# Analysis of the Model on the Effect of Seasonal Factors on Malaria Transmission Dynamics

Authors: Victor Yiga; Hasifa Nampala; Julius Tumwiine
Journal: Journal of Applied Mathematics
 (2020)

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## Abstract

Malaria is one of the world’s most prevalent epidemics. Current control and eradication efforts are being frustrated by rapid changes in climatic factors such as temperature and rainfall. This study is aimed at assessing the impact of temperature and rainfall abundance on the intensity of malaria transmission. A human host-mosquito vector deterministic model which incorporates temperature and rainfall dependent parameters is formulated. The model is analysed for steady states and their stability. The basic reproduction number is obtained using the next-generation method. It was established that the mosquito population depends on a threshold valueθ, defined as the number of mosquitoes produced by a female Anopheles mosquito throughout its lifetime, which is governed by temperature and rainfall. The conditions for the stability of the equilibrium points are investigated, and it is shown that there exists a unique endemic equilibrium which is locally and globally asymptotically stable whenever the basic reproduction number exceeds unity. Numerical simulations show that both temperature and rainfall affect the transmission dynamics of malaria; however, temperature has more influence.


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## Body

## 1. Introduction

Malaria is one of the world’s most prevalent epidemic despite a series of control and eradication measures. It is caused by thePlasmodium parasite transmitted between humans through the bite of a female Anopheles mosquito as it seeks blood necessary for ovipositon [1]. The malaria parasites of humans are Plasmodium falciparum, Plasmodium malariae, Plasmodium ovale, and Plasmodium vivax [2]. Plasmodium falciparum and Plasmodium vivax are the most prevalent species in the tropical areas and temperate regions, respectively [3]. The life cycle of a mosquito begins as an egg, it hatches into a larva which turns into a pupa, then after about two to four days of pupation, the mosquito emerges as an adult [4]. On biting a human host, the female Anopheles mosquito injects sporozoites into the blood of the human host. The sporozoite form of the Plasmodium parasite multiplies in the host’s liver before developing into the gametocyte form which is released in the bloodstream and is ingested by a female Anopheles mosquito during a future blood meal [5]. Malaria has a long incubation period so symptoms can occur 7-30 days after the infection. Symptoms of malaria include fever, headache, body aches, chills, and vomiting [6]. Severe malaria can develop when the infection is not treated and may result in organ failure or even death. Examples of severe malaria include cerebral malaria, severe anemia, distress, kidney failure, acidosis, and hypoglycemia [7]. Pregnant women and children aged under 5 years are the most vulnerable groups affected by malaria [8].There are about 450 species of theAnopheles mosquito; however, only about 35-40 transmit malaria. The Anopheles gambiae, Anopheles arabiensis, and Anopheles coluzzii of the Anopheleles gambiae species complex, and Anopheles funestus of the Anopheles funestus species are major mosquito vector species of malaria in sub-Sahara Africa [9–11]. Unlike humans, mosquitoes are ectotherms (they do not regulate their own body temperatures) [12]. Both the Anopheles mosquito vector and Plasmodium malaria parasites have highly temperature-dependent life cycles [13]. The aquatic immature Anopheles habitats are also strongly dependent upon rainfall and local hydrodynamics. Change in climatic factors may establish conditions favourable for the malaria parasite and vector development and reproduction leading to the occurrence of malaria in previously disease-free areas, or change the intensity of malaria transmission due to changes in biting patterns determined by seasonal factors [14]. Malaria prevalence in the African tropics has been attributed to favourable environmental conditions for larval development, and parasite maturation within the infected mosquito [15–17]. Temperature plays a major role in the life cycle of both the Anopheles mosquito vector and the Plasmodium malaria parasites. Numerous studies have shown that mosquito vectors are more active at warmer temperatures [12, 18–20]. Rainfall provides breeding sites for the mosquitoes thus increasing the number of mosquito larval habitants [17, 21]. The impact of temperature and rainfall is therefore significant in the transmission dynamics of malaria.Mathematical models have been developed over the years to gain insight into malaria transmission dynamics and aid its control and eradication. Ross [22] developed a simple susceptible-infective-susceptible (SIS) malaria model which explained the relationship between the number of mosquitoes and the incidence of malaria in humans. It was noted that there is a threshold for the number of mosquitoes below which malaria can be sufficiently eliminated. Macdonald [23] proposed a model in which it was shown that reducing the number of mosquitoes is not a sufficient control strategy, with the assumption that the amount of infective material to which a population is exposed remains unchanged. Mosquito vector longevity was identified as the single most important variable in the force of transmission. Aron [24] and Bailey [25] considered models with acquired immunity to malaria that depends on exposure to malaria infection. Tumwiine et al. [7] considered a host-vector malaria model with delays in the development of immature mosquitoes into adult mosquitoes that transmit malaria, based on susceptible-infective-susceptible (SIS) for humans and susceptible-infective (SI) for mosquito vectors. It was established that the bigger the proportion of young mosquitoes that survives the developmental period, the higher the susceptible vector population and the lower the susceptible human host population. It was suggested that the infected human population can be reduced if the adult mosquito population is controlled. Martens et al. [3], Craig et al. [26] and Bouma et al. [27] showed that environmental and climatic factors play an important role in the geographical distribution and transmission of malaria.The majority of the malaria models ignore the role of aquatic mosquito stages since they are not involved in the spread of malaria. However, the survival of the aquatic mosquitoes increases the adult mosquito population that is responsible for the spread of malaria. It is therefore important to include the aquatic mosquito population in the study of the effect of temperature and rainfall on malaria transmission since they are highly affected by these factors.The dynamic process-based mathematical models play a significant role that can provide strategic insights into the effects of seasonal factors on malaria transmission. Several studies have investigated the impact of seasonality and climate factors on malaria transmission [12, 19, 20, 28–32]. Beck-Johnson et al. [12] used a temperature-dependent, stage-structured delayed differential equation model to investigate how climate determines malaria risk and found out that adult mosquito dynamics is highly affected by temperature sensitivities and juvenile dynamics influences adult age structure. Their model combined with the Detinova curve predicts the peak temperature for potentially infectious mosquitoes at 30°C, whereas when combined with the Paaijman’s curve, it predicts peak temperature at 28°C. Ngarakana-Gwasira et al. [31] assessed the impact of temperature on malaria transmission dynamics. It was shown that the malaria burden increases with the increase in temperature with an optimum temperature window of 30°C-32°C. Mukhtar et al. [30] developed and analysed a human host-mosquito vector disease-based model that included temperature and rainfall. The model was used to investigate the potential impact of climatic conditions on malaria prevalence in two climatically distinct regions of South Sudan. It was found out that malaria is more severe in the tropical region than in the hot semiarid.In this paper, a malaria transmission model with temperature- and rainfall-dependent parameters is studied. The present model differs from the models proposed by Mukhtar et al. [30] and Bhuju et al. [33] in that it assumes that interaction coefficients between humans and mosquitoes are constants and also ignores the exposed class in the mosquito population. In addition, the stability analysis of the steady states is also carried out.This paper is organised as follows: Section2 presents the model formulation. In Section 3, the stability of equilibria, sensitivity, and bifurcation analysis are presented. In Section 4, numerical simulation is performed. The discussion of results is presented in Section 5.


## 2. Model Formulation

A human host-mosquito vector model is formulated to study the transmission dynamics of malaria using a deterministic model. The total human populationNHt is divided into the epidemiological classes: susceptible humans SHt, exposed humans EHt, infectious humans IHt, and recovered humans RHt. Individuals are recruited into the susceptible class through birth and immigration at a constant rate ΛH. It is assumed that there is no recruitment of infective humans and vertical transmission due to malaria. Susceptible humans enter the exposed class with the interaction coefficient βH after being bitten by an infected female Anopheles mosquito. This is because the sporozoites injected by the infected female Anopheles mosquito have not yet developed into gametocytes in the bloodstream of the human and so cannot infect susceptible mosquitoes. Exposed humans progress to the infectious class at the rate ρ. Infected humans either cure at the rate ν to join the recovered class or die due to malaria at a rate δ. Recovered humans lose their immunity at a rate σ. Humans in all compartments die due to natural causes at the rate μH.The total mosquito populationMTt is divided into the aquatic mosquito population MAt and the adult mosquito population NVt. The aquatic mosquito population consists of the eggs, larvae, and pupae stages. The total adult female Anopheles mosquito population is divided into susceptible SVt and infective IVt mosquitoes. There is no recovered class for mosquitoes because they do not cure from malaria throughout their lifetime. It is assumed that mosquitoes do not die from malaria due to their short lifespan. Adult female Anopheles mosquitoes lay eggs at a temperature-dependent rate LT, and the aquatic mosquito population increase is constrained by the carrying capacity of the environment K [33]. Aquatic mosquitoes mature and develop into adult mosquitoes at a temperature- and rainfall-dependent rate λT,R. Susceptible mosquitoes become infected with interaction coefficient βV through biting infected humans. It is assumed that aquatic mosquitoes die at a temperature-dependent death rate μAT while adult mosquitoes die at a temperature-dependent natural death rate μVT. It is also assumed that all variables presented in each compartment are differentiable with respect to time and all parameters are nonnegative except that δ≥0.


### 2.1. Model Equations

The human and mosquito populations are governed by the following system of ordinary differential equations.(1)dSHdt=ΛH−βHSHIV−μHSH+σRH,(2)dEHdt=βHSHIV−ρ+μHEH,(3)dIHdt=ρEH−μH+ν+δIH,(4)dRHdt=νIH−σ+μHRH,(5)dMAdt=LT1−MAKSV+IV−λT,R+μATMA,(6)dSVdt=λT,RMA−βVSVIH−μVTSV,(7)dIVdt=βVSVIH−μVTIV,together with
(8)NHt=SHt+EHt+IHt+RHt,NVt=SVt+IVt,and
(9)MTt=NVt+MAt.It can be shown that the total human population is bounded byΛH/μH and the total adult mosquito population is bounded by (λT,RMA)/μVT. Therefore, the solution set of the system (1)–(7) is bounded in D=SH,EH,IH,RH∈ℝ+4:0≤NH≤ΛH/μH,MA,SV,IV∈ℝ+3:0≤NV≤λT,RMA/μVT.Theorem 1.
For system (1)–(7) if SH0>0,EH0>0,IH0>0,RH0>0,MA0>0,SV0>0,IV0>0, then SHt>0,EHt>0,IHt>0,RHt>0,MAt>0,SVt>0 and IVt>0 for all t>0.Proof.
Define a setH=t>0:SHt>0,EHt>0,IHt>0,RHt>0,MAt>0,SVt>0,IVt>0.It is assumed by contradiction that if the setH defined above is bounded, then H has a supremum τ. Now, define τ as
(10)τ=supt>0:SHt>0,EHt>0,IHt>0,RHt>0,MAt>0,SVt>0,IVt>0,0≤t≤τ.SinceSHt,EHt,IHt,RHt,MAt,SVt and IVt are continuous, then τ>0. If τ<∞, then it is necessary that SHτ=0 or EHτ=0 or IHτ=0 or RHτ=0 or MAτ=0 or SVτ=0 or IVτ=0.From equation (1),
(11)dSHdt=ΛH+σRH−βHIV+μHSH.LetPt=expμHt+∫0tβHIVsds and note that P0=1 and Pt>0 for all t>0.Consider(12)∫0τddtSHtPtdt=∫0τPtΛH+σRHdt,SHτPτ−SH0P0=∫0τΛH+σRHtPtdt,SHτ=Pτ−1SH0+∫0τΛH+σRHtPtdt.Therefore,SHτ>0 since all parameters are positive. Applying the above reasoning to the remaining equations of system (1)–(7) shows that EHτ>0,IHτ>0,RHτ>0,MAτ>0,SVτ>0,IVτ>0; thus, τ=∞. This contradicts τ being a supremum of H; thus, H is not bounded.This confirms the positivity of solutions for allt>0. The model is epidemiological and mathematically well posed.


