ρF−θE+ϕ+ζ+μGνB,F=βS0βS00βS0Sβck0βhSβhSβhS000000000000000000000000000000000000000000000000000r,V=a00000000α+μ0−η000000b0000000−κc00000−α00δ+μ0−ϕ000−ξ00d00000−θ0−ρe00000000ν,FV−1=βSa0βSb+βξSdb00βSd0βcτμkν−r0βhSfβhSb+βhSκcb+βhSκηcbfβhSc+βhSηcf0000000000000000000000000000000000000000000000000000,where a=ω+φ+μ,b=ψ+ξ+κ+μ,c=η+θ+μ,d=ρ+ς+μ,e=ϕ+ζ+μ,andf=α+μ. The foregoing next-generation matrix shows that the reproduction number produced by HIV infected individual is βhS/f. The reproduction number produced by cholera-infected individual and environment is βS/a. Therefore, the basic reproduction number R0 of the full model is given by(41)R0=maxR0h,R0c.


## 4.5. Local Stability of Disease-Free Equilibrium of the Full Model

Theorem 16.
The disease-free equilibrium of the co-infection model is locally asymptotically stable ifR0<1 and unstable if R0>1.Proof.
To prove the theorem, we constructed a Jacobian matrix evaluated at disease-free equilibrium from the co-infection model as follows:(42)J=−μ−βγ−βh−β−βh−βh0−β0−βcκ0β−a00β00β0βcκ0ω−b0000000000βh−cβhβh+η00000000−d000000000κ−e0000000α00−f0ϕ00000ξ00−g0000000θ0ρ−h0000000000r−ν,a=ω+φ+μ,b=γ+μ,c=α+μ,d=ψ+ξ+κ+μ,e=η+θ+μ,f=δ+μ,g=ρ+ς+μ,h=ϕ+ζ+μ.
The computed eigenvalues of the preceding Jacobian matrix are(43)λ1=−μ,λ2=r−ν,λ3=aR0c−1,λ4=−b,λ5=−d,λ6=−e,λ7=−f,λ8=−g,λ9=−h,λ10=cR0h−1.
Here, all computed eigenvalues are negative ifr<ν,R0c<1,andR0h<1. Therefore, by stability theory, the disease-free equilibrium of the full model is locally asymptotically stable if R0<1 and unstable if R0>1.


## 5. Global Stability of Disease-Free Equilibrium

Theorem 17.
A disease-free equilibrium of a full model is globally asymptotically stable if0<R0<1 and otherwise unstable.Proof.
LetX=S,R∈ℜ+2 represent uninfected individuals in the full model and Y=CI,D,E,A,F,G,B∈ℜ+8 be individuals in the infected class. Hence, we write full model (2) as(44)dXdt=HX,Y,dYdt=GX,Y,GX,0=0.
The disease-free equilibrium of the foregoing system is computed as(45)E01=X0,0.
To guarantee global stability, the method we applied must met conditionsH1 and H2 as stated below:H1: for dX/dt=HX,0,X0 is globally asymptotically stableH2: GX,Y=PY−G^X,Y,G^X,Y≥0 for X,Y∈Ω
Here,P=DYGX,0 be Metzler matrix and Ω is a region where solutions are acceptable.
Considering the state variable of noninfected compartment in model (2), we have(46)dXdt=dSdtdRdt=τ−λ1+λ2S−λ3S+γR−μSωC−λ3R−γ+μR=HX,Y.
Now, setting(47)HX,0=0and solving, for X = S,R, we obtain(48)X=λμ,0.
Hence,(49)X0=λμ,0.
Ast⟶∞, we observe that X=S⟶λ/μ. Thus, X0 is globally asymptotically stable equilibrium of the subsequent equation of the form:(50)dXdt=HX,0.
Furthermore, considering the state variable of the infected compartments in model (2), we have(51)dCdt=βcBk+B+βC+D+FS−βhI+D+EC−ω+φ+μC,dIdt=βhI+D+ES−βcBk+B+βC+D+FI+ηE−α+μI,dDdt=βcBk+B+βC+D+FI+βhI+D+EC−ψ+ξ+κ+μD,dEdt=βhI+D+ER+κD−η+θ+μE,dAdt=αI−βcBk+B+βC+D+FA+ϕG−δ+μA,dFdt=βcBk+B+βC+D+FA+ξD−ρ+ς+μF,dGdt=ρF+θE−ϕ+ζ+μG,dBdt=rB1−BK−νB.
Also, at disease-free equilibrium, the foregoing equations can be written as(52)GX,Y=β−a0β00β0βcκ0βh−cβhβh+η000000−d0000000κ−e00000α00−f0ϕ000ξ00−g00000θ0ρ−h00000000r−νCIDEAFGB−00000000.
Now, comparing the required condition and the preceding equation, we obtain(53)P=β−a0β00β0βcκ0βh−cβhβh+η000000−d0000000κ−e00000α00−f0ϕ000ξ00−g00000θ0ρ−h00000000r−ν,Y=CIDEAFGB,G^X,Y=00000000.
Now, we observe thatGX,Y=PY−G^X,Y, where G^X,Y≥0,∀X,Y. Moreover, P is Metzler matrix. Therefore, a disease-free equilibrium E0 is globally asymptotically stable if R0<1.


