# Fractional-Order Derivative Model of Rift Valley Fever in Urban Peridomestic Cycle

Authors: Saul C. Mpeshe
Journal: Discrete Dynamics in Nature and Society
 (2021)

---

## Abstract

Rift Valley fever is a zoonotic disease which is mainly transmitted by mosquitoes and has potential to affect humans and animals. To gain some understanding on its dynamics in an urban peridomestic cycle, a fractional-order derivative model is formulated and analysed. The basic reproduction numberℛ0 is computed and used in analysing the stability of disease when an outbreak occurs. Numerical simulations are performed in order to the variation of each population at order α=1,0.75,0.5, and 0.25. Results from simulations show that there is an increase in susceptible and exposed population in both human and mosquitoes as the value of α decreases. The infected population decreases with a decrease in the value of α. However, a rapid increase in susceptible mosquitoes is observed just after the first 30 days and a rapid decrease in infected human and mosquitoes after the first 30 days for α=1. Hence, fractional-order derivative also plays a significant role in providing insight on disease transmission and dynamics.


---

## Body

## 1. Introduction

Rift Valley fever (RVF) is a mosquito-borne zoonotic viral disease which affects both humans and animals. The main causative agent of RVF is the Rift Valley fever virus (RVFV) of the genusPhlebovirus in the family Bunyaviridae [1]. The virus was first isolated in sheep in 1930 during an outbreak in Kenya [2].Domestic animals are highly susceptible to RVFV infection and are considered to be the major amplifiers of the virus [3]. Humans are also at a high risk to contract RVFV infection, and they have the ability to develop sufficient viraemia to infect mosquitoes [3].The transmission of RVFV to humans can result through different cycles: the sylvatic cycle and the urban peridomestic cycle [1]. In a sylvatic cycle, humans become infected mostly from contact with the blood or organs of infected animals, while, in the urban peridomestic cycle, humans become infected through the bites of mosquitoes [1]. The Egyptian outbreak of 1977-1978 is known to be peridomestic and anthropophilic [4]. There are possibilities also for human infections to result from other blood-feeding insects, which might be serve as mechanical transmitters of infection. The current urban peridomestic outbreaks of RVF include the 2019 outbreak in Sudan where 1,129 case patients and 19 deaths were reported [5] and that of Mayotte (France) where 143 human cases were reported [6].


## 2. The RVF Model

Mathematical modelling has played a major role in understanding better the global behaviour of infectious diseases. Modelling of infectious diseases using ordinary differential equations (ODEs) also known as integer-order differential equations (IDEs) has proven to be valuable in analysing the dynamics of various diseases including Rift Valley fever [7–9]. However, ODE or IDE cannot store memory.Memory is important in the transmission dynamics and control of infectious diseases. The experience and/or knowledge we have about the spread of a certain disease in the past affects our response during an epidemic outbreak. Knowing the history of a certain disease in our area may help us to think of different control strategies. Therefore, it is crucial to incorporate memory in our epidemic models for a better understanding of the disease dynamics and control. The best way to do this is to use fractional-order derivative models.Fractional-order derivatives are a generalization of the integer-order derivative and integrals to an arbitrary real or complex order. They are powerful mathematical tools which can be used to model the real-world phenomenon which involve memory, wide range of interactions, and hereditary properties, a property which is neglected by integer-order derivatives [10].Modelling with fractional-order derivatives has motivated many people involved in research in science including epidemiology. This is evident from the work by Silva and Torres [11], Boukhouima et al. [10], Ghanbari et al. [12], Alkahtani and Alzaid [13], Tuan et al. [14], and Khan and Atangana [15], just to mention a few. Motivated by their work, a fractional-order derivative model of RVF infection in Caputo derivative is presented in this paper. Other works involving Caputo derivative include works of Ndaïrou et al. [16], Aba Oud et al. [17], and Chu et al. [18] just to mention a few. The paper begins with the formulation of a model using ODEs, and then the model is converted to a fractional-order derivative model using the Caputo derivative. The global stability analysis of the model is established using suitable construction of fractional Lyapunov function, and numerical simulation is done using the predictor-corrector method as outlined by Diethelm et al. [19].