## 2.1. Model Equations

The human and mosquito populations are governed by the following system of ordinary differential equations.(1)dSHdt=ΛH−βHSHIV−μHSH+σRH,(2)dEHdt=βHSHIV−ρ+μHEH,(3)dIHdt=ρEH−μH+ν+δIH,(4)dRHdt=νIH−σ+μHRH,(5)dMAdt=LT1−MAKSV+IV−λT,R+μATMA,(6)dSVdt=λT,RMA−βVSVIH−μVTSV,(7)dIVdt=βVSVIH−μVTIV,together with
(8)NHt=SHt+EHt+IHt+RHt,NVt=SVt+IVt,and
(9)MTt=NVt+MAt.It can be shown that the total human population is bounded byΛH/μH and the total adult mosquito population is bounded by (λT,RMA)/μVT. Therefore, the solution set of the system (1)–(7) is bounded in D=SH,EH,IH,RH∈ℝ+4:0≤NH≤ΛH/μH,MA,SV,IV∈ℝ+3:0≤NV≤λT,RMA/μVT.Theorem 1.
For system (1)–(7) if SH0>0,EH0>0,IH0>0,RH0>0,MA0>0,SV0>0,IV0>0, then SHt>0,EHt>0,IHt>0,RHt>0,MAt>0,SVt>0 and IVt>0 for all t>0.Proof.
Define a setH=t>0:SHt>0,EHt>0,IHt>0,RHt>0,MAt>0,SVt>0,IVt>0.It is assumed by contradiction that if the setH defined above is bounded, then H has a supremum τ. Now, define τ as
(10)τ=supt>0:SHt>0,EHt>0,IHt>0,RHt>0,MAt>0,SVt>0,IVt>0,0≤t≤τ.SinceSHt,EHt,IHt,RHt,MAt,SVt and IVt are continuous, then τ>0. If τ<∞, then it is necessary that SHτ=0 or EHτ=0 or IHτ=0 or RHτ=0 or MAτ=0 or SVτ=0 or IVτ=0.From equation (1),
(11)dSHdt=ΛH+σRH−βHIV+μHSH.LetPt=expμHt+∫0tβHIVsds and note that P0=1 and Pt>0 for all t>0.Consider(12)∫0τddtSHtPtdt=∫0τPtΛH+σRHdt,SHτPτ−SH0P0=∫0τΛH+σRHtPtdt,SHτ=Pτ−1SH0+∫0τΛH+σRHtPtdt.Therefore,SHτ>0 since all parameters are positive. Applying the above reasoning to the remaining equations of system (1)–(7) shows that EHτ>0,IHτ>0,RHτ>0,MAτ>0,SVτ>0,IVτ>0; thus, τ=∞. This contradicts τ being a supremum of H; thus, H is not bounded.This confirms the positivity of solutions for allt>0. The model is epidemiological and mathematically well posed.


## 3. Model Analysis

In this section, the equilibrium points of the system (1)–(7) are obtained and analysed for their stability. It is established that system (1)–(7) has two disease-free equilibrium points and one endemic equilibrium point.Theorem 2.
System (1)–(7) has two disease-free equilibrium points
(13)E01SH,EH,IH,RH,MA,SV,IV=ΛHμH,0,0,0,0,0,0,E02SH,EH,IH,RH,MA,SV,IV=ΛHμH,0,0,0,KLTλT,R−μVTλT,R+μATLTλT,R,KLTλT,R−μVTλT,R+μATLTμVT,0,whose existence depends on parameter θ, where
(14)θ=LTλT,RμVTλT,R+μAT.Proof.
Setting the right-hand side (RHS) of system (1)–(7) equal to zero gives either MA=0 or MA=KLTλT,R−μVTλT,R+μAT/LTλT,R.ForMA=0, it implies that SV=IV=EH=IH=RH=0 and SH=ΛH/μH. Therefore, there exists a disease-free equilibrium point E01SH,EH,IH,RH,MA,SV,IV=ΛH/μH,0,0,0,0,0,0. For MA=KLTλT,R−μVTλT,R+μAT/LTλT,R with IH=0, it implies that IV=RH=EH=0,SH=ΛH/μH and SV=KLTλT,R−μVTλT,R+μAT/LTμVT. Therefore, there exists a disease-free equilibrium point
(15)E02SH,EH,IH,RH,MA,SV,IV=ΛHμH,0,0,0,KLTλT,R−μVTλT,R+μATLTλT,R,KLTλT,R−μVTλT,R+μATLTμVT,0.E02 is positive and exists only if LTλT,R>μVTλT,R+μAT, which gives the condition of existence of E02 as θ>1, where θ=LTλT,R/μVTλT,R+μAT.


### 3.1. Basic Reproduction Number

According to Diekmann et al. [34], the basic reproduction number is the expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population.The basic reproduction number is obtained using the next-generation matrix method as described by Diekmann et al. [34].The basic reproduction number is the spectral radius of the next-generation matrixFV−1, where F=∂Fj and V=∂Vj are computed at the disease-free equilibrium point of the system.
(16)R0=σFV−1.Consider the infected subsystem of system (1)–(7) below.
(17)dEHdt=βHSHIV−ρEH−μHEH,dIHdt=ρEH−μH+ν+δIH,dIVdt=βVSVIH−μVTIV.The vector of new infectionsF and the vector formed by other transfers V are given by
(18)F=βHSHIV0βVSVIH,and
(19)V=μH+ρEH−ρEH+μH+ν+δIHμVTIV.For the disease-free equilibrium pointE01, the matrix FV−1 has only one eigenvalue equal to zero; thus, the basic reproduction number is zero for this case. This implies that if an infective individual is introduced into the population at the steady-state E01, the disease will not spread due to the absence of the parasite-transmitting mosquito vectors.For the disease-free equilibrium pointE02, the matrices F and V are computed as follows:
(20)F=00βHΛHμH0000KβVLTλT,R−μVTλT,R+μATLTμVT0,and
(21)V=μH+ρ00−ρμH+ν+δ000μVT,V−1=1μH+ρ00ρρ+μHμH+ν+δ1μH+ν+δ0001μVT.Thus, the next-generation matrix is given by(22)FV−1=00βHΛHμHμVT000ρKβVLTλT,R−μVTλT,R+μATLTμVTμH+ν+δμH+ρKβVLTλT,R−μVTλT,R+μATLTμVTμH+ν+δ0.The eigenvalues ofFV−1 are
(23)0,−1μVTρβHβVΛHKLTλT,R−μVTλT,R+μATLTμHμH+ν+δμH+ρ,1μVTρβHβVΛHKLTλT,R−μVTλT,R+μATLTμHμH+ν+δμH+ρ.Therefore, the basic reproduction numberR0 is given by
(24)R0=1μVTρβHβVΛHKLTλT,R−μVTλT,R+μATLTμHμH+ν+δρ+μH.R0 exists only if LTλT,R>μVTλT,R+μAT.ExpressingR0 in terms of θ gives
(25)R0=ρβHβVΛHKλT,R+μATθ−1LTμV2TμHμH+ν+δμH+ρ.R0 exists only if the term under the square root is nonnegative, that is, θ≥1; otherwise, if θ<1, there is no growth of the mosquito population, and malaria will not develop in the community since mosquito vectors are important for the spread of malaria.


### 3.2. Sensitivity Analysis

Malaria control and eradication strategies should target important parameters which have a high impact on the basic reproduction number. A sensitivity analysis ofR0 to the various parameters is thus presented in this section. The basic reproduction number is explicitly determined by the parameters ρ,βH,βV,ΛH,LT,λT,R,μVT,μH,μAT,ν and δ. The sensitivity indices of R0 to these parameters are computed using the approach in Chitnis et al. [35].Definition 3.
The sensitivity index of a variableu that depends continuously on a parameter p is defined as
(26)⋎pu=∂u∂p.pu,
whereu is a differentiable function of p.Thus, by the definition above, the formula used to derive an expression for the sensitivity ofR0 to a parameter p is given by
(27)⋎pR0=∂R0∂p.pR0.Table1 shows the sensitivity indices of R0 to the parameters (independent of temperature and rainfall variations) determining its value. Parameter values ΛH=0.031, μH=1/23178, βH=βV=0.00021, ρ=1/20, ν=1/30, δ=0.00000638, and K=1000000 are used.Table 1
Numerical values of sensitivity indices ofR0.
Parameter symbolSensitivity indexμH-0.5011βH0.5βV0.5ΛH0.5ν-0.4993ρ0.0004δ-0.0001The MATLAB computer software program is used in the simulation of the sensitivity ofR0 with respect to temperature and rainfall since the temperature-rainfall-dependent parameters μVT, LT, λT,R, and μAT are given by nonlinear functions. The results are shown in Figure 1.Figure 1
(a) Sensitivity indices of the basic reproduction number with respect to temperature. (b) Sensitivity indices of the basic reproduction number with respect to rainfall.
(a)(b)


#### 3.2.1. Interpretation of Sensitivity Analysis

The natural death rateμH and the disease-induced death rate δ of the human population are the most and least sensitive parameters, respectively. A positive value in the sensitivity index shows that if the parameter is increased when all other parameters are kept constant, the value of R0 increases, while for a negative sensitivity index, when the parameter value is increased with all other parameters kept constant, the value of R0 decreases. In Figure 1(a), rainfall is fixed at 10 mm. It is observed that R0 is most sensitive to temperatures within the ranges 17°C-20°C and 37°C-40°C. This is in agreement with Githeko et al. [16] in which the lower-end range and the upper-end range of disease transmission are established at 14°C-18°C and 35°C-40°C, respectively. The sensitivity indices for temperatures between 17°C and 25°C are positive, whereas those for temperatures between 25°C and 35°C are negative. In Figure 1(b), the temperature is fixed at 25°C. It is observed that the sensitivity to rainfall reduces with more rainfall received. Indices for daily rainfall below 25 mm are positive, whereas indices for rainfall above 25 mm are negative.


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

The Jacobian matrix of the system (1)–(7) evaluated at the disease-free equilibrium point E01 is shown below.
(28)JE01=−μH00σ00−βHΛHμH0−ρ+μH0000βHΛHμH0ρ−μH+ν+δ000000ν−μH+σ0000000−λT,R+μATLTLT0000λT,R−μVT0000000−μVT.The eigenvalues ofJE01 are −μH, −μVT, −μH+σ, −μH+ν+σ, −ρ+μH and the zero points of the polynomial.Z2+μVT+μAT+λT,RZ+μVTλT,R+μAT−λT,RLT=0, where Z is the eigenvalue.The zero points of a polynomial of order two have negative real parts if and only if its coefficients and constant terms are positive. Thus, the disease-free equilibriumE01 is stable only if
(29)λT,RLTμVTλT,R+μAT<1.Theorem 4.
The disease-free equilibrium pointE02 of system (1)–(7) is locally asymptotically stable if R0<1 and unstable if R0>1.Proof.
It has already been shown that the disease-free equilibriumE02 only exists if LTλT,R/μVTλT,R+μAT>1. If this condition is satisfied, then R0 is real and positive. Thus, the basic reproduction number R0 is biologically consistent. Using the theorem by van den Driessche and Watmough [36], the disease-free equilibrium E02 is locally asymptotically stable if R0<1 and unstable if R0>1.