## 6. A New Generalized Fractional-Order Derivative

In this section, we apply a new generalized fractional-order derivative using the works in [15, 16]. We study the dynamic behavior of the population study using a new generalized definition of derivative with fractional order.First, let us consider only the rate of change of human population, from model (2), that can be described as(54)dSdt=τ−λ1+λ2S−λ3S+γR−μS,dCdt=λ1+λ2S−λ3C−ω+φ+μC,dRdt=ωC−λ3R−γ+μR,dIdt=λ3S−λ1+λ2I+ηE−α+μI,dDdt=λ1+λ2I+λ3C−ψ+ξ+κ+μD,dEdt=λ3R+κD−η+θ+μE,dAdt=αI−λ1+λ2A+ϕG−δ+μA,dFdt=λ1+λ2A+ξD−ρ+ς+μF,dGdt=ρF+θE−ϕ+ζ+μG,with nonnegative initial conditions, where λ1=βcB/k+B,λ2=βC+D+F,andλ3=βhI+D+E.LetNHt be the size of human populations at time t such that(55)NHt=St+Ct+Rt+It+Dt+Et+At+Ft+Gt.Differentiating both sides of the foregoing equation over time interval [0,t] and using (54), we obtain(56)dNHdt=τ–μNh−φI−ψD−δA−ςF−ζG≤τ–μNh.Now, considering the upper bound, the preceding inequality reduces to the form(57)dNHdt=τ–μNH.The solution of the preceding equation over time interval [0, t] is given by(58)NH=τμ−τμ−NH0e−μt.Definition 1.
(a new generalized fractional derivative [15, 16]). Let α∈0,1,β,γ>0, and g∈H1a,b. In the sense given by Caputo, the α order constructed new generalized fractional derivative of some function gt with respect to the weight function wt is defined as(59)DCa,t,wα,β,γgt=Nα1−α1wt∫atEβ−μαt−xγddxwgxdx,where w∈C1a,b,w,w′>0 on a,b, Nα is defined as a normalized function that satisfy the conditions N0=N1=1,μα=α/1−α, and Eβt=∑k=0+∞tk/Γβk+1 is the Mittag–Leffler function of parameter β. For simplicity, we symbolize DCa,t,wα,β,γgt by Da,t,wα,β,γgt. Assuming that when HIV and cholera infections spread within a population, the individuals get knowledge about the infection; we replace the applied classical derivative by Da,t,wα,β,γ. Then, (57) becomes(60)Da,t,wα,β,γNht=τ–μNht.
The basic properties of a new generalized fractional derivative are shown in [36]. Applying the Laplace transform on both sides of the foregoing equation, we obtain(61)ℒwtDa,t,wα,β,γNHt=τℒwt−μℒwtNHt.
Using Theorem2 [19], we have(62)ℒwtNHt=Nαw0T0sβ−1Nα+μ1−αsβ+αμ+τ1−αsβ+λαNα+μ1−αsβ+αμℒwt.
Letaα=Nα+μ1−α. Then,(63)ℒwtNHt=Nαw0NH0ααsβ−1sβ+αμ/aα+τ1−αaαsβ−1sβ+αμ/aαsℒwt+ταaα1sβ+αμ/aαℒwt.
Therefore,(64)ℒwtNHt=Nαw0NH0ααℒEβ−αμaαtβ+τ1−αaαℒEβ−αμaαtβℒw′t+w0−λμℒddtEβ−αμaαtβℒwt.
Applying inverse Laplace, the preceding equation reduces to(65)wtNHt=Nαw0NH0ααEβ−αμaαtβ+τ1−αaαEβ−αμaαtβw′t−τμddtEβ−αμaαtβwt+τ1−αw0aαEβ−αμaαtβ.
Lettingwt=1 and applying the integration by parts, the preceding equation reduces to(66)NHt=τμ−Nαaατμ−NH0Eβ−αμaαtβ.
In Section7, using numerical simulations, we will analyze the impact of order of new fractional derivative on the behavior of solutions of the HIV and cholera co-infection model.
Next, we consider the rate of change ofV. cholerae population, from model (2), that can be described as(67)dBdt=rB1−BK−νB,under nonnegative initial condition.
To get the size ofV. cholerae populations Bt at time t, we set and solve (57) as follows:(68)dBdt=rB1−BK−νB.
We rewrite the foregoing equation in the form(69)dBdt=rB1−νK−BK.
Rearranging the terms, the foregoing equation reduces to the form(70)dBB1−νK−B=rKdt.
Integrating both sides of the preceding equation over time interval [0, t], we obtain(71)∫0tdBB1−νK−B=∫0trKdt.
Applying partial decomposition, we obtain(72)∫0t11−νK1B+11−νK−BdB=∫0trKdt.
Then, the computation of the preceding integral gives(73)lnBt1−νK−Bt×1−νK−B0B0=1−νrt.
Now, applying the logarithmic property, the preceding equation reduces to(74)Bt1−νK−Bt×1−νK−B0B0=e1−νrt.
Rearranging the terms of the preceding equation, we obtain(75)Bt1−νK−Bt=B0e1−νrt1−νK−B0.
This implies(76)Bt=1−νK−BtB0e1−νrt1−νK−B0.
It follows that(77)Bt1+B0e1−νrt1−νK−B0=1−νKB0e1−νrt1−νK−B0.
Now, solving forBt, from the preceding equation, we obtain(78)Bt=1−νKB0e1−νrt1−νK−B0+B0e1−νrt.
In Section7, using numerical simulations, we will analyze the impact of order of new fractional derivative and rank of control measure on the behavior of solutions of the population under the study.