### 2.1. Model Formulation

The model formulation considers two populations, namely: humans and mosquitoes. There is natural death rate in each stage because the infection may take long time, and therefore individual may die naturally. The mode of transmission considered in this model is from mosquito to human and human to mosquito as shown in Figure1. The mosquito population is divided into three compartments, namely: susceptible Sm, exposed Em, and infectious Im. The total mosquito population is given by Nm=Sm+Em+Im. Mosquitoes are also capable of transmitting the virus directly to their offspring (vertical transmission) via eggs leading to new generations of infected mosquitoes hatching from eggs. Therefore, we include the vertical transmission rate parameter q in the model. The human population has four compartments, namely, susceptible Sh, exposed Eh, infectious Ih, and recovered Rh. The total human population is given by Nh=Sh+Eh+Ih+Rh.Figure 1
Transmission diagram for RVF.The parameters and their description as they have been used in this paper are presented in Table1.Table 1
Description of parameters.
ParameterDescriptionbmBirth rate in mosquitoesbhBirth rate in humanλhmAdequate contact rate: human to mosquitoλmhAdequate contact rate: mosquito to humanεmRate of progression from latency to infectious mosquitoεhRate of progression from latency to infectious humanμmNatural death rate of mosquitoesμhNatural death rate of humanqRate of vertical transmissiondDisease-induced death rate in humanγhDisease recovery rate in humanFigure1 represents the flow of diagram of RVF infection in each population involved.From Table1 and Figure 1, an SEIR model is formulated using first-order nonlinear ordinary differential equations as follows:(1a)dShdt=bhNh−μhSh−λmhImNmSh,(1b)dEhdt=λmhImNmSh−εh+μhEh,(1c)dIhdt=εhEh−μh+γh+dIh,(1d)dRhdt=γhIh−μhRh,(1e)dSmdt=bmNm−qIm−μmSm−λhmIhNhSm,(1f)dEmdt=λhmIhNhSm−εm+μmEm,(1g)dImdt=εmEm−μm−bmqIm.Definition 1.
Letft be a continuous function differentiable n-times. The Caputo fractional derivative of ft is defined as in the study by Petras [20] as follows:(2)DaCtαft=1Γn−α∫atfnτt−τα−n+1dτ,for n−1<α<n.For0<α≤1 and a=0, the definition can be written as in [16] as(3)D0Ctαft=1Γ1−α∫0tf′τt−ταdτ.Definition 2.
The integral corresponding to Caputo fractional derivative is defined as in [17] as(4)Itαft=1Γα∫0tfτt−τα−1dτ,for α∈0,1,t>0.The fractional-order derivative model using Caputo derivative is now defined as(5a)D0CtαSh=bhNh−μhSh−λmhImNmSh,(5b)D0CtαEh=λmhImNmSh−εh+μhEh,(5c)D0CtαIh=εhEh−μh+γh+dIh,(5d)D0CtαRh=γhIh−μhRh,(5e)D0CtαSm=bmNm−qIm−μmSm−λhmIhNhSm,(5f)D0CtαEm=λhmIhNhSm−εm+μmEm,(5g)D0CtαIm=εmEm−μm−bmqIm,where D0Ctα is the Caputo derivative of order α∈0,1, with initial conditions as follows:(6)Sh0=Sh0≥0,Eh0=Eh0≥0,Ih0=Ih0≥0,Rh0=Rh0≥0,Sh0=Sm0≥0,Em0=Em0≥0,Im0=Im0≥0.The orderα indicates the index of memory in the system.Let(7)sh=ShNh,eh=EhNh,ih=IhNh,rh=RhNh,sm=SmNm,em=EmNm,im=ImNm,where sh,eh,ih,rh,sm,em,im are population proportions. Then, sh+eh+ih+rh=1 and sm+em+im=1. Since RVF outbreak usually occurs over a short period, then Nh and Nm can be considered to be constant at the time of the outbreak. Therefore, D0Ctαxi=1/NhD0CtαXi for xi=sh,eh,ih,rh and Xi=Sh,Eh,Ih,Rh and D0Ctαxj=1/NhD0CtαXj for xj=sm,em,im and Xj=Sm,Em,Im. Thus, the model system (5a)–(5g) can now be rescaled to(8a)D0Ctαsh=bh−μhsh−λmhimsh,(8b)D0Ctαeh=λmhimsh−εh+μheh,(8c)D0Ctαih=εheh−μh+γh+dih,(8d)D0Ctαrh=γhih−μhrh,(8e)D0Ctαsm=bm1−qim−μmsm−λhmihsm,(8f)D0Ctαem=λhmihsm−εm+μmem,(8g)D0Ctαim=εmem−μm−bmqim.Sincerh is not involved in deriving other equations, then the equation for rh is omitted from the analysis for its value which can be calculated from the values of sh,eh, and ih. Thus, the model system is reduced to(9a)D0Ctαsh=bh−μhsh−λmhimsh,(9b)D0Ctαeh=λmhimsh−εh+μheh,(9c)D0Ctαih=εheh−μh+γh+dih,(9d)D0Ctαsm=bm1−qim−μmsm−λhmihsm,(9e)D0Ctαem=λhmihsm−εm+μmem,(9f)D0Ctαim=εmem−μm−bmqim.


### 2.2. Feasibility of the Model Solution

To show that the model solution exists and is in the positive regionℝ+6, we show that D0Ctαxi≥0 in the region ℝ+6. Using the model system (9a)–(9f), we have(10a)D0Ctαsh|sh=0=bh>0,(10b)D0Ctαeh|eh=0=λmhimsh≥0,(10c)D0Ctαih|ih=0=εheh≥0,(10d)D0Ctαsm|sm=0=bm1−qim>0,(10e)D0Ctαem|em=0=λhmihsm≥0,(10f)D0Ctαim|im=0=εmem≥0.Hence, the model solution is feasible and positive inΩ=sh,eh,ih,sm,em,im≥0∈ℝ+6.To show the solution is positively invariant in the feasible region, we show thatsht+eht+iht≤bh/μh and smt+emt+imt≤bm/μm as t⟶∞. Since sh+eh+ih+rh=1, it follows that(11)D0Ctαsh+eh+ih≤bh−μhsh+eh+ih.Solving this inequality gives(12)sht+eht+iht≤sh0+eh0+ih0Eα−μhtα+bhμh1−Eα−μhtα,where Eα⋅ is a Mittag-Leffler operator.Ast⟶∞, we have(13)sht+eht+iht≤bhμh.Similarly,(14)D0Ctαsm+em+im≤bm−μmsm+em+im.Solving this equation gives(15)smt+emt+imt≤sm0+em0+im0Eα−μmtα+bmμm1−Eα−μmtα,where also Eα⋅ is a Mittag-Leffler operator.Ast⟶∞, we have(16)smt+emt+imt≤bmμm.Hence, the model solution is positively invariant inℝ+6. This means that the model solution will remain in the feasible region Ω if it starts in Ω.