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

The global stability of the disease-free equilibrium is investigated using a theorem by Castillo-Chavez et al. [37]. System (1)–(7) can be expressed in terms of
(30)dXdt=FX,ZdZdt=GX,Z,GX,0=0,where X∈ℝm denotes the number of uninfected individuals and Z∈ℝn denotes the number of infected individuals.The following conditions (H1) and (H2) must be satisfied providedR0<1 to guarantee global asymptotic stability.(H1) FordX/dt=FX,0,X∗is globally asymptotically stable(H2)GX,Z=MZ−G^X,Z,G^X,Z≥0forX,Z∈Ω, where M=DZGX∗,0 is an M-matrix and Ω is the region where the model makes biological sense. From system (1)–(7)
(31)FX,Z=ΛH−βHSHIV−μHSH+σRHνIH−σ+μHRHLT1−MAKSV+IV−λT,R+μATMAλT,RMA−βVSVIH−μVTSV,and
(32)GX,Z=βHSHIV−ρ+μHEHρEH−μH+ν+δIHβVSVIH−μVTIV.To investigate condition (H1),(33)FX,0=ΛH−μHSH0LT1−MAKSV−λT,R+μATMAλT,RMA−μVTSV.It has already been established that the threshold values for the human and mosquito populations areΛH/μH and λT,RMA/μVT, respectively; thus, there is convergence in Ω. Therefore, X∗ is globally asymptotically stable.To investigate condition (H2) for the disease-free equilibrium pointE01(34)M=−ρ+μH0a2ρ−μH+ν+δ000−μVT,where a2=βHΛH/μH(35)G^X,Z=βHIVΛHμH−SH0−βVSVIH.SinceG^3X,Z<0, condition (H2) is violated. Therefore, the disease-free equilibrium point E01 may not be globally asymptotically stable.For the disease-free equilibrium pointE02(36)M=−ρ+μH0a2ρ−μH+ν+δ00a1−μVT,where a1=βVKLTλT,R−μVTλT,R+μAT/LTμVT and a2=βHΛH/μH(37)G^X,Z=βHIVΛHμH−SH0βVIHKLTλT,R−μVTλT,R+μATLTμVT−SV.It has already been established that the threshold value of the human population isΛH/μH. Therefore, SH≤ΛH/μH. Similarly, SV≤KLTλT,R−μVTλT,R+μAT/LTμVT. This shows that G^T,R≥0; thus, equilibrium point E02 is globally asymptotically stable.


### 3.5. Bifurcation Analysis

In this subsection, the centre manifold theorem by Castillo-Chavez and Song [38] is used to investigate the bifurcation behaviour of system (1)–(7) when the basic reproduction number R0=1.Theorem 5.
Consider a general system of ODEs with a parameterϕ(38)dxdt=fx,ϕ,f:ℝn×ℝ⟶ℝn,f∈ℂ2ℝn×ℝ.
Without loss of generality, it is assumed that 0 is an equilibrium for system (38) for all values of the parameter ϕ, that is
(39)f0,ϕ≡0forallϕ.AssumeA1.
A=Dxf0,0=∂fi/∂xj is the linearisation matrix of system (38) around the equilibrium 0 with ϕ evaluated at 0. Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts.A2.
MatrixA has a nonnegative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue.Letfk be the kth component of f and
(40)a=∑k,i,j=1nvkwiwj∂2fk∂xi∂xj0,0,b=∑k,i=1nvkwi∂2fk∂xi∂ϕ0,0.The local dynamics of (38) around 0 are totally determined by a and b.
(i)
a>0, b>0, when ϕ<0 with ∣ϕ∣<<1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0<ϕ<<1, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium.(ii)
a<0, b<0, when ϕ<0 with ∣ϕ∣<<1, 0 is unstable; when 0<ϕ<<1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium.(iii)
a>0, b<0, when ϕ<0 with ∣ϕ∣<<1, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0<ϕ<<1, 0 is stable, and a positive unstable equilibrium.(iv)
a<0, b>0, when ϕ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable.A change in notations of the variables is used such thatSH=x1, EH=x2, IH=x3, RH=x4, MA=x5, SV=x6, and IV=x7. For R0=1, let βH be a bifurcation parameter with bifurcation value βH∗,
(41)βH∗=LTμHμVT2μH+ν+δρ+μHρβVΛHKLTλT,R−μVTλT,R+μAT.System (1)–(7) becomes
(42)f1x1,x2,x3,x4,x5,x6,x7=ΛH−βHx1x7−μHx1+σx4,(43)f2x1,x2,x3,x4,x5,x6,x7=βHx1x7−ρ+μHx2,(44)f3x1,x2,x3,x4,x5,x6,x7=ρx2−μH+ν+δx3,(45)f4x1,x2,x3,x4,x5,x6,x7=νx3−σ+μHx4,(46)f5x1,x2,x3,x4,x5,x6,x7=LT1−x5Kx6+x7−λT,R+μATx5,(47)f6x1,x2,x3,x4,x5,x6,x7=λT,Rx5−βVx6x3−μVTx6,(48)f7x1,x2,x3,x4,x5,x6,x7=βVx6x3−μVTx7.E02x1,x2,x3,x4,x5,x6,x7=ΛH/μH,0,0,0,Kd/LTλT,R,Kd/LTμVT,0is a disease free equilibrium point for system (42)–(48), where d=LTλT,R−μVTλT,R+μVT.The Jacobian matrix of system (42)–(48) evaluated at E02 is given by
(49)JE02=−J100σ00−βHΛHμH0−J20000βHΛHμH0ρ−J3000000ν−J40000000−J5μVTλT,R+μATλT,RμVTλT,R+μATλT,R00−βVKdLTμVT0λT,R−J6000βVKdLTμVT000−J7whereJ1=μH, J2=ρ+μH, J3=μH+ν+δ, J4=μH+σ, J5=LTλT,R/μVT, and J6=J7=μVT.The right eigenvectorw=w1,w2,w3,w4,w5,w6,w7T, corresponding to the zero eigenvalue, is computed using JE02w=0 which yields
(50)−J1w1+σw4−βHΛHμHw7=0,−J2w2+βHΛHμHw7=0,ρw2−J3w3=0,νw3−J4w4=0,−J5w5+μVTλT,R+μATλT,Rw6+μVTλT,R+μATλT,Rw7=0,−βVKdLTμVTw3+λT,Rw5−J6w6=0,βVKdLTμVTw3−J7w7=0.Settingw7=1 gives,
(51)w3=LTμvT2βVKd,w4=LTνμVT2βVKdμH+σ,w2=LTμVT2μH+ν+δρβVKd,w1=1μHLTνσμVT2βVKdμH+σ−βHΛHμH,and
(52)−LTλT,R2μVTw5+μVTλT,R+μATw6=−μVTλT,R+μAT,(53)λT,Rw5−μVTw6=μVT.Solving equations (52) and (53) simultaneously gives
(54)w5=0,w6=−1.The left eigenvectorv=v1,v2,v3,v4,v5,v6,v7, corresponding to the zero eigenvalue, is computed using vJE02=0 which yields
(55)−J1v1=0,−J2v2+ρv3=0,−J3v3+νv4−βVKdLTμVTv6+βVKdLTμVTv7=0,σv1−J4v4=0,μVTλT,R+μATλT,Rv5−J6v6=0,βHΛHμHv1+βHΛHμHv2+μVTλT,R+μATλT,Rv5−J7v7=0.It follows thatv1=v4=0, setting v3=v6=1 gives
(56)v2=ρρ+μH,v5=λT,RλT,R+μAT,v7=LTμVTμH+ν+δ+βvKdβVKd.Sincev1=v4=0, the values of a and b are obtained from
(57)a=∑i,j=17v2wiwj∂2f2∂xi∂xj+∑i,j=17v3wiwj∂2f3∂xi∂xj+∑i,j=17v5wiwj∂2f5∂xi∂xj+∑i,j=17v6wiwj∂2f6∂xi∂xj+∑i,j=17v7wiwj∂2f7∂xi∂xj,b=∑k,i=17vkwi∂2fk∂xi∂βH∗,where the partial derivatives computed at E02 are
(58)∂2f2∂x1∂x7=βH,∂2f5∂x5∂x6=−LTK,∂2f5∂x5∂x7=−LTK,∂2f6∂x3∂x6=−βV,∂2f7∂x3∂x6=βV,∂2f2∂x7∂βH∗=ΛHμH.It follows that(59)a=2ρβHμHρ+μHLTνσμVT2βVKdμH+σ−βHΛHμH+2LTμVT2Kd−2LTμVT2βVKd+LTμVTμH+ν+δβVK2d2,b=ρβHΛHμHρ+μH>0.Sinceb>0, the model undergoes a backward bifurcation if a>0, that is
(60)2ρβHLTνσμVT2μHβVKdμH+ρμH+σ+2LTμVT2Kd>2ρβH2ΛHμH2μH+ρ+2LTμVT2αβVK2d2,where α=βVKd+LTμVTμH+ν+δ.Remark 6.
Existence of a backward bifurcation whena>0 means that there is a possibility of coexistence of an endemic equilibrium and the disease-free equilibrium when R0<1. In this case, the strategy of reducing the basic reproduction number to a value less than unity would not be sufficient for the eradication of malaria.


