# Global Stability of Malaria Transmission Dynamics Model with Logistic Growth

Authors: Abadi Abay Gebremeskel
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

Mathematical models become an important and popular tools to understand the dynamics of the disease and give an insight to reduce the impact of malaria burden within the community. Thus, this paper aims to apply a mathematical model to study global stability of malaria transmission dynamics model with logistic growth. Analysis of the model applies scaling and sensitivity analysis and sensitivity analysis of the model applied to understand the important parameters in transmission and prevalence of malaria disease. We derive the equilibrium points of the model and investigated their stabilities. The results of our analysis have shown that ifR0≤1, then the disease-free equilibrium is globally asymptotically stable, and the disease dies out; if R0>1, then the unique endemic equilibrium point is globally asymptotically stable and the disease persists within the population. Furthermore, numerical simulations in the application of the model showed the abrupt and periodic variations.


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## Body

## 1. Introduction

Malaria is a mosquito-borne disease caused by Plasmodium parasite, which is transmitted through the bites of an infected mosquito. In 2017, the World Health Organization report reveals estimations of 216 million malaria cases and 445 thousand deaths due to malaria were registered worldwide in 2016. However, the most malaria cases and deaths were shared by the WHO Africa region, which account for 90% of cases and 91% deaths. The most predominant malaria parasite in the WHO Africa region isPlasmodium falciparum, accounting for 99% of malaria cases in 2016 [1].Malaria is entirely preventable and treatable disease if the recommended interventions are properly applied. Individuals should have taken some aggressive measurements to decline malaria burden. Personal protection measures are the first line of defense against mosquito-borne diseases. Mosquito repellent is a method used for personal protection; and these are the substances used for exposed skin to prevent human-mosquito contact. Insecticide Treated Bed Nets (ITNs) are used for individuals against malaria to reduce the morbidity of childhood malaria (below five years of age) by 50% and global child mortality by20–30% [2, 3]. When used on a large scale, ITNs are supposed to represent efficient tools for malaria vector control but there is a limitation of resistance to insecticides used for a saturated net. The resistance of the most important African malariaAnopheles gambiae to protrude is already widespread in several West African countries [4, 5].Nowadays, mathematical models become an important and popular tools to understand the transmission dynamics of the disease and give an insight to reduce the impact of malaria burden in the society. This is because mathematical modeling can answer the following questions raised by the public health authorities and policy makers to make the correct decisions: (1) how severe will the epidemics be? (2) How long will it last? (3) How effective will an intervention be? (4) What are the effective measures to control and eliminate an endemic disease? The earliest malaria model study originated from Ross in 1911 [6] and later modification made by Macdonald [7]. Some further extensions of Ross-Macdonald models for malaria were described in [8–13]. Tumwiine et al. [13] define the reproduction number, R0, and show the existence and stability of the disease-free equilibrium and an endemic equilibrium.Recently, many works on host-vector interaction models have been done in [14–24]. In [18, 20, 22, 25], global stability of equilibria has been investigated using suitable Lyapunov functions; and their results show that the disease-free and endemic equilibrium points become globally asymptotically stable if R0≤1 and R0>1, respectively. Application of the optimal control theory becomes an important tool for investigating the efficiency of joint control intervention strategies to minimize the impact of malaria disease and cost-effectiveness of implementing them [19, 21, 23, 24]. Their studies suggest that the optimal control strategies can effectively reduce the malaria disease.Motivated by the above studies, we extend the model presented in [14] by taking into account a logistic model with population dependent birth rates for both human and vector populations that describes self-limiting growth of both the human host and mosquito vector populations. We consider logistic malaria model as no population can grow exponentially at all time, in general. A number of populations initially grow exponentially, but, due to competition and limited resources availability, their population size decline, after some time, to a stable size K, called the maximumcarrying capacity. The competition for limited resources (including food, territory, light, water, and oxygen) decreases the fertility or survival of individuals. Furthermore, this paper presents application of the model to study abrupt and periodic variations of malaria and sensitivity analysis applied to understand the important parameters in transmission and prevalence of the malaria disease. The purpose of this work is to investigate the global stability of both disease-free and endemic equilibrium points.