### 2.3. The Basic Reproduction Number

The computation of basic reproduction numberℛ0 is done by applying the next generation method as described by van den Driessche and Watmough [21]. From the model system (9a)–(9f), it is found that(17)F=000λmhsh∗00000λhmsm∗000000,V=εh+μh000−εhμh+γh+d0000εm+μm000−εmμm−bmq,FV−1=00εmλmhsh∗εm+μmμm−bmqλmhsh∗μm−bmq0000εhλhmsm∗εh+μhμh+γh+dλhmsm∗μh+γh+d000000.The largest eigenvalue of theFV−1 gives the expression for the basic reproduction number ℛ0. Computing the eigenvalues of FV−1 and substituting sh∗=bh/μh and sm∗=bm/μm at disease-free equilibrium give(18)ℛ0=bhbmεhεmλmhλhmμhμmεh+μhμh+γh+dεm+μmμm−bmq,with μm−bmq>0. Notice that μm−bmq>0=μm1−bmq/μm where the quantity bmq/μm is the mean number of mosquitoes infected as a result of vertical transmission during its life span.Observe thatℛ0 has six components: the component bm/μm represents the mean number of births of mosquitoes during its lifespan; bh/μh represents the mean number of births of human during the lifespan; εh/εh+μh represents the probability that an infected human survives throughout the incubation period and becomes infectious; εm/εm+μm represents the probability that an infected mosquito survives throughout the incubation period and becomes infectious; λhm/μh+γh+d represents the mean number of mosquitoes infected by human during the infectious period, and λmh/μm−bmq represents the mean number of human infected by mosquitoes during its infectious period.


## 2.1. Model Formulation

The model formulation considers two populations, namely: humans and mosquitoes. There is natural death rate in each stage because the infection may take long time, and therefore individual may die naturally. The mode of transmission considered in this model is from mosquito to human and human to mosquito as shown in Figure1. The mosquito population is divided into three compartments, namely: susceptible Sm, exposed Em, and infectious Im. The total mosquito population is given by Nm=Sm+Em+Im. Mosquitoes are also capable of transmitting the virus directly to their offspring (vertical transmission) via eggs leading to new generations of infected mosquitoes hatching from eggs. Therefore, we include the vertical transmission rate parameter q in the model. The human population has four compartments, namely, susceptible Sh, exposed Eh, infectious Ih, and recovered Rh. The total human population is given by Nh=Sh+Eh+Ih+Rh.Figure 1
Transmission diagram for RVF.The parameters and their description as they have been used in this paper are presented in Table1.Table 1
Description of parameters.
ParameterDescriptionbmBirth rate in mosquitoesbhBirth rate in humanλhmAdequate contact rate: human to mosquitoλmhAdequate contact rate: mosquito to humanεmRate of progression from latency to infectious mosquitoεhRate of progression from latency to infectious humanμmNatural death rate of mosquitoesμhNatural death rate of humanqRate of vertical transmissiondDisease-induced death rate in humanγhDisease recovery rate in humanFigure1 represents the flow of diagram of RVF infection in each population involved.From Table1 and Figure 1, an SEIR model is formulated using first-order nonlinear ordinary differential equations as follows:(1a)dShdt=bhNh−μhSh−λmhImNmSh,(1b)dEhdt=λmhImNmSh−εh+μhEh,(1c)dIhdt=εhEh−μh+γh+dIh,(1d)dRhdt=γhIh−μhRh,(1e)dSmdt=bmNm−qIm−μmSm−λhmIhNhSm,(1f)dEmdt=λhmIhNhSm−εm+μmEm,(1g)dImdt=εmEm−μm−bmqIm.Definition 1.
Letft be a continuous function differentiable n-times. The Caputo fractional derivative of ft is defined as in the study by Petras [20] as follows:(2)DaCtαft=1Γn−α∫atfnτt−τα−n+1dτ,for n−1<α<n.For0<α≤1 and a=0, the definition can be written as in [16] as(3)D0Ctαft=1Γ1−α∫0tf′τt−ταdτ.Definition 2.
The integral corresponding to Caputo fractional derivative is defined as in [17] as(4)Itαft=1Γα∫0tfτt−τα−1dτ,for α∈0,1,t>0.The fractional-order derivative model using Caputo derivative is now defined as(5a)D0CtαSh=bhNh−μhSh−λmhImNmSh,(5b)D0CtαEh=λmhImNmSh−εh+μhEh,(5c)D0CtαIh=εhEh−μh+γh+dIh,(5d)D0CtαRh=γhIh−μhRh,(5e)D0CtαSm=bmNm−qIm−μmSm−λhmIhNhSm,(5f)D0CtαEm=λhmIhNhSm−εm+μmEm,(5g)D0CtαIm=εmEm−μm−bmqIm,where D0Ctα is the Caputo derivative of order α∈0,1, with initial conditions as follows:(6)Sh0=Sh0≥0,Eh0=Eh0≥0,Ih0=Ih0≥0,Rh0=Rh0≥0,Sh0=Sm0≥0,Em0=Em0≥0,Im0=Im0≥0.The orderα indicates the index of memory in the system.Let(7)sh=ShNh,eh=EhNh,ih=IhNh,rh=RhNh,sm=SmNm,em=EmNm,im=ImNm,where sh,eh,ih,rh,sm,em,im are population proportions. Then, sh+eh+ih+rh=1 and sm+em+im=1. Since RVF outbreak usually occurs over a short period, then Nh and Nm can be considered to be constant at the time of the outbreak. Therefore, D0Ctαxi=1/NhD0CtαXi for xi=sh,eh,ih,rh and Xi=Sh,Eh,Ih,Rh and D0Ctαxj=1/NhD0CtαXj for xj=sm,em,im and Xj=Sm,Em,Im. Thus, the model system (5a)–(5g) can now be rescaled to(8a)D0Ctαsh=bh−μhsh−λmhimsh,(8b)D0Ctαeh=λmhimsh−εh+μheh,(8c)D0Ctαih=εheh−μh+γh+dih,(8d)D0Ctαrh=γhih−μhrh,(8e)D0Ctαsm=bm1−qim−μmsm−λhmihsm,(8f)D0Ctαem=λhmihsm−εm+μmem,(8g)D0Ctαim=εmem−μm−bmqim.Sincerh is not involved in deriving other equations, then the equation for rh is omitted from the analysis for its value which can be calculated from the values of sh,eh, and ih. Thus, the model system is reduced to(9a)D0Ctαsh=bh−μhsh−λmhimsh,(9b)D0Ctαeh=λmhimsh−εh+μheh,(9c)D0Ctαih=εheh−μh+γh+dih,(9d)D0Ctαsm=bm1−qim−μmsm−λhmihsm,(9e)D0Ctαem=λhmihsm−εm+μmem,(9f)D0Ctαim=εmem−μm−bmqim.