### 3.6. Existence and Stability of the Endemic Equilibrium

Theorem 7.
System (1)–(7) has a unique endemic equilibrium when R0>1.Proof.
Setting the right-hand side of the equations in system (1)–(7) to zero shows that either IH=0 or IH=KρβHβVΛHσ+μHd−LTμHμVT2c/KdβHβVc−ρσν+LTβVμHμVTc, where d=LTλT,R−μVTλT,R+μAT and c=μH+ν+δσ+μHρ+μH. IH=0 corresponds to the disease-free equilibrium.ExpressingIH above in terms of R0 using d=R02μV2μHLTc/KρβHβVΛHσ+μH gives
(61)IH=ρΛHμVTσ+μHR02−1R02μVTc−ρσν+βVρΛHσ+μH.Thus, there exists a unique endemic equilibrium pointE1=SH∗,EH∗,IH∗,RH∗,MA∗,SV∗,IV∗ only if R02>1, that is, R0>1 since c>ρσν. The endemic equilibrium is given by
(62)SH∗=LTμVTΛHσ+μH+σνIH∗βVIH∗+μVTσ+μHKβHd+LTμHμVTβVIH∗+LTμHμVT2,EH∗=KβHβVdIH∗ΛHσ+μH+σνIH∗ρ+μHσ+μHKβHd+LTμHμVTβVIH∗+LT,RμHμVT2,IH∗=ρΛHμVTσ+μHR02−1R02μVTc−ρσν+βVΛHρσ+μH,RH∗=νIH∗σ+μH,MA∗=KdLTλT,R,SV∗=KdLTβVIH∗+μVT,IV∗=KβVdIH∗LTμVTβVIH∗+μVT.The Jacobian matrix of system of equations (1)–(7) evaluated at the endemic equilibrium point E1 is given by
(63)JE1=−e100σ00−e2e3−e40000e20ρ−e5000000ν−e60000000−e7e8e800−e90e10−e11000e900e12−e13,where e1=βHIH∗+μH, e2=βHSH∗, e3=βHIV∗, e4=ρ+μH, e5=μH+ν+δ, e6=σ+μH, e7=LTSV∗+IV∗/K−λT,R+μAT, e8=LT1−MA∗/K, e9=βVSV∗, e10=λT,R, e11=βVIH∗+μV, e12=βVIH∗, and e13=μVT.The characteristic polynomial of the Jacobian matrixJE1 is obtained as
(64)Z7+a1Z6+a2Z5+a3Z4+a4Z3+a5Z2+a6Z+a7=0,where
(65)a1=e1+e4+e5+e6+e7+e11+e13,a2=e1e4+e5+e6+e7+e11+e13+e4e5+e6+e7+e11+e5e6+e7+e11+e13+e6e7+e11+e13+e7e11+e13+e13e4+e7−e8e10,a3=e1e4e5+e6+e7+e11+e13+e1e5e7+e11+e13+e1e6e7+e11+e13+e4e5e6+e7+e11+e13+e4e6e7+e11+e13+e5e6e7+e11+e7e11e1+e4+e5+e6+e7−e8e10e1+e4+e5+e12+e11e13e1+e4+e5+e6+e7−e8e13e4+e5+e6−e2e9ρ+e1e7e13−e8e10e13,a4=e2e9ρe3+e6−e7−e11+e12−e1+e1e6e13e4+e5+e7+e11+e1e4e13e5+e7+e1e7e5e13+e4e11+e11e13+e4e6+e4e5e6e13+e7e13+e11e13+e1e7+e1e11+e6e11−e8e10+e6e7+e4e7e6e13+e11e13+e1e6+e6e11+e5e13e6e7+e6e11+e8e10+e7e11+e1e11−e1e10e8e12+e8e13+e6e8+e5e8−e6e11e1e7+e5e7−e4e8+e1e4+e1e5+e8e10e4e12−e4e13−e5e12−e6e12−e4e13−e5e6−e1e4+e6e11e13e4+e7−e3νρσ+e1e5e7e11,a5=e1e4e5e6e7+e6e11+e6e13+e7e11−e8e10+e7e13+e11e13+e1e5e6e7e11−e8e10+e7e13+e11e13+e4e6e8e10e13−e5e10+e2e9ρe3e6−e1e6+e1e7−e1e11+e3e11−e6e11−e7e11−e8e10+e1e12−e3e12+e6e12+e7e12+e4e5e6e7e11+e7e13−e8e10e1e4e12+e1e5e12+e1e5e13+e1e6e13+e1e6e12+e4e5e12+e1e4e13+e5e6e13+e7e11e1e4e13+e4e6e13+e11e13e4e5e6+e4e5e7+e1e6e7+e1e5e7−e3νρσe7+e11+e13,a6=e1e4e5e6e7e11−e6e8e10+e6e7e13+e6e11e13−e8e10e12−e8e10e13+e7e11e13−e2e9ρe3e6e7−e1e6e11−e1e7e11+e1e8e10+e3e6e11+e3e7e11−e3e8e10−e6e7e11+e6e8e10+e1e6e12+e1e7e12−e3e6e12−e3e7e12+e1e4e6e7e11e13−e8e10e12−e8e10e13−e1e5e6e8e10e12−e7e11e13+e8e10e13−e4e5e6e8e10e12−e7e11e13−e4e5e7e8e10e13−e1e4e6e7e9ρ−νρσe3e7e11−e3e8e10+e3e7e13+e3e11e13,a7=e1e2e6e9ρe8e10−e7e11+e2e3e6e9e12ρe7e11−e8e10+e1e4e5e6e7e11e13−e8e10e12+e2e6e7e9e12ρe1−e3+e3ρσe8e10e12ν−e7e11e13ν+e8e10e13−e1e4e5e6e8e10e13.By the Routh-Hurwitz criteria, the endemic equilibrium pointE1 is locally stable provided
(66)a1>0,a2>0,a3>0,a4>0,a5>0,a6>0,a7>0,a1a2>a3,a1a2a3−a12a4−a32>0,2a1a4a5−a12a42−a1a22a5+a2a3a5−a32a4−a52>0,2a1a2a5a6+a12a3a4a6−a13a63−a12a42a5−a1a22a52−a1a2a32a6+a1a2a3a4a5−3a1a3a5a6+2a1a4a52+a2a3a52+a3a6−a3a4a5−a53>0,and
(67)a12a3a4a62−a13a63−3a12a2a4a6a7+2a12a2a5a62+a12a43a7−a12a42a5a6+3a12a62a7−a1a23a72+2a1a22a3a6a7+a1a22a4a5a7−a1a22a52a6−a1a2a32a62−a1a2a3a42a7+a1a2a3a4a5a6+3a1a2a4a72−a1a2a5a6a7+a1a3a4a6a7−3a1a3a5a62−2a1a42a5a7+2a1a4a52a6−3a1a6a72+a22a32a6a7−a2a3a4a5a7+a2a3a52a6−a2a5a72+a33a62+a32a42a7−a32a4a5a6−2a3a4a72+3a3a5a6a7+a4a52a7−a53a6+a73>0.The stability of the endemic equilibriumE1 is demonstrated in the numerical simulations (see Section 4).Theorem 8.
The endemic equilibrium pointE1 for system (1)–(7) is globally asymptotically stable in D if R0>1.Proof.
Consider a Lyapunov function of the form(68)Lt=SH−SH∗−SH∗lnSHSH∗+EH−EH∗−EH∗lnEHEH∗+IH−IH∗−IH∗lnIHIH∗+RH−RH∗−RH∗lnRHRH∗+MA−MA∗−MA∗lnMAMA∗+SV−SV∗−SV∗lnSVSV∗+IV−IV∗−IV∗lnIVIV∗.DifferentiatingLt with respect to time gives
(69)dLdt=SH−SH∗SHdSHdt+EH−EH∗EHdEHdt+IH−IH∗IHdIHdt+RH−RH∗RHdRHdt+MA−MA∗MAdMAdt+SV−SV∗SVdSVdt+IV−IV∗IVdIVdt,dLdt=SH−SH∗SHΛH−βHSHIV−μHSH+σRH+EH−EH∗EHβHSHIV−ρ+μHEH+IH−IH∗IHρEH−μH+ν+δIH+RH−RH∗RHνIH−σ+μHRH+MA−MA∗MALTSV+LTIV−LTSVMAK−LTIVMAK−λT,R+μATMA+SV−SV∗SVλT,RMA−βVIHSV−μVTSV+IV−IV∗IVβVSVIH−μVTIV,dLdt=SH−SH∗SHΛH−βHSH−SH∗IV−IV∗−μHSH−SH∗+σRH−RH∗+EH−EH∗EHβHSH−SH∗IV−IV∗−ρ+μHEH−EH∗+IH−IH∗IHρEH−EH∗−μH+ν+δIH−IH∗+RH−RH∗RHνIH−IH∗−σ+μHRH−RH∗+MA−MA∗MALTSV−SV∗+LTIV−IV∗−LTSV−SV∗MA−MA∗K−LTIV−IV∗MA−MA∗K−λT,R+μATMA−MA∗+SV−SV∗SVλT,RMA−MA∗−βVIH−IH∗SV−SV∗−μVTSV−SV∗+IV−IV∗IVβVSV−SV∗IH−IH∗−μVTIV.dLdt=ΛHSH−SH∗SH−βHIVSH−SH∗2SH+βHIV∗SH−SH∗2SH−μHSH−SH∗2SH+σRHSH−SH∗SH−σRH∗SH−SH∗SH+βHSHIVEH−EH∗EH−βHSHIV∗EH−EH∗EH−βHSH∗IVEH−EH∗EH+βHSH∗IV∗EH−EH∗EH−ρ+μHEH−EH∗2EH+ρEHIH−IH∗IH−ρEH∗IH−IH∗IH−μH+ν+δIH−IH∗2IH+νIHRH−RH∗RH−νIH∗RH−RH∗RH−σ+μHRH−RH∗2RH+LTSVMA−MA∗MA−LTSV∗MA−MA∗MA+LTIVMA−MA∗MA−LTIV∗MA−MA∗MA−LTSVMA−MA∗2KMA+LTSV∗MA−MA∗2KMA−LTIVMA−MA∗2KMA+LTIV∗MA−MA∗2KMA−λT,R+μATMA−MA∗2MA+λT,RMASV−SV∗SV−λT,RMA∗SV−SV∗SV−βVIHSV−SV∗2SV+βVIH∗SV−SV∗2SV−μVTSV−SV∗2SV+βVSVIHIV−IV∗IV−βVSVIH∗IV−IV∗IV−βVSV∗IHIV−IV∗IV+βVSV∗IH∗IV−IV∗IV−μVTIV−IV∗2IV.Collecting positive terms together and negative parts together gives(70)dLdt=A−B,where
(71)A=ΛHSH−SH∗SH+βHIV∗SH−SH∗2SH+σRHSH−SH∗SH+βHIVSHEH−EH∗EH+βHIV∗SH∗EH−EH∗EH+ρEHIH−IH∗IH+νIHRH−RH∗RH+LTSVMA−MA∗MA+LTIVMA−MA∗MA+LTSV∗MA−MA∗2KMA+LTIV∗MA−MA∗2KMA+λT,RMASV−SV∗SV+βVIH∗SV−SV∗2SV+βVSVIHIV−IV∗IV+βVSV∗IH∗IV−IV∗IV,and
(72)B=βHIVSH−SH∗2SH+μHSH−SH∗2SH+σRH∗SH−SH∗SH+βHSHIV∗EH−EH∗EHβHSH∗IVEH−EH∗EH+ρ+μHEH−EH∗2EH+ρEH∗IH−IH∗IH+μH+ν+δIH−IH∗2IHνIH∗RH−RH∗RH+σ+μHRH−RH∗2RH+LTSV∗MA−MA∗MA+LTIV∗MA−MA∗MALTSVMA−MA∗2KMA+LTIVMA−MA∗2KMA+λT,R+μATMA−MA∗2MA+λT,RMA∗SV−SV∗SV+βVIHSV−SV∗2SV+μVTSV−SV∗SV+βVSVIH∗IV−IV∗IV+βVSV∗IHIV−IV∗IV+μVTIV−IV∗2IV.Hence, ifA<B, then dL/dt≤0. Note that dL/dt=0 if and only if SH=SH∗, EH=EH∗, IH=IH∗, RH=RH∗, MA=MA∗, SV=SV∗, and IV=IV∗. Thus, the largest compact invariant set in SH∗,EH∗,IH∗,RH∗,MA∗,SV∗,IV∗∈D:dL/dt=0 is the singleton set E1. By Lasalle’s invariant principles [39], it implies that E1 is globally asymptotically stable in D if A<B.