## 2. Malaria Model

We consider that the total human host population,Nh, at a time t∗, is divided into three disjoint compartments: susceptible Sh, infectious Ih, and recovered Rh. The total vector population, Nv, at a time t∗, is divided into two mutually exclusive subpopulations of individuals who are susceptible, Sv, and infectious, Iv. The susceptible human and vector populations are recruited at the rates Bh and Bv, respectively, where Bh=μhNh1-Nh/Kh and Bv=μvNv1-Nv/Kv. The susceptible human and vector populations decrease due to natural death at a rate αh for humans and αv for vectors, and those that move to the infected classes at a rate βhIv and βvIh, respectively. The infected human population grows as a result of new infection at a rate βhShIv and decline due to natural mortality, disease induced death, and recovery at a rate αh, ρh, and γh, respectively. For details, see the schematic diagram of the model in Figure 1. The state variables and parameters for the model are described in “State Variables, Parameters, Descriptions, and Their Dimensions of Malaria Model” section.Figure 1
The compartmental model for malaria transmission.The model has made the following assumptions: both the total sizes of human and vector populations not being constant; all variables and parameters involving the model assumed to be nonnegative; all newborns susceptible to infection; mosquitoes not dying because of infection; no recovery compartment for infected mosquitoes; and the recovered human population developing permanent immunity. From the schematics diagram of transmission of malaria between human and mosquito (see Figure1), we have the governed differential equations which describe the dynamics of malaria,(1)dShdt∗=μhNh1-NhKh-βhShIv-αhSh,dIhdt∗=βhShIv-αhIh-ρhIh-γhIh,dRhdt∗=γhIh-αhRh,dSvdt∗=μvNv1-NvKv-βvSvIh-αvSv,dIvdt∗=βvSvIh-αvIv,with initial conditions (2)Sh0=Sh0,Ih0=Ih0,Rh0=Rh0,Sv0=Sv0,Iv0=Iv0.At all times,Nh=Sh+Ih+Rh and Nv=Sv+Iv. Moreover, their differential equations are satisfying(3)dNhdt∗=μhNh1-NhKh-αhNh-ρhIh,dNvdt∗=μvNv1-NvKv-αvNv,respectively. We may notice that the vector population equation is completely decoupled from the human equations which is physically reasonable.If we eliminateRh and Sv and add the total population equations, then we finally have(4)dShdt∗=μhNh1-NhKh-βhShIv-αhSh,dIhdt∗=βhShIv-αhIh-ρhIh-γhIh,dIvdt∗=βvNv-IvIh-αvIv,dNhdt∗=μhNh1-NhKh-αhNh-ρhIh,dNvdt∗=μvNv1-NvKv-αvNv.


### 2.1. Basic Properties

Since the model system (1) involves human and mosquito populations, all its associated variables and parameters are nonnegative.Theorem 1.
Solutions of the model system (1) with positive initial data will remain nonnegative for all time t≥0.Proof.
LetΩ={(Sh,Ih,Rh,Sv,Iv∈R5:Sh0>0,Ih0>0,Rh0>0,Iv0>0,Sv0>0)}. Then it follows from the second equation of malaria model (1) that (5)dShdt=μhNh1-NhKh-ShβhIv-αhSh≥-ShβhIv+αh,so that (6)dShSh≥-βhIv+αhdt.Hence, (7)Sht≥Sh0exp⁡-βhIv-αht≥0.Similarly, it can be shown that Ih≥0, Rh≥0, Sv≥0, and Iv≥0 for all time t≥0.
This completes the proof.


#### 2.1.1. Invariant Region

The malaria model (1) will be analyzed in biologically feasible region. Thus, the feasible solutions set for the model written by (8)Ω=Sh,Ih,Rh,Sv,Iv∈R+5:Sh+Ih+Rh≤Khμh-αhμh,Sv+Iv≤Kvμv-αvμvis positively invariant and then the model is biologically meaningful and mathematically well posed in the domain Ω. The proof is omitted for simplicity.


## 2.1. Basic Properties

Since the model system (1) involves human and mosquito populations, all its associated variables and parameters are nonnegative.Theorem 1.
Solutions of the model system (1) with positive initial data will remain nonnegative for all time t≥0.Proof.
LetΩ={(Sh,Ih,Rh,Sv,Iv∈R5:Sh0>0,Ih0>0,Rh0>0,Iv0>0,Sv0>0)}. Then it follows from the second equation of malaria model (1) that (5)dShdt=μhNh1-NhKh-ShβhIv-αhSh≥-ShβhIv+αh,so that (6)dShSh≥-βhIv+αhdt.Hence, (7)Sht≥Sh0exp⁡-βhIv-αht≥0.Similarly, it can be shown that Ih≥0, Rh≥0, Sv≥0, and Iv≥0 for all time t≥0.
This completes the proof.