## 2.2. Feasibility of the Model Solution

To show that the model solution exists and is in the positive regionℝ+6, we show that D0Ctαxi≥0 in the region ℝ+6. Using the model system (9a)–(9f), we have(10a)D0Ctαsh|sh=0=bh>0,(10b)D0Ctαeh|eh=0=λmhimsh≥0,(10c)D0Ctαih|ih=0=εheh≥0,(10d)D0Ctαsm|sm=0=bm1−qim>0,(10e)D0Ctαem|em=0=λhmihsm≥0,(10f)D0Ctαim|im=0=εmem≥0.Hence, the model solution is feasible and positive inΩ=sh,eh,ih,sm,em,im≥0∈ℝ+6.To show the solution is positively invariant in the feasible region, we show thatsht+eht+iht≤bh/μh and smt+emt+imt≤bm/μm as t⟶∞. Since sh+eh+ih+rh=1, it follows that(11)D0Ctαsh+eh+ih≤bh−μhsh+eh+ih.Solving this inequality gives(12)sht+eht+iht≤sh0+eh0+ih0Eα−μhtα+bhμh1−Eα−μhtα,where Eα⋅ is a Mittag-Leffler operator.Ast⟶∞, we have(13)sht+eht+iht≤bhμh.Similarly,(14)D0Ctαsm+em+im≤bm−μmsm+em+im.Solving this equation gives(15)smt+emt+imt≤sm0+em0+im0Eα−μmtα+bmμm1−Eα−μmtα,where also Eα⋅ is a Mittag-Leffler operator.Ast⟶∞, we have(16)smt+emt+imt≤bmμm.Hence, the model solution is positively invariant inℝ+6. This means that the model solution will remain in the feasible region Ω if it starts in Ω.


## 2.3. The Basic Reproduction Number

The computation of basic reproduction numberℛ0 is done by applying the next generation method as described by van den Driessche and Watmough [21]. From the model system (9a)–(9f), it is found that(17)F=000λmhsh∗00000λhmsm∗000000,V=εh+μh000−εhμh+γh+d0000εm+μm000−εmμm−bmq,FV−1=00εmλmhsh∗εm+μmμm−bmqλmhsh∗μm−bmq0000εhλhmsm∗εh+μhμh+γh+dλhmsm∗μh+γh+d000000.The largest eigenvalue of theFV−1 gives the expression for the basic reproduction number ℛ0. Computing the eigenvalues of FV−1 and substituting sh∗=bh/μh and sm∗=bm/μm at disease-free equilibrium give(18)ℛ0=bhbmεhεmλmhλhmμhμmεh+μhμh+γh+dεm+μmμm−bmq,with μm−bmq>0. Notice that μm−bmq>0=μm1−bmq/μm where the quantity bmq/μm is the mean number of mosquitoes infected as a result of vertical transmission during its life span.Observe thatℛ0 has six components: the component bm/μm represents the mean number of births of mosquitoes during its lifespan; bh/μh represents the mean number of births of human during the lifespan; εh/εh+μh represents the probability that an infected human survives throughout the incubation period and becomes infectious; εm/εm+μm represents the probability that an infected mosquito survives throughout the incubation period and becomes infectious; λhm/μh+γh+d represents the mean number of mosquitoes infected by human during the infectious period, and λmh/μm−bmq represents the mean number of human infected by mosquitoes during its infectious period.