## 3.1. Basic Reproduction Number

According to Diekmann et al. [34], the basic reproduction number is the expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population.The basic reproduction number is obtained using the next-generation matrix method as described by Diekmann et al. [34].The basic reproduction number is the spectral radius of the next-generation matrixFV−1, where F=∂Fj and V=∂Vj are computed at the disease-free equilibrium point of the system.
(16)R0=σFV−1.Consider the infected subsystem of system (1)–(7) below.
(17)dEHdt=βHSHIV−ρEH−μHEH,dIHdt=ρEH−μH+ν+δIH,dIVdt=βVSVIH−μVTIV.The vector of new infectionsF and the vector formed by other transfers V are given by
(18)F=βHSHIV0βVSVIH,and
(19)V=μH+ρEH−ρEH+μH+ν+δIHμVTIV.For the disease-free equilibrium pointE01, the matrix FV−1 has only one eigenvalue equal to zero; thus, the basic reproduction number is zero for this case. This implies that if an infective individual is introduced into the population at the steady-state E01, the disease will not spread due to the absence of the parasite-transmitting mosquito vectors.For the disease-free equilibrium pointE02, the matrices F and V are computed as follows:
(20)F=00βHΛHμH0000KβVLTλT,R−μVTλT,R+μATLTμVT0,and
(21)V=μH+ρ00−ρμH+ν+δ000μVT,V−1=1μH+ρ00ρρ+μHμH+ν+δ1μH+ν+δ0001μVT.Thus, the next-generation matrix is given by(22)FV−1=00βHΛHμHμVT000ρKβVLTλT,R−μVTλT,R+μATLTμVTμH+ν+δμH+ρKβVLTλT,R−μVTλT,R+μATLTμVTμH+ν+δ0.The eigenvalues ofFV−1 are
(23)0,−1μVTρβHβVΛHKLTλT,R−μVTλT,R+μATLTμHμH+ν+δμH+ρ,1μVTρβHβVΛHKLTλT,R−μVTλT,R+μATLTμHμH+ν+δμH+ρ.Therefore, the basic reproduction numberR0 is given by
(24)R0=1μVTρβHβVΛHKLTλT,R−μVTλT,R+μATLTμHμH+ν+δρ+μH.R0 exists only if LTλT,R>μVTλT,R+μAT.ExpressingR0 in terms of θ gives
(25)R0=ρβHβVΛHKλT,R+μATθ−1LTμV2TμHμH+ν+δμH+ρ.R0 exists only if the term under the square root is nonnegative, that is, θ≥1; otherwise, if θ<1, there is no growth of the mosquito population, and malaria will not develop in the community since mosquito vectors are important for the spread of malaria.


## 3.2. Sensitivity Analysis

Malaria control and eradication strategies should target important parameters which have a high impact on the basic reproduction number. A sensitivity analysis ofR0 to the various parameters is thus presented in this section. The basic reproduction number is explicitly determined by the parameters ρ,βH,βV,ΛH,LT,λT,R,μVT,μH,μAT,ν and δ. The sensitivity indices of R0 to these parameters are computed using the approach in Chitnis et al. [35].Definition 3.
The sensitivity index of a variableu that depends continuously on a parameter p is defined as
(26)⋎pu=∂u∂p.pu,
whereu is a differentiable function of p.Thus, by the definition above, the formula used to derive an expression for the sensitivity ofR0 to a parameter p is given by
(27)⋎pR0=∂R0∂p.pR0.Table1 shows the sensitivity indices of R0 to the parameters (independent of temperature and rainfall variations) determining its value. Parameter values ΛH=0.031, μH=1/23178, βH=βV=0.00021, ρ=1/20, ν=1/30, δ=0.00000638, and K=1000000 are used.Table 1
Numerical values of sensitivity indices ofR0.
Parameter symbolSensitivity indexμH-0.5011βH0.5βV0.5ΛH0.5ν-0.4993ρ0.0004δ-0.0001The MATLAB computer software program is used in the simulation of the sensitivity ofR0 with respect to temperature and rainfall since the temperature-rainfall-dependent parameters μVT, LT, λT,R, and μAT are given by nonlinear functions. The results are shown in Figure 1.Figure 1
(a) Sensitivity indices of the basic reproduction number with respect to temperature. (b) Sensitivity indices of the basic reproduction number with respect to rainfall.
(a)(b)


### 3.2.1. Interpretation of Sensitivity Analysis

The natural death rateμH and the disease-induced death rate δ of the human population are the most and least sensitive parameters, respectively. A positive value in the sensitivity index shows that if the parameter is increased when all other parameters are kept constant, the value of R0 increases, while for a negative sensitivity index, when the parameter value is increased with all other parameters kept constant, the value of R0 decreases. In Figure 1(a), rainfall is fixed at 10 mm. It is observed that R0 is most sensitive to temperatures within the ranges 17°C-20°C and 37°C-40°C. This is in agreement with Githeko et al. [16] in which the lower-end range and the upper-end range of disease transmission are established at 14°C-18°C and 35°C-40°C, respectively. The sensitivity indices for temperatures between 17°C and 25°C are positive, whereas those for temperatures between 25°C and 35°C are negative. In Figure 1(b), the temperature is fixed at 25°C. It is observed that the sensitivity to rainfall reduces with more rainfall received. Indices for daily rainfall below 25 mm are positive, whereas indices for rainfall above 25 mm are negative.


## 3.2.1. Interpretation of Sensitivity Analysis

The natural death rateμH and the disease-induced death rate δ of the human population are the most and least sensitive parameters, respectively. A positive value in the sensitivity index shows that if the parameter is increased when all other parameters are kept constant, the value of R0 increases, while for a negative sensitivity index, when the parameter value is increased with all other parameters kept constant, the value of R0 decreases. In Figure 1(a), rainfall is fixed at 10 mm. It is observed that R0 is most sensitive to temperatures within the ranges 17°C-20°C and 37°C-40°C. This is in agreement with Githeko et al. [16] in which the lower-end range and the upper-end range of disease transmission are established at 14°C-18°C and 35°C-40°C, respectively. The sensitivity indices for temperatures between 17°C and 25°C are positive, whereas those for temperatures between 25°C and 35°C are negative. In Figure 1(b), the temperature is fixed at 25°C. It is observed that the sensitivity to rainfall reduces with more rainfall received. Indices for daily rainfall below 25 mm are positive, whereas indices for rainfall above 25 mm are negative.


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

The Jacobian matrix of the system (1)–(7) evaluated at the disease-free equilibrium point E01 is shown below.
(28)JE01=−μH00σ00−βHΛHμH0−ρ+μH0000βHΛHμH0ρ−μH+ν+δ000000ν−μH+σ0000000−λT,R+μATLTLT0000λT,R−μVT0000000−μVT.The eigenvalues ofJE01 are −μH, −μVT, −μH+σ, −μH+ν+σ, −ρ+μH and the zero points of the polynomial.Z2+μVT+μAT+λT,RZ+μVTλT,R+μAT−λT,RLT=0, where Z is the eigenvalue.The zero points of a polynomial of order two have negative real parts if and only if its coefficients and constant terms are positive. Thus, the disease-free equilibriumE01 is stable only if
(29)λT,RLTμVTλT,R+μAT<1.Theorem 4.
The disease-free equilibrium pointE02 of system (1)–(7) is locally asymptotically stable if R0<1 and unstable if R0>1.Proof.
It has already been shown that the disease-free equilibriumE02 only exists if LTλT,R/μVTλT,R+μAT>1. If this condition is satisfied, then R0 is real and positive. Thus, the basic reproduction number R0 is biologically consistent. Using the theorem by van den Driessche and Watmough [36], the disease-free equilibrium E02 is locally asymptotically stable if R0<1 and unstable if R0>1.


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

The global stability of the disease-free equilibrium is investigated using a theorem by Castillo-Chavez et al. [37]. System (1)–(7) can be expressed in terms of
(30)dXdt=FX,ZdZdt=GX,Z,GX,0=0,where X∈ℝm denotes the number of uninfected individuals and Z∈ℝn denotes the number of infected individuals.The following conditions (H1) and (H2) must be satisfied providedR0<1 to guarantee global asymptotic stability.(H1) FordX/dt=FX,0,X∗is globally asymptotically stable(H2)GX,Z=MZ−G^X,Z,G^X,Z≥0forX,Z∈Ω, where M=DZGX∗,0 is an M-matrix and Ω is the region where the model makes biological sense. From system (1)–(7)
(31)FX,Z=ΛH−βHSHIV−μHSH+σRHνIH−σ+μHRHLT1−MAKSV+IV−λT,R+μATMAλT,RMA−βVSVIH−μVTSV,and
(32)GX,Z=βHSHIV−ρ+μHEHρEH−μH+ν+δIHβVSVIH−μVTIV.To investigate condition (H1),(33)FX,0=ΛH−μHSH0LT1−MAKSV−λT,R+μATMAλT,RMA−μVTSV.It has already been established that the threshold values for the human and mosquito populations areΛH/μH and λT,RMA/μVT, respectively; thus, there is convergence in Ω. Therefore, X∗ is globally asymptotically stable.To investigate condition (H2) for the disease-free equilibrium pointE01(34)M=−ρ+μH0a2ρ−μH+ν+δ000−μVT,where a2=βHΛH/μH(35)G^X,Z=βHIVΛHμH−SH0−βVSVIH.SinceG^3X,Z<0, condition (H2) is violated. Therefore, the disease-free equilibrium point E01 may not be globally asymptotically stable.For the disease-free equilibrium pointE02(36)M=−ρ+μH0a2ρ−μH+ν+δ00a1−μVT,where a1=βVKLTλT,R−μVTλT,R+μAT/LTμVT and a2=βHΛH/μH(37)G^X,Z=βHIVΛHμH−SH0βVIHKLTλT,R−μVTλT,R+μATLTμVT−SV.It has already been established that the threshold value of the human population isΛH/μH. Therefore, SH≤ΛH/μH. Similarly, SV≤KLTλT,R−μVTλT,R+μAT/LTμVT. This shows that G^T,R≥0; thus, equilibrium point E02 is globally asymptotically stable.