### 2.1.1. Invariant Region

The malaria model (1) will be analyzed in biologically feasible region. Thus, the feasible solutions set for the model written by (8)Ω=Sh,Ih,Rh,Sv,Iv∈R+5:Sh+Ih+Rh≤Khμh-αhμh,Sv+Iv≤Kvμv-αvμvis positively invariant and then the model is biologically meaningful and mathematically well posed in the domain Ω. The proof is omitted for simplicity.


## 2.1.1. Invariant Region

The malaria model (1) will be analyzed in biologically feasible region. Thus, the feasible solutions set for the model written by (8)Ω=Sh,Ih,Rh,Sv,Iv∈R+5:Sh+Ih+Rh≤Khμh-αhμh,Sv+Iv≤Kvμv-αvμvis positively invariant and then the model is biologically meaningful and mathematically well posed in the domain Ω. The proof is omitted for simplicity.


## 3. Model Analysis

To analyze the malaria model in system (4), we use the normalized quantities instead of the actual populations. Since Nh and Nv may vary, these scales are not suitable for use in the scaling. However, the typical choice for logistic models is to use the sustainable populations Kh and Kv for the scales. In the present case, we shall also consider varying Kh and Kv. It is, therefore, convenient to write Kh=Kh0ght and Kv=Kv0gvt, where Kh0 and Kv0 are typical sizes and where gh and gv take care of the time variations. At the moment, we just assume that ght=gvt=1. We shall scale the time t∗ with the quantity 1/μh by setting t=μht∗. The scaling is then Sh=Kh0sh, Ih=Kh0ih, Rh=Kh0rh, Nh=Kh0nh, Sv=Kv0sv, Iv=Kv0iv, Nv=Kv0nv, and t∗=(1/μh)t. The dimensionless parameters of the model become β=βhNv/μh, α=αh/μh, γ=(ρh+γh)/μh, ν=βvNh/μv, δ=αv/μv, and ε=μh/μv≪1.The scaled equations then become(9)dnhdt=nh1-nh-αnh-ρih,dshdt=nh1-nh-βshiv-αsh,dihdt=βshiv-α+ρ+γih,εdnvdt=nv1-nv-δnv,εdivdt=νnv-ivih-δiv,subject to suitable initial conditions, (10)nh0=nh0,ih0=ih0,sh0=sh0,nv0=nv0,iv0=iv0.


### 3.1. Stability of Disease-Free Equilibrium and Reproduction Number

Our model (9) admits two disease-free equilibria, namely, E0=(0,0,0,0,0) and E1=(1-α,1-α,0,1-δ,0), where α and δ lie between 0 and 1, and a unique endemic equilibrium, E2=(nh∗,sh∗,ih∗,nv∗,iv∗). The equilibria of the system (9) are obtained by setting the right side equal to zero.The basic reproduction number,R0, is the single most important parameter in epidemiological modeling. It measures the average number of the secondary infections caused by a single infective in an entirely susceptible population during its whole infectious period [26]. To derive the basic reproduction number R0 of model (9), we use the next generation matrix approach described in [27–29]. The infected compartments of system (9) are ih and iv. Following [29], the new infection matrix F and the transition matrix V are given, respectively, by(11)F=0β1-αν1-δ0,V=α+ρ+γ00δ.Hence, the basic reproduction number,R0, is the dominant eigenvalue of the next generation matrix FV-1 and becomes(12)R0=ρFV-1,=β1-αα+ρ+γν1-δδ=R0vh×R0hv.From (12), it is noted that the reproduction number depends on the product of the number of humans that one mosquito infects through its infectious lifetime, R0vh, and the number of mosquitoes that one human infects through its infectious lifetime, R0hv.Theorem 2.
The disease-free equilibrium point,E0, is locally asymptotically stable if all eigenvalues of the characteristic equation of the variational matrix lie below zero.Proof.
At the equilibrium point,E0, the variational matrix is given by (13)JE0=1-α0-ρ001-α00000-α+ρ+γ000001-δ00000-δ.The characteristic equation may be written as det⁡J(E0)-λI=0. It implies (14)1-α-λ-α-λ-α-ρ-γ-λ1-δ-λ-δ-λ=0.Clearly, we have (15)λ1=1-α>0,asα<1,λ2=-α<0,λ3=-α+ρ+γ<0,λ4=1-δ>0asδ<1,λ5=-δ<0.Thus, this shows that the solution, E0, is unstable since λ1 and λ4 lie above zero.Theorem 3.
The disease-free equilibrium point,E1, is locally asymptotically stable if R0<1.Proof.
The variational matrix at the equilibrium point,E1, becomes (16)JE1=α-10-ρ002α-1-α00βα-100-α+ρ+γ0β1-α000δ-1000ν1-δ0-δ.Thus, the characteristic equation of the variational matrix is given by(17)α-1-λ-α-λδ-1-λλ2+a1λ+a0=0,where(18)a0=δα+ρ+γ1-R02,a1=δ+α+γ+ρ.The characteristic polynomial in (17) has roots λ1=α-1, λ2=δ-1, λ3=-α, with negative real parts since 0<α,δ<1. By Routh-Hurwitz criterion [28], the other roots λ4 and λ5 have negative real parts if both a0 and a1 lie above zero. From the second equation of (18), a1>0, and from the first equation of (18), a0>0, when R0<1. Hence, the disease-free equilibrium point, E1, is locally asymptotically stable.Theorem 4.
The disease-free equilibrium point,E1, is globally asymptotically stable in Ω if R0≤1; otherwise it is unstable.Proof.
Consider the Lyapunov function(19)Vih,iv=aih+biv,where (20)a=ν1-δδα+γ+ρ,b=εδ.The time derivative of the function V along the solutions of (9) becomes (21)V˙ih,iv=adihdt+bdivdt=aβshiv-α+γ+ρih+bνnv-ivih-δivε=aβ1-α-bδεiv+bνε1-δ-iv-aα+γ+ρih=βν1-α1-δδα+γ+ρ-1iv-νδihiv=R02-1iv-νδihiv≤0.Thus V˙≤0 if R0≤1 and the equality V˙=0 holds if and only if ih=iv=0. Therefore, the largest compact invariant set in {(ih,iv)∈Ω:V˙=0} is the singleton {E1}, where E1 is the disease-free equilibrium. LaSalle’s Invariant Principle [30] implies that E1 is globally asymptotically stable in Ω.