## 3. Stability Analysis of Equilibrium Points

To compute the equilibrium points, the left-hand side of model system (9a)–(9f) is equated to zero. For the disease-free equilibrium P0, we have eh=ih=em=im=0, giving(19)P0=bhμh,0,0,bmμm,0,0.Further computation gives the endemic equilibriumP∗ as P∗=sh∗,eh∗,ih∗,sm∗,em∗,im∗ where(20a)sh∗=bhμh+λmhim∗,(20b)eh∗=λmhim∗sh∗εh+μh,(20c)ih∗=εheh∗μh+γh+d,(20d)sm∗=bm1−qim∗μm+λhmih∗,(20e)em∗=λhmih∗sm∗εm+μm,(20f)im∗=εmem∗μm−bmq.


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

Theorem 1.
The disease-free equilibrium of the RVF models (9a)–(9f) is locally asymptotically stable if ℛ0<1 and unstable if ℛ0>1.Proof.
The theorem is proved by showing that all the eigenvalues of the Jacobian matrixJP0 of the fractional-order RVF model (9a)–(9f) evaluated at the disease-free equilibrium satisfy the condition argeigJ=argλi>απ/2. That is, the Jacobian matrix J has negative real eigenvalues. From the model (9a)–(9f), the Jacobian matrix at P0 is given by(21)JP0=−μa0′000−λmhsh∗0−εh+μh000λmhsh∗0εh−μh+γh+d00000−λhmsm∗−μm0−bmq00λhmsm∗0−εm+μm00000εm−μm−bmq.
FromJP0, the diagonal entries −μh and −μm form the first two eigenvalues of the matrix. If the columns and their corresponding rows of these two eigenvalues are eliminated from the matrix, the remaining is a 4×4 matrix given by(22)J∗P0=−εh+μh00λmhsh∗εh−μh+γh+d000λhmsm∗−εm+μm000εm−μm−bmq.
It is easy to see that the remaining matrixJ∗P0 is a Metzler stable matrix whose eigenvalues are all negative. Hence, the Jacobian matrix JP0 has all its eigenvalues negative, and thus the disease-free equilibrium is locally asymptotically stable.


### 3.2. Global Stability of the Disease-Free Equilibrium

Theorem 2.
The disease-free equilibrium of the fractional-order RVF model (9a)–(9f) is globally asymptotically stable if ℛ0<1 and unstable if ℛ0>1.Proof.
We apply the Lyapunov functional approach for fractional differential equations.Definition 3.
LetVxit=∑inωiϕixit be a C1 function defined on some domain in ℝ+n and xit be a solution of the model system D0Ctαxit=fxit,xi0=x0,α∈0,1. Then, the Caputo derivative of V along xit is given by(23)D0CtαVxit=∑inωiD0Ctαϕixit.Now, consider the Lyapunov function(24)V=ω1eh+ω2ih+ω3em+ω4im.The Caputo derivative ofV is then given by(25)D0CtαV=ω1D0Ctαeh+ω2D0Ctαih+ω3D0Ctαem+ω4D0Ctαim.From the model system (9a)–(9f), we have(26)D0CtαV=ω1λmhimsh−εh+μheh+ω2εheh−μh+γh+dih+ω3λhmihsm−εm+μmem+ω4εmem−μm−bmqim≤ω1λmhimsh∗−εh+μheh+ω2εheh−μh+γh+dih+ω3λhmihsm∗−εm+μmem+ω4εmem−μm−bmqim.If we chooseω1=εh,ω2=εh+μh,ω3=εm,ω4=εm+μm and simplify the equation, we have(27)D0CtαV≤εhλmhimsh∗−εh+μhμh+γh+dih+εmλhmihsm∗−εm+μmμm−bmqim=−εh+μhμh+γh+dih1−εmλhmsm∗εh+μhμh+γh+d−εm+μmμm−bmqim1−εhλmhsh∗εm+μmμm−bmq.At disease-free equilibrium,sh∗=bh/μh and sm∗=bm/μm. Therefore, we have(28)D0CtαV≤−εh+μhμh+γh+dih1−εmλhmbmεh+μhμh+γh+dμm−εm+μmμm−bmqim1−εhλmhbhεm+μmμm−bmqμh=−εh+μhμh+γh+dih1−Rhm−εm+μmμm−bmqim1−Rmh,where Rhm=εhλmhbh/εh+μhμh+γh+dμh is the mean number of the infected mosquitoes due to one infected human and Rmh=εhλmhbh/εm+μmμm−bmqμh is the mean number of infected human due one infected mosquito. Observe that the product of Rmh and Rhm gives ℛ02 which is less than 1 at disease-free equilibrium. Consequently, Rmh and Rhm are less than, and therefore we have D0CtαV≤0. Thus, the disease-free equilibrium is globally asymptotically stable at ℛ0<1.