## 3.5. Bifurcation Analysis

In this subsection, the centre manifold theorem by Castillo-Chavez and Song [38] is used to investigate the bifurcation behaviour of system (1)–(7) when the basic reproduction number R0=1.Theorem 5.
Consider a general system of ODEs with a parameterϕ(38)dxdt=fx,ϕ,f:ℝn×ℝ⟶ℝn,f∈ℂ2ℝn×ℝ.
Without loss of generality, it is assumed that 0 is an equilibrium for system (38) for all values of the parameter ϕ, that is
(39)f0,ϕ≡0forallϕ.AssumeA1.
A=Dxf0,0=∂fi/∂xj is the linearisation matrix of system (38) around the equilibrium 0 with ϕ evaluated at 0. Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts.A2.
MatrixA has a nonnegative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue.Letfk be the kth component of f and
(40)a=∑k,i,j=1nvkwiwj∂2fk∂xi∂xj0,0,b=∑k,i=1nvkwi∂2fk∂xi∂ϕ0,0.The local dynamics of (38) around 0 are totally determined by a and b.
(i)
a>0, b>0, when ϕ<0 with ∣ϕ∣<<1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0<ϕ<<1, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium.(ii)
a<0, b<0, when ϕ<0 with ∣ϕ∣<<1, 0 is unstable; when 0<ϕ<<1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium.(iii)
a>0, b<0, when ϕ<0 with ∣ϕ∣<<1, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0<ϕ<<1, 0 is stable, and a positive unstable equilibrium.(iv)
a<0, b>0, when ϕ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable.A change in notations of the variables is used such thatSH=x1, EH=x2, IH=x3, RH=x4, MA=x5, SV=x6, and IV=x7. For R0=1, let βH be a bifurcation parameter with bifurcation value βH∗,
(41)βH∗=LTμHμVT2μH+ν+δρ+μHρβVΛHKLTλT,R−μVTλT,R+μAT.System (1)–(7) becomes
(42)f1x1,x2,x3,x4,x5,x6,x7=ΛH−βHx1x7−μHx1+σx4,(43)f2x1,x2,x3,x4,x5,x6,x7=βHx1x7−ρ+μHx2,(44)f3x1,x2,x3,x4,x5,x6,x7=ρx2−μH+ν+δx3,(45)f4x1,x2,x3,x4,x5,x6,x7=νx3−σ+μHx4,(46)f5x1,x2,x3,x4,x5,x6,x7=LT1−x5Kx6+x7−λT,R+μATx5,(47)f6x1,x2,x3,x4,x5,x6,x7=λT,Rx5−βVx6x3−μVTx6,(48)f7x1,x2,x3,x4,x5,x6,x7=βVx6x3−μVTx7.E02x1,x2,x3,x4,x5,x6,x7=ΛH/μH,0,0,0,Kd/LTλT,R,Kd/LTμVT,0is a disease free equilibrium point for system (42)–(48), where d=LTλT,R−μVTλT,R+μVT.The Jacobian matrix of system (42)–(48) evaluated at E02 is given by
(49)JE02=−J100σ00−βHΛHμH0−J20000βHΛHμH0ρ−J3000000ν−J40000000−J5μVTλT,R+μATλT,RμVTλT,R+μATλT,R00−βVKdLTμVT0λT,R−J6000βVKdLTμVT000−J7whereJ1=μH, J2=ρ+μH, J3=μH+ν+δ, J4=μH+σ, J5=LTλT,R/μVT, and J6=J7=μVT.The right eigenvectorw=w1,w2,w3,w4,w5,w6,w7T, corresponding to the zero eigenvalue, is computed using JE02w=0 which yields
(50)−J1w1+σw4−βHΛHμHw7=0,−J2w2+βHΛHμHw7=0,ρw2−J3w3=0,νw3−J4w4=0,−J5w5+μVTλT,R+μATλT,Rw6+μVTλT,R+μATλT,Rw7=0,−βVKdLTμVTw3+λT,Rw5−J6w6=0,βVKdLTμVTw3−J7w7=0.Settingw7=1 gives,
(51)w3=LTμvT2βVKd,w4=LTνμVT2βVKdμH+σ,w2=LTμVT2μH+ν+δρβVKd,w1=1μHLTνσμVT2βVKdμH+σ−βHΛHμH,and
(52)−LTλT,R2μVTw5+μVTλT,R+μATw6=−μVTλT,R+μAT,(53)λT,Rw5−μVTw6=μVT.Solving equations (52) and (53) simultaneously gives
(54)w5=0,w6=−1.The left eigenvectorv=v1,v2,v3,v4,v5,v6,v7, corresponding to the zero eigenvalue, is computed using vJE02=0 which yields
(55)−J1v1=0,−J2v2+ρv3=0,−J3v3+νv4−βVKdLTμVTv6+βVKdLTμVTv7=0,σv1−J4v4=0,μVTλT,R+μATλT,Rv5−J6v6=0,βHΛHμHv1+βHΛHμHv2+μVTλT,R+μATλT,Rv5−J7v7=0.It follows thatv1=v4=0, setting v3=v6=1 gives
(56)v2=ρρ+μH,v5=λT,RλT,R+μAT,v7=LTμVTμH+ν+δ+βvKdβVKd.Sincev1=v4=0, the values of a and b are obtained from
(57)a=∑i,j=17v2wiwj∂2f2∂xi∂xj+∑i,j=17v3wiwj∂2f3∂xi∂xj+∑i,j=17v5wiwj∂2f5∂xi∂xj+∑i,j=17v6wiwj∂2f6∂xi∂xj+∑i,j=17v7wiwj∂2f7∂xi∂xj,b=∑k,i=17vkwi∂2fk∂xi∂βH∗,where the partial derivatives computed at E02 are
(58)∂2f2∂x1∂x7=βH,∂2f5∂x5∂x6=−LTK,∂2f5∂x5∂x7=−LTK,∂2f6∂x3∂x6=−βV,∂2f7∂x3∂x6=βV,∂2f2∂x7∂βH∗=ΛHμH.It follows that(59)a=2ρβHμHρ+μHLTνσμVT2βVKdμH+σ−βHΛHμH+2LTμVT2Kd−2LTμVT2βVKd+LTμVTμH+ν+δβVK2d2,b=ρβHΛHμHρ+μH>0.Sinceb>0, the model undergoes a backward bifurcation if a>0, that is
(60)2ρβHLTνσμVT2μHβVKdμH+ρμH+σ+2LTμVT2Kd>2ρβH2ΛHμH2μH+ρ+2LTμVT2αβVK2d2,where α=βVKd+LTμVTμH+ν+δ.Remark 6.
Existence of a backward bifurcation whena>0 means that there is a possibility of coexistence of an endemic equilibrium and the disease-free equilibrium when R0<1. In this case, the strategy of reducing the basic reproduction number to a value less than unity would not be sufficient for the eradication of malaria.