## 7. Numerical Simulations

In this section, the numerical simulations of the solutions obtained using fractional derivative are carried out. The parameters values used in the simulations are taken in reference to [15]:(79)τ=548peopleday,μ=0.0000548day,v=0.033day,r=0.004,βc=1.7×10−8,β=1.48×10−8,βh=4×10−10,α=0.001,ω=0.2day,φ=6×10−6,K=1000000,B0=10000.


## 8. Result and Discussion

In this study, we have investigated the behavior of population based on the new fractional-order derivative developed in [16]. In Figure 2, the forward bifurcation behavior of only HIV model is simulated. The forward bifurcation behavior of the only HIV model shows that the disease-free equilibrium is globally asymptotically stable if R0<1 and unstable if R0>1. However, endemic equilibrium is globally asymptotically stable whenever R0>1 and unstable if R0<1. The results obtained from numerical simulations described in Figure 3 show that increasing the fractional order increases the behavior of population toward controlling the infection. Also, the memory effects toward the invading disease are less whenever order of fractional derivative is near 0 but higher whenever the order of fractional derivative is near 1. In Figure 4, the dynamics of V. cholerae concentration is simulated. The size of V. cholerae concentration in the environment increases significantly whenever intrinsic growth rate increases.Figure 2
Illustration of forward bifurcation atR0=1.Figure 3
Simulation of dynamics of total human population size with order of derivative.Figure 4
Dynamics ofV. cholerae concentration in the environment with different intrinsic growth rates.


## 9. Conclusion

The order of new fractional derivative is directly related to the behavior population toward the disease invading the population. The higher order of fractional derivative depicts that memory of population toward the disease increase so that the disease is controlled with applying possible controlling strategies and increasing population awareness toward HIV and cholera infections. The progression of the disease is described by classical derivative normally. Based on simulated forward bifurcation, the stability behavior of the steady state of the only HIV model is determined as stable or unstable. However, as the order of fractional derivative approaches to, 1 the number of infection-free population increases with time.



---

*Source: 7965145-2022-02-23.xml*