### 3.3. Global Stability of the Endemic Equilibrium

In section3.1, we established that the disease-free equilibrium is locally stable. The local stability of the disease-free equilibrium implies local stability of the endemic equilibrium in counterpart [21]. Therefore, we only investigate the global stability of the endemic equilibrium by Lyapunov functional approach.From Definition3, define ϕixit by ϕixit=xit−xi∗−xi∗lnxit/xi∗ where x∗ is the equilibrium point of the model systems (9a)–(9f). Then, we have the following corollary as in the study by Bourkhouima et al. [10].Corollary 1.
Letxt∈ℝ+ be a continuous differentiable function. Then, for any t≥0,α∈0,1 and x∗≥0, we have(29)D0CtαVxit=∑inD0Ctαxit−xi∗−xi∗lnxitxi∗≤∑inωi1−xi∗xitD0Ctαxit.Now, using Definition3 and Corollary 1, we consider the Lyapunov function(30)V=ω1sh−sh∗−sh∗lnshsh∗+ω2eh−eh∗−eh∗lneheh∗+ω3ih−ih∗−ih∗lnihih∗+ω4sm−sm∗−sm∗lnsmsm∗+ω5em−em∗−em∗lnemem∗+ω6im−im∗−im∗lnimim∗.The Caputo derivative ofV is then given by(31)D0CtαV≤ω11−sh∗sh0CDtαsh+ω21−eh∗eh0CDtαeh+ω31−ih∗ih0CDtαih+ω41−sm∗sm0CDtαsm+ω51−em∗em0CDtαem+ω61−im∗im0CDtαim.From the model systems (9a)–(9f), we have(32)D0CtαV≤ω11−sh∗shbh−μhsh−λmhimsh+ω21−eh∗ehλmhimsh−εh+μheh+ω31−ih∗ihεheh−μh+γh+dih+ω41−sm∗smbm1−qim−μmsm−λhmihsm+ω51−em∗emλhmihsm−εm+μmem+ω61−im∗imεmem−μm−bmqim.Further simplification gives(33)D0CtαV≤−ω1μhsh1−sh∗sh2−ω4μmsm1−sm∗sm2+FP,where P=sh,eh,ih,sm,em,im≥0 and(34)FP=ω11−sh∗sh1−imshim∗sh∗λmhim∗sh∗+ω21−eh∗ehimshim∗sh∗−eheh∗λmhim∗sh∗+ω31−ih∗iheheh∗−ihih∗εheh∗+ω41−sm∗sm1−imim∗bmqim∗+ω41−sm∗sm1−ihsmih∗sm∗λhmih∗sm∗+ω51−em∗emihsmih∗sm∗−emem∗λhmih∗sm∗+ω61−im∗imemem∗−imim∗εmem∗.If we choose(35)ω1=ω2=1λmhim∗sh∗,ω3=1εheh∗,ω4=ω5=1λhmih∗sm∗,ω5=1εmem∗,and perform further simplification, we have(36)FP=3−sh∗sh−ih∗ehiheh∗−eh∗imshehim∗sh∗+3−sm∗sm−im∗emimem∗−em∗ihsmemih∗sm∗+1−sm∗sm1−imim∗bmqim∗λhmih∗sm∗.In general,1−x∗/x≥0 and 1−x/x∗≤0 if the x∗ is the equilibrium point. Therefore, 1−sm∗/sm1−im/im∗≤0, and hence FP is nonpositive in P. Thus, D0CtαV≤0 in P and is zero when P=P∗. Following LaSalle’s invariant principle, it is concluded that P∗ is globally asymptotically stable.


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

Theorem 1.
The disease-free equilibrium of the RVF models (9a)–(9f) is locally asymptotically stable if ℛ0<1 and unstable if ℛ0>1.Proof.
The theorem is proved by showing that all the eigenvalues of the Jacobian matrixJP0 of the fractional-order RVF model (9a)–(9f) evaluated at the disease-free equilibrium satisfy the condition argeigJ=argλi>απ/2. That is, the Jacobian matrix J has negative real eigenvalues. From the model (9a)–(9f), the Jacobian matrix at P0 is given by(21)JP0=−μa0′000−λmhsh∗0−εh+μh000λmhsh∗0εh−μh+γh+d00000−λhmsm∗−μm0−bmq00λhmsm∗0−εm+μm00000εm−μm−bmq.
FromJP0, the diagonal entries −μh and −μm form the first two eigenvalues of the matrix. If the columns and their corresponding rows of these two eigenvalues are eliminated from the matrix, the remaining is a 4×4 matrix given by(22)J∗P0=−εh+μh00λmhsh∗εh−μh+γh+d000λhmsm∗−εm+μm000εm−μm−bmq.
It is easy to see that the remaining matrixJ∗P0 is a Metzler stable matrix whose eigenvalues are all negative. Hence, the Jacobian matrix JP0 has all its eigenvalues negative, and thus the disease-free equilibrium is locally asymptotically stable.