## 3.6. Existence and Stability of the Endemic Equilibrium

Theorem 7.
System (1)–(7) has a unique endemic equilibrium when R0>1.Proof.
Setting the right-hand side of the equations in system (1)–(7) to zero shows that either IH=0 or IH=KρβHβVΛHσ+μHd−LTμHμVT2c/KdβHβVc−ρσν+LTβVμHμVTc, where d=LTλT,R−μVTλT,R+μAT and c=μH+ν+δσ+μHρ+μH. IH=0 corresponds to the disease-free equilibrium.ExpressingIH above in terms of R0 using d=R02μV2μHLTc/KρβHβVΛHσ+μH gives
(61)IH=ρΛHμVTσ+μHR02−1R02μVTc−ρσν+βVρΛHσ+μH.Thus, there exists a unique endemic equilibrium pointE1=SH∗,EH∗,IH∗,RH∗,MA∗,SV∗,IV∗ only if R02>1, that is, R0>1 since c>ρσν. The endemic equilibrium is given by
(62)SH∗=LTμVTΛHσ+μH+σνIH∗βVIH∗+μVTσ+μHKβHd+LTμHμVTβVIH∗+LTμHμVT2,EH∗=KβHβVdIH∗ΛHσ+μH+σνIH∗ρ+μHσ+μHKβHd+LTμHμVTβVIH∗+LT,RμHμVT2,IH∗=ρΛHμVTσ+μHR02−1R02μVTc−ρσν+βVΛHρσ+μH,RH∗=νIH∗σ+μH,MA∗=KdLTλT,R,SV∗=KdLTβVIH∗+μVT,IV∗=KβVdIH∗LTμVTβVIH∗+μVT.The Jacobian matrix of system of equations (1)–(7) evaluated at the endemic equilibrium point E1 is given by
(63)JE1=−e100σ00−e2e3−e40000e20ρ−e5000000ν−e60000000−e7e8e800−e90e10−e11000e900e12−e13,where e1=βHIH∗+μH, e2=βHSH∗, e3=βHIV∗, e4=ρ+μH, e5=μH+ν+δ, e6=σ+μH, e7=LTSV∗+IV∗/K−λT,R+μAT, e8=LT1−MA∗/K, e9=βVSV∗, e10=λT,R, e11=βVIH∗+μV, e12=βVIH∗, and e13=μVT.The characteristic polynomial of the Jacobian matrixJE1 is obtained as
(64)Z7+a1Z6+a2Z5+a3Z4+a4Z3+a5Z2+a6Z+a7=0,where
(65)a1=e1+e4+e5+e6+e7+e11+e13,a2=e1e4+e5+e6+e7+e11+e13+e4e5+e6+e7+e11+e5e6+e7+e11+e13+e6e7+e11+e13+e7e11+e13+e13e4+e7−e8e10,a3=e1e4e5+e6+e7+e11+e13+e1e5e7+e11+e13+e1e6e7+e11+e13+e4e5e6+e7+e11+e13+e4e6e7+e11+e13+e5e6e7+e11+e7e11e1+e4+e5+e6+e7−e8e10e1+e4+e5+e12+e11e13e1+e4+e5+e6+e7−e8e13e4+e5+e6−e2e9ρ+e1e7e13−e8e10e13,a4=e2e9ρe3+e6−e7−e11+e12−e1+e1e6e13e4+e5+e7+e11+e1e4e13e5+e7+e1e7e5e13+e4e11+e11e13+e4e6+e4e5e6e13+e7e13+e11e13+e1e7+e1e11+e6e11−e8e10+e6e7+e4e7e6e13+e11e13+e1e6+e6e11+e5e13e6e7+e6e11+e8e10+e7e11+e1e11−e1e10e8e12+e8e13+e6e8+e5e8−e6e11e1e7+e5e7−e4e8+e1e4+e1e5+e8e10e4e12−e4e13−e5e12−e6e12−e4e13−e5e6−e1e4+e6e11e13e4+e7−e3νρσ+e1e5e7e11,a5=e1e4e5e6e7+e6e11+e6e13+e7e11−e8e10+e7e13+e11e13+e1e5e6e7e11−e8e10+e7e13+e11e13+e4e6e8e10e13−e5e10+e2e9ρe3e6−e1e6+e1e7−e1e11+e3e11−e6e11−e7e11−e8e10+e1e12−e3e12+e6e12+e7e12+e4e5e6e7e11+e7e13−e8e10e1e4e12+e1e5e12+e1e5e13+e1e6e13+e1e6e12+e4e5e12+e1e4e13+e5e6e13+e7e11e1e4e13+e4e6e13+e11e13e4e5e6+e4e5e7+e1e6e7+e1e5e7−e3νρσe7+e11+e13,a6=e1e4e5e6e7e11−e6e8e10+e6e7e13+e6e11e13−e8e10e12−e8e10e13+e7e11e13−e2e9ρe3e6e7−e1e6e11−e1e7e11+e1e8e10+e3e6e11+e3e7e11−e3e8e10−e6e7e11+e6e8e10+e1e6e12+e1e7e12−e3e6e12−e3e7e12+e1e4e6e7e11e13−e8e10e12−e8e10e13−e1e5e6e8e10e12−e7e11e13+e8e10e13−e4e5e6e8e10e12−e7e11e13−e4e5e7e8e10e13−e1e4e6e7e9ρ−νρσe3e7e11−e3e8e10+e3e7e13+e3e11e13,a7=e1e2e6e9ρe8e10−e7e11+e2e3e6e9e12ρe7e11−e8e10+e1e4e5e6e7e11e13−e8e10e12+e2e6e7e9e12ρe1−e3+e3ρσe8e10e12ν−e7e11e13ν+e8e10e13−e1e4e5e6e8e10e13.By the Routh-Hurwitz criteria, the endemic equilibrium pointE1 is locally stable provided
(66)a1>0,a2>0,a3>0,a4>0,a5>0,a6>0,a7>0,a1a2>a3,a1a2a3−a12a4−a32>0,2a1a4a5−a12a42−a1a22a5+a2a3a5−a32a4−a52>0,2a1a2a5a6+a12a3a4a6−a13a63−a12a42a5−a1a22a52−a1a2a32a6+a1a2a3a4a5−3a1a3a5a6+2a1a4a52+a2a3a52+a3a6−a3a4a5−a53>0,and
(67)a12a3a4a62−a13a63−3a12a2a4a6a7+2a12a2a5a62+a12a43a7−a12a42a5a6+3a12a62a7−a1a23a72+2a1a22a3a6a7+a1a22a4a5a7−a1a22a52a6−a1a2a32a62−a1a2a3a42a7+a1a2a3a4a5a6+3a1a2a4a72−a1a2a5a6a7+a1a3a4a6a7−3a1a3a5a62−2a1a42a5a7+2a1a4a52a6−3a1a6a72+a22a32a6a7−a2a3a4a5a7+a2a3a52a6−a2a5a72+a33a62+a32a42a7−a32a4a5a6−2a3a4a72+3a3a5a6a7+a4a52a7−a53a6+a73>0.The stability of the endemic equilibriumE1 is demonstrated in the numerical simulations (see Section 4).Theorem 8.
The endemic equilibrium pointE1 for system (1)–(7) is globally asymptotically stable in D if R0>1.Proof.
Consider a Lyapunov function of the form(68)Lt=SH−SH∗−SH∗lnSHSH∗+EH−EH∗−EH∗lnEHEH∗+IH−IH∗−IH∗lnIHIH∗+RH−RH∗−RH∗lnRHRH∗+MA−MA∗−MA∗lnMAMA∗+SV−SV∗−SV∗lnSVSV∗+IV−IV∗−IV∗lnIVIV∗.DifferentiatingLt with respect to time gives
(69)dLdt=SH−SH∗SHdSHdt+EH−EH∗EHdEHdt+IH−IH∗IHdIHdt+RH−RH∗RHdRHdt+MA−MA∗MAdMAdt+SV−SV∗SVdSVdt+IV−IV∗IVdIVdt,dLdt=SH−SH∗SHΛH−βHSHIV−μHSH+σRH+EH−EH∗EHβHSHIV−ρ+μHEH+IH−IH∗IHρEH−μH+ν+δIH+RH−RH∗RHνIH−σ+μHRH+MA−MA∗MALTSV+LTIV−LTSVMAK−LTIVMAK−λT,R+μATMA+SV−SV∗SVλT,RMA−βVIHSV−μVTSV+IV−IV∗IVβVSVIH−μVTIV,dLdt=SH−SH∗SHΛH−βHSH−SH∗IV−IV∗−μHSH−SH∗+σRH−RH∗+EH−EH∗EHβHSH−SH∗IV−IV∗−ρ+μHEH−EH∗+IH−IH∗IHρEH−EH∗−μH+ν+δIH−IH∗+RH−RH∗RHνIH−IH∗−σ+μHRH−RH∗+MA−MA∗MALTSV−SV∗+LTIV−IV∗−LTSV−SV∗MA−MA∗K−LTIV−IV∗MA−MA∗K−λT,R+μATMA−MA∗+SV−SV∗SVλT,RMA−MA∗−βVIH−IH∗SV−SV∗−μVTSV−SV∗+IV−IV∗IVβVSV−SV∗IH−IH∗−μVTIV.dLdt=ΛHSH−SH∗SH−βHIVSH−SH∗2SH+βHIV∗SH−SH∗2SH−μHSH−SH∗2SH+σRHSH−SH∗SH−σRH∗SH−SH∗SH+βHSHIVEH−EH∗EH−βHSHIV∗EH−EH∗EH−βHSH∗IVEH−EH∗EH+βHSH∗IV∗EH−EH∗EH−ρ+μHEH−EH∗2EH+ρEHIH−IH∗IH−ρEH∗IH−IH∗IH−μH+ν+δIH−IH∗2IH+νIHRH−RH∗RH−νIH∗RH−RH∗RH−σ+μHRH−RH∗2RH+LTSVMA−MA∗MA−LTSV∗MA−MA∗MA+LTIVMA−MA∗MA−LTIV∗MA−MA∗MA−LTSVMA−MA∗2KMA+LTSV∗MA−MA∗2KMA−LTIVMA−MA∗2KMA+LTIV∗MA−MA∗2KMA−λT,R+μATMA−MA∗2MA+λT,RMASV−SV∗SV−λT,RMA∗SV−SV∗SV−βVIHSV−SV∗2SV+βVIH∗SV−SV∗2SV−μVTSV−SV∗2SV+βVSVIHIV−IV∗IV−βVSVIH∗IV−IV∗IV−βVSV∗IHIV−IV∗IV+βVSV∗IH∗IV−IV∗IV−μVTIV−IV∗2IV.Collecting positive terms together and negative parts together gives(70)dLdt=A−B,where
(71)A=ΛHSH−SH∗SH+βHIV∗SH−SH∗2SH+σRHSH−SH∗SH+βHIVSHEH−EH∗EH+βHIV∗SH∗EH−EH∗EH+ρEHIH−IH∗IH+νIHRH−RH∗RH+LTSVMA−MA∗MA+LTIVMA−MA∗MA+LTSV∗MA−MA∗2KMA+LTIV∗MA−MA∗2KMA+λT,RMASV−SV∗SV+βVIH∗SV−SV∗2SV+βVSVIHIV−IV∗IV+βVSV∗IH∗IV−IV∗IV,and
(72)B=βHIVSH−SH∗2SH+μHSH−SH∗2SH+σRH∗SH−SH∗SH+βHSHIV∗EH−EH∗EHβHSH∗IVEH−EH∗EH+ρ+μHEH−EH∗2EH+ρEH∗IH−IH∗IH+μH+ν+δIH−IH∗2IHνIH∗RH−RH∗RH+σ+μHRH−RH∗2RH+LTSV∗MA−MA∗MA+LTIV∗MA−MA∗MALTSVMA−MA∗2KMA+LTIVMA−MA∗2KMA+λT,R+μATMA−MA∗2MA+λT,RMA∗SV−SV∗SV+βVIHSV−SV∗2SV+μVTSV−SV∗SV+βVSVIH∗IV−IV∗IV+βVSV∗IHIV−IV∗IV+μVTIV−IV∗2IV.Hence, ifA<B, then dL/dt≤0. Note that dL/dt=0 if and only if SH=SH∗, EH=EH∗, IH=IH∗, RH=RH∗, MA=MA∗, SV=SV∗, and IV=IV∗. Thus, the largest compact invariant set in SH∗,EH∗,IH∗,RH∗,MA∗,SV∗,IV∗∈D:dL/dt=0 is the singleton set E1. By Lasalle’s invariant principles [39], it implies that E1 is globally asymptotically stable in D if A<B.


## 4. Numerical Simulations

In this section, initial conditionsSH0=5000000,EH0=200,IH0=500,RH0=300,MA0=200000,SV0=300000,IV0=1000 are used to perform numerical simulations of system (1)–(7) using the MATLAB computer software program. Parameter values used are given in Table 2. The replication temperature range 16°C-42°C used for analysis is established from the condition given in Section 3, that is, LT>μVT as shown in Figure 2.Table 2
Parameter values for the human and mosquito populations.
ParameterSymbolValueUnitSourceRecruitment rate of humansΛH0.031day-1[40]Natural death rate of humansμH1/23178day-1[41]Interaction coefficient between susceptible humans and infected mosquitoesβH0.00021-[42]Rate of loss of immunityσ1/20×365day-1[43]Progression rate from exposed classρ1/20day-1[43]Recovery rateν1/30day-1[43]Disease-induced death rateδ0.00000638day-1[33]Egg deposition rateLT−0.153T2+8.61T−97.7day-1[30]Aquatic mosquito death rateμAT1.0257−0.094T+0.0025T2day-1[30]Interaction coefficient between susceptible mosquitoes and infected humansβV0.00021-[42]Adult mosquito death rateμVT−ln0.522−0.000828T2+0.0367Tday-1[30]Carrying capacity of the environmentK1000000Dimensionless[30]Figure 2
Egg deposition rate and adult mosquito death rate versus temperature.


### 4.1. Parameters Determining the Aquatic Mosquito Maturation Rate

According to Mukhtar et al. [30], the maturation rate of mosquitoes to adulthood is temperature-rainfall dependent governed by the total number of eggs laid per adult mosquito per oviposition ωT, daily survival probability of rainfall-dependent eggs P1R, daily survival probability of rainfall-dependent larvae P2R, daily survival probability of the rainfall-dependent pupae P3R, daily survival probability of the temperature-dependent larvae P2T, and the temperature-dependent duration of the immature mosquito development TEAT given by
(73)ωT=−0.153T2+8.61T−97.7μVT,P1R=4×0.932500R50−R,P2R=4×0.252500R50−R,P3R=4×0.752500R50−R,P2T=exp0.06737−0.00554T,TEAT=1−0.00094T2+0.049T−0.552.It is assumed that aquatic mosquitoes cannot survive at daily rainfall beyond 50 mm. The maturation rate is given by(74)λT,R=ωTP1RP2RP3RP2TTEAT.In Figure3, daily temperature and rainfall are fixed at T=25°C and R=10mm, respectively. Results in Figure 3(a) show a sharp fall in the susceptible human population and a rise in the infectious human population within the first 50 days. The infectious human population then falls due to recovery and death. Figure 3(b) shows a fall in the susceptible vector population with a rise in the infectious vector population. It is observed from Figure 3 that the steady state in both populations is stable. Thus, the endemic equilibrium SH∗,EH∗,IH∗,RH∗,MA∗,SV∗,IV∗ of the model is locally stable.Figure 3
(a) Change in the human population with time over a period of 1000 days. (b) Change in mosquito population with time over a period of 1000 days.
(a)(b)In Figure4, temperatures within the 16°C-42°C range are taken at a fixed amount of daily rainfall 10 mm to investigate the impact of temperature on the various compartments of the mosquito population. Temperature values of 20°C, 25°C, 30°C, 35°C, and 40°C are considered. In Figure 4(a), it is observed that the aquatic mosquito population is lowest at 40°C at all times and highest at 25°C. Thus, a temperature of 25°C is favourable for replication in the mosquito population. Figure 4(a) also shows a fall in the aquatic mosquito population with time at temperature 40°C. This shows that the aquatic mosquitoes hardly survive at very high temperatures. A sharp fall in the susceptible mosquito population is observed at all temperatures in Figure 4(b). Figure 4(c) shows the change in the infectious mosquito population with time. The highest rate of infection in the vector population is observed at 25°C and the lowest rate at 40°C. The comparison of Figures 4(a) and 4(c) shows that although the aquatic mosquito population grows faster at 30°C than 20°C, the infection rate on the other hand is higher at 20°C compared to 30°C.Figure 4
Change in the mosquito population over time for temperatures valuesT=20°C, T=25°C, T=30°C, T=35°C, and T=40°C. (a) The number of aquatic mosquitoes over a period of 100 days. (b) The number of susceptible mosquitoes over a period of 100 days. (c) The number of infectious mosquitoes over a period of 100 days.
(a)(b)(c)Therefore, an increase in the aquatic mosquito population does not necessarily imply that there will be a corresponding rise in the infection rate. These results show that temperature greatly affects the transmission dynamics of malaria as there is a significant difference in the number of infected mosquitoes at different temperature values. Malaria is more effectively transmitted at 25°C as compared to other temperature values considered.Figure5 shows the impact of daily rainfall at a fixed temperature. According to Mukhtar et al. [30], aquatic mosquitoes cannot survive at daily rainfall beyond 50 mm which limits the growth of the vector population. Therefore, values below 50 mm, that is, (10 mm, 20 mm, 30 mm, 40 mm) are considered at a fixed temperature 25°C. The aquatic vector population growth is observed to be lowest at rainfall value 30 mm and highest at 40 mm as shown in Figure 5(a). This is because more rainfall received provides breeding sites for the mosquitoes. Variations in the aquatic mosquito population due to temperature differences are observed to be more than the variations due to rainfall differences; thus, temperature affects the aquatic mosquito population more than rainfall. A rise followed by a sharp fall in the susceptible mosquito population is observed in Figure 5(b) at all values of rainfall. The infection rate in the vector population is highest at 30 mm of daily rainfall. This shows that the malaria parasite is more efficiently transmitted in the mosquito population at 30 mm as compared to other values considered. Similar to the observation from Figure 5, comparing Figure 5(a) and Figure 5(c), the aquatic mosquito population is higher at 40 mm than it is at 30 mm; however, the infection is seen to be higher at 30 mm compared to 40 mm. This suggests that a higher aquatic mosquito population does not necessarily lead to higher infection rates. From Figure 5, there is a variation in the infection rate for the different rainfall values; thus, rainfall affects the transmission of malaria.Figure 5
Change in mosquito population over time for rainfall valuesR=10 mm, R=20 mm, R=30 mm, and R=40 mm. (a) The number of aquatic mosquitoes over a period 10 days. (b) The number of susceptible mosquitoes over a period of 10 days. (c) The number of infectious mosquitoes over a period of 10 days.
(a)(b)(c)