### 3.2. Stability of Endemic Equilibrium

To find the endemic equilibriumE2, we shall keep the assumptions about nv∗ and iv∗ from the singular perturbation equations (see the fourth and fifth equations of system (9)) and then focus on the first three equations of system (9). We consider the equilibrium solutions, using ih∗ as the basic quantity. From the third equation of (9), we have an expression for sh∗:(22)sh∗=α+γ+ρβν1-δδ+νih∗.Addition of the second and third equations of system (9) gives(23)nh∗-γαih∗-ih∗-sh∗=0.Now, let us consider the first equation of (9) which connects nh∗ and ih∗. That is,(24)nh∗1-nh∗-αnh∗-ρih∗=0.Equation (24) has only real and positive solutions for nh∗ if ρih∗<(1-α)/22, since the maximum value of nh∗1-nh∗-αnh∗ is (1-α)2/4. If ρih∗<(1-α)2/4, then (24) has two solutions:(25)nh∗+,nh∗-=1-α2±1-α24-ρih∗,for each value of ρih∗.One might wonder what happens when the inequality is violated andρih∗>(1-α)/22. The solution of (24) will then be majorized by the equation (26)dydt=y1-y-αy-1-α22whenρih∗=1-α22,which has only one unstable equilibrium point at y=(1-α)/2 and otherwise tends to 0. Moreover, it is clear that nh∗ needs to be larger or equal to ih∗. Let us, therefore, consider nh∗+ and nh∗- in the light of this restriction. The various situations are best described in terms of the graph shown in Figure 2.Figure 2
Graph illustrating the behaviour of the solutions of the equation betweennh∗ and ih∗.The parabola is the locus of the solutions fornh∗+ and nh∗- when 0≤ρih∗≤1-α2/4. The line, ih∗=(1/ρ)ρih∗, defines the limit of the region for solutions fulfilling ih∗≤nh∗. In the graph, the upper solution (upper red dot) is acceptable since nh∗+>ih∗, whereas the lower solution is outside the region and hence unacceptable. Further inspection of the graph shows the following.(1)
If1-α≤ρ, both solutions, nh∗- and nh∗+, are larger than ih∗ (consider the line y2).(2)
Ifρ<1-α/2, only nh∗+ is larger than ih∗ when ih∗ is positive (consider the line y1).(3)
For1-α/2<ρ<1-α, one or both of the solutions are acceptable.The main conclusion is that there are acceptable solutions with respect to size for allih∗ where 0≤ρih∗≤1-α2/4.Assume thatρih∗<1-α2/4, where only nh∗+ is acceptable with respect to size. Then substituting expressions (