## 3.2. Global Stability of the Disease-Free Equilibrium

Theorem 2.
The disease-free equilibrium of the fractional-order RVF model (9a)–(9f) is globally asymptotically stable if ℛ0<1 and unstable if ℛ0>1.Proof.
We apply the Lyapunov functional approach for fractional differential equations.Definition 3.
LetVxit=∑inωiϕixit be a C1 function defined on some domain in ℝ+n and xit be a solution of the model system D0Ctαxit=fxit,xi0=x0,α∈0,1. Then, the Caputo derivative of V along xit is given by(23)D0CtαVxit=∑inωiD0Ctαϕixit.Now, consider the Lyapunov function(24)V=ω1eh+ω2ih+ω3em+ω4im.The Caputo derivative ofV is then given by(25)D0CtαV=ω1D0Ctαeh+ω2D0Ctαih+ω3D0Ctαem+ω4D0Ctαim.From the model system (9a)–(9f), we have(26)D0CtαV=ω1λmhimsh−εh+μheh+ω2εheh−μh+γh+dih+ω3λhmihsm−εm+μmem+ω4εmem−μm−bmqim≤ω1λmhimsh∗−εh+μheh+ω2εheh−μh+γh+dih+ω3λhmihsm∗−εm+μmem+ω4εmem−μm−bmqim.If we chooseω1=εh,ω2=εh+μh,ω3=εm,ω4=εm+μm and simplify the equation, we have(27)D0CtαV≤εhλmhimsh∗−εh+μhμh+γh+dih+εmλhmihsm∗−εm+μmμm−bmqim=−εh+μhμh+γh+dih1−εmλhmsm∗εh+μhμh+γh+d−εm+μmμm−bmqim1−εhλmhsh∗εm+μmμm−bmq.At disease-free equilibrium,sh∗=bh/μh and sm∗=bm/μm. Therefore, we have(28)D0CtαV≤−εh+μhμh+γh+dih1−εmλhmbmεh+μhμh+γh+dμm−εm+μmμm−bmqim1−εhλmhbhεm+μmμm−bmqμh=−εh+μhμh+γh+dih1−Rhm−εm+μmμm−bmqim1−Rmh,where Rhm=εhλmhbh/εh+μhμh+γh+dμh is the mean number of the infected mosquitoes due to one infected human and Rmh=εhλmhbh/εm+μmμm−bmqμh is the mean number of infected human due one infected mosquito. Observe that the product of Rmh and Rhm gives ℛ02 which is less than 1 at disease-free equilibrium. Consequently, Rmh and Rhm are less than, and therefore we have D0CtαV≤0. Thus, the disease-free equilibrium is globally asymptotically stable at ℛ0<1.


## 3.3. Global Stability of the Endemic Equilibrium

In section3.1, we established that the disease-free equilibrium is locally stable. The local stability of the disease-free equilibrium implies local stability of the endemic equilibrium in counterpart [21]. Therefore, we only investigate the global stability of the endemic equilibrium by Lyapunov functional approach.From Definition3, define ϕixit by ϕixit=xit−xi∗−xi∗lnxit/xi∗ where x∗ is the equilibrium point of the model systems (9a)–(9f). Then, we have the following corollary as in the study by Bourkhouima et al. [10].Corollary 1.
Letxt∈ℝ+ be a continuous differentiable function. Then, for any t≥0,α∈0,1 and x∗≥0, we have(29)D0CtαVxit=∑inD0Ctαxit−xi∗−xi∗lnxitxi∗≤∑inωi1−xi∗xitD0Ctαxit.Now, using Definition3 and Corollary 1, we consider the Lyapunov function(30)V=ω1sh−sh∗−sh∗lnshsh∗+ω2eh−eh∗−eh∗lneheh∗+ω3ih−ih∗−ih∗lnihih∗+ω4sm−sm∗−sm∗lnsmsm∗+ω5em−em∗−em∗lnemem∗+ω6im−im∗−im∗lnimim∗.The Caputo derivative ofV is then given by(31)D0CtαV≤ω11−sh∗sh0CDtαsh+ω21−eh∗eh0CDtαeh+ω31−ih∗ih0CDtαih+ω41−sm∗sm0CDtαsm+ω51−em∗em0CDtαem+ω61−im∗im0CDtαim.From the model systems (9a)–(9f), we have(32)D0CtαV≤ω11−sh∗shbh−μhsh−λmhimsh+ω21−eh∗ehλmhimsh−εh+μheh+ω31−ih∗ihεheh−μh+γh+dih+ω41−sm∗smbm1−qim−μmsm−λhmihsm+ω51−em∗emλhmihsm−εm+μmem+ω61−im∗imεmem−μm−bmqim.Further simplification gives(33)D0CtαV≤−ω1μhsh1−sh∗sh2−ω4μmsm1−sm∗sm2+FP,where P=sh,eh,ih,sm,em,im≥0 and(34)FP=ω11−sh∗sh1−imshim∗sh∗λmhim∗sh∗+ω21−eh∗ehimshim∗sh∗−eheh∗λmhim∗sh∗+ω31−ih∗iheheh∗−ihih∗εheh∗+ω41−sm∗sm1−imim∗bmqim∗+ω41−sm∗sm1−ihsmih∗sm∗λhmih∗sm∗+ω51−em∗emihsmih∗sm∗−emem∗λhmih∗sm∗+ω61−im∗imemem∗−imim∗εmem∗.If we choose(35)ω1=ω2=1λmhim∗sh∗,ω3=1εheh∗,ω4=ω5=1λhmih∗sm∗,ω5=1εmem∗,and perform further simplification, we have(36)FP=3−sh∗sh−ih∗ehiheh∗−eh∗imshehim∗sh∗+3−sm∗sm−im∗emimem∗−em∗ihsmemih∗sm∗+1−sm∗sm1−imim∗bmqim∗λhmih∗sm∗.In general,1−x∗/x≥0 and 1−x/x∗≤0 if the x∗ is the equilibrium point. Therefore, 1−sm∗/sm1−im/im∗≤0, and hence FP is nonpositive in P. Thus, D0CtαV≤0 in P and is zero when P=P∗. Following LaSalle’s invariant principle, it is concluded that P∗ is globally asymptotically stable.