## 4.1. Parameters Determining the Aquatic Mosquito Maturation Rate

According to Mukhtar et al. [30], the maturation rate of mosquitoes to adulthood is temperature-rainfall dependent governed by the total number of eggs laid per adult mosquito per oviposition ωT, daily survival probability of rainfall-dependent eggs P1R, daily survival probability of rainfall-dependent larvae P2R, daily survival probability of the rainfall-dependent pupae P3R, daily survival probability of the temperature-dependent larvae P2T, and the temperature-dependent duration of the immature mosquito development TEAT given by
(73)ωT=−0.153T2+8.61T−97.7μVT,P1R=4×0.932500R50−R,P2R=4×0.252500R50−R,P3R=4×0.752500R50−R,P2T=exp0.06737−0.00554T,TEAT=1−0.00094T2+0.049T−0.552.It is assumed that aquatic mosquitoes cannot survive at daily rainfall beyond 50 mm. The maturation rate is given by(74)λT,R=ωTP1RP2RP3RP2TTEAT.In Figure3, daily temperature and rainfall are fixed at T=25°C and R=10mm, respectively. Results in Figure 3(a) show a sharp fall in the susceptible human population and a rise in the infectious human population within the first 50 days. The infectious human population then falls due to recovery and death. Figure 3(b) shows a fall in the susceptible vector population with a rise in the infectious vector population. It is observed from Figure 3 that the steady state in both populations is stable. Thus, the endemic equilibrium SH∗,EH∗,IH∗,RH∗,MA∗,SV∗,IV∗ of the model is locally stable.Figure 3
(a) Change in the human population with time over a period of 1000 days. (b) Change in mosquito population with time over a period of 1000 days.
(a)(b)In Figure4, temperatures within the 16°C-42°C range are taken at a fixed amount of daily rainfall 10 mm to investigate the impact of temperature on the various compartments of the mosquito population. Temperature values of 20°C, 25°C, 30°C, 35°C, and 40°C are considered. In Figure 4(a), it is observed that the aquatic mosquito population is lowest at 40°C at all times and highest at 25°C. Thus, a temperature of 25°C is favourable for replication in the mosquito population. Figure 4(a) also shows a fall in the aquatic mosquito population with time at temperature 40°C. This shows that the aquatic mosquitoes hardly survive at very high temperatures. A sharp fall in the susceptible mosquito population is observed at all temperatures in Figure 4(b). Figure 4(c) shows the change in the infectious mosquito population with time. The highest rate of infection in the vector population is observed at 25°C and the lowest rate at 40°C. The comparison of Figures 4(a) and 4(c) shows that although the aquatic mosquito population grows faster at 30°C than 20°C, the infection rate on the other hand is higher at 20°C compared to 30°C.Figure 4
Change in the mosquito population over time for temperatures valuesT=20°C, T=25°C, T=30°C, T=35°C, and T=40°C. (a) The number of aquatic mosquitoes over a period of 100 days. (b) The number of susceptible mosquitoes over a period of 100 days. (c) The number of infectious mosquitoes over a period of 100 days.
(a)(b)(c)Therefore, an increase in the aquatic mosquito population does not necessarily imply that there will be a corresponding rise in the infection rate. These results show that temperature greatly affects the transmission dynamics of malaria as there is a significant difference in the number of infected mosquitoes at different temperature values. Malaria is more effectively transmitted at 25°C as compared to other temperature values considered.Figure5 shows the impact of daily rainfall at a fixed temperature. According to Mukhtar et al. [30], aquatic mosquitoes cannot survive at daily rainfall beyond 50 mm which limits the growth of the vector population. Therefore, values below 50 mm, that is, (10 mm, 20 mm, 30 mm, 40 mm) are considered at a fixed temperature 25°C. The aquatic vector population growth is observed to be lowest at rainfall value 30 mm and highest at 40 mm as shown in Figure 5(a). This is because more rainfall received provides breeding sites for the mosquitoes. Variations in the aquatic mosquito population due to temperature differences are observed to be more than the variations due to rainfall differences; thus, temperature affects the aquatic mosquito population more than rainfall. A rise followed by a sharp fall in the susceptible mosquito population is observed in Figure 5(b) at all values of rainfall. The infection rate in the vector population is highest at 30 mm of daily rainfall. This shows that the malaria parasite is more efficiently transmitted in the mosquito population at 30 mm as compared to other values considered. Similar to the observation from Figure 5, comparing Figure 5(a) and Figure 5(c), the aquatic mosquito population is higher at 40 mm than it is at 30 mm; however, the infection is seen to be higher at 30 mm compared to 40 mm. This suggests that a higher aquatic mosquito population does not necessarily lead to higher infection rates. From Figure 5, there is a variation in the infection rate for the different rainfall values; thus, rainfall affects the transmission of malaria.Figure 5
Change in mosquito population over time for rainfall valuesR=10 mm, R=20 mm, R=30 mm, and R=40 mm. (a) The number of aquatic mosquitoes over a period 10 days. (b) The number of susceptible mosquitoes over a period of 10 days. (c) The number of infectious mosquitoes over a period of 10 days.
(a)(b)(c)


## 5. Discussion

In this paper, a malaria transmission model with temperature and rainfall dependent parameters is formulated. The analysis of the model reveals that the model is mathematically and epidemiologically well posed. Further analysis shows that there are two disease-free equilibrium points, one without the mosquito population (E01) and the other with the mosquito population (E02). It is found out that the existence and stability of the disease-free equilibria are dependent on the vector reproduction number (θ). θ is a threshold parameter defined as the number of mosquitoes produced by a female Anopheles mosquito throughout its lifetime, which is entirely governed by temperature-rainfall-dependent parameters (that is egg deposition rate, maturation rate of aquatic mosquitoes to adulthood, and the death rates of both adult and aquatic mosquitoes). Seasonal factors highly determine the population size of mosquito vectors which transmit malaria because mosquito replication depends on the value of θ. It was shown that the mosquito population replicates only if θ>1. The basic reproduction number R0 for the model is computed using the next-generation method, and it was shown that R0 only exists if θ>1. Malaria transmission depends on the mosquito vectors which survive only if θ>1. The disease-free equilibrium point E01 is stable if θ<1 which means that if an infective individual is introduced into the community at this point, malaria does not spread due to the absence of the mosquito vectors. The disease-free equilibrium point E02 exists if θ>1 and is stable if additionally R0<1. This means that if an infected individual is introduced into the community, malaria will not spread if R0<1; otherwise, there will be an outbreak. There is a unique endemic equilibrium if R0>1. This would suggest that malaria remains in the community as long as R0>1; thus, it is important to keep the basic reproduction number below unity.In order to establish the temperature range within which mosquitoes replicate, the threshold parameterθ was investigated. It was revealed that a replication range of 16°C-42°C is favourable, and it is this range that was used in the proceeding analysis, because any temperature outside this range was assumed not to be favourable for mosquito reproduction. Sensitivity analysis of the model revealed that the basic reproduction number R0 is highly sensitive to temperature variations within 17°C-20°C and 37°C-40°C temperature ranges. The sensitivity of the basic reproduction number to rainfall variations reduces with more rainfall received. It was observed that sensitivity indices are positive below 25 mm and negative above 25 mm. This result shows that when the rainfall received is below 25 mm, a reduction in the amount of rainfall reduces malaria endemicity while an increase in the amount of rainfall received leads to a rise in malaria endemicity. Rainfall increments in this case create more breeding sites for the mosquitoes in the form of water pools which aid mosquito population increase. Reduction in rainfall would reduce the breeding sites and thus reduce the mosquito population. On the other hand, when the rainfall received is above 25 mm, a reduction in the amount of rainfall received increases malaria endemicity whereas an increase in rainfall reduces malaria endemicity. This is because excessive rainfall flushes out breeding sites thus reduces the mosquito population. Sensitivity indices due to temperature variations are observed to be greater than those due to rainfall variations. This implies that malaria transmission is more sensitive to temperature changes than rainfall changes; thus, more attention should be directed to temperature variations.Numerical simulations of the model were performed to investigate the effect of temperature and rainfall on malaria transmission. Temperature values 20°C, 25°C, 30°C, 35°C, and 40°C were considered; it was revealed that malaria is more effectively transmitted at temperature 25°C (this is in agreement with [30, 31]). Daily rainfall below 50 mm, that is, 10 mm, 20 mm, 30 mm, and 40 mm were considered, since it was assumed that mosquitoes hardly survive rainfall above 50 mm as breeding sites are flushed out. It was noted that that daily rainfall of 30 mm is favourable for malaria transmission as compared to other values considered. Mass malaria control programmes such as the distribution of mosquito nets should be implemented when the temperature is 25 C and daily rainfall is 30 mm. In agreement with sensitivity analysis, it was observed that variations in malaria infection due to temperature differences were more than the variations due to rainfall differences. Therefore, temperature affects the transmission dynamics of malaria more than rainfall. It was also shown that the growth of the aquatic mosquito population does not necessarily lead to higher infections. Therefore, vector control measures should target adult mosquitoes more, since most of the aquatic mosquitoes do not survive to adulthood to participate in malaria transmission.



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*Source: 8885558-2020-12-19.xml*