## 4. Numerical Simulations

This section presents results from numerical simulations for the model system (9a)–(9f). The simulation is done using the predictor-corrector method implemented in Matlab fde12 [22]. The initial values used in simulations are sh=0.6,eh=0.3,ih=0.1,sm=0.5,em=0.3,andim=0.2. The mosquito birth rate bh has been estimated from the number of eggs laid per day, and the adequate contact rate from mosquito to human λmh has been estimated from the biting rate and the probability that a mosquito can bite human and cause infection. Other parameter values and their source are as given in Table 2.Table 2
Parameters and their description.
ParameterDescriptionValue per daySourcebhBirth rate in human0.0015[8]bmBirth rate in mosquitoes0.01EstimatedλhmAdequate contact rate: human to mosquito0.375[7]λmhAdequate contact rate: mosquito to human0.245Estimatedεm1/(incubation period of mosquitoes in days)1/8[9]εh1/(incubation period of human in days)1/6[9]μm1/(life span of mosquitoes in days)1/60[9]μhNatural death rate in human0.000042[23]qRate of vertical transmission0.05[8]dDisease-induced death rate in human0.01[7]γh1/(infectious period of human in days)1/7[9]Figure2 shows the variations of susceptible human and mosquito animal population. In Figure 2(a), the human susceptible population decreases with increase in the order α, while, in Figure 2(b), there is a rapid increase in susceptible mosquitoes just after the first 30 days for order α=1, surpassing all orders.Figure 2
Variations of susceptible population atα = 1, 0.75, 0.50, and 0.25: (a) susceptible human; (b) susceptible mosquitoes.
(a)(b)Figure3 shows the variations of exposed human and mosquito animal population. In Figure 3(a), the human exposed population decreases with increase in the order α, where a similar trend is shown also in Figure 3(b) of exposed mosquitoes.Figure 3
Variations of exposed population atα = 1, 0.75, 0.50, and 0.25: (a) exposed human; (b) exposed mosquitoes.
(a)(b)Figure4 shows the variations of infected human and mosquito animal population. In Figure 4(a), the human infected population decreases with the decrease in the order α. However, there is a rapid decrease for α=1 and α=0.75 towards the disease-free equilibrium. In Figure 2(b), a rapid decrease in infected mosquitoes is seen for the order α=1, surpassing all orders.Figure 4
Variations of infected population atα = 1, 0.75, 0.50, and 0.25: (a) infected human; (b) infected mosquitoes.
(a)(b)


## 5. Discussion

In this paper, a fractional-order derivative approach using Caputo derivative has been used to investigate the dynamics of RVF in an urban peridomestic cycle. The basic reproduction numberℛ0 of the model was computed to study the effect of initial transmission of the disease and used to analyse the stability of the disease-free equilibrium and endemic equilibrium. The analysis of the stability of equilibrium points indicated that both the disease-free equilibrium and endemic equilibrium of the model system are locally and globally asymptotically stable. The stability of the disease-free equilibrium means that the RVF outbreak can be controlled provided that ℛ0<1.Numerical simulations were performed to analyse the population variation with respect to the time and orderα=1,0.75,0.5,and0.25. In general, the results from simulations show that whenever there is an increase in susceptible and exposed population in both human and mosquitoes, the value of α decreases. A rapid increase in susceptible mosquitoes is observed just after the first 30 days for order α=1, surpassing all orders. The infected population decreases with a decrease in the value of α. However, there is a rapid decrease in infected human and mosquitoes after the first 30 days for the order α=1, surpassing all orders.


## 6. Conclusion

Rift Valley fever will remain endemic in many countries in the world because of its dynamical behaviour and RVF virus maintenance during the period of no outbreak in the endemic areas. Its ability to emerge and re-emerge after several years makes RVF infection a disease of global concern. The ability of the RVF viruses to adopt transmission cycles involving different mosquito species poses a greater risk in human health and poses need to looking into the social and environmental factors which favour the growth of mosquito vectors in urban areas. The results of the analysis show that the disease is persistent when introduced in the society causing threat to human health. Effective educational campaign about the transmission dynamics of RVF and proper hygiene may help to reduce the infection.



---

*Source: 2941961-2021-05-28.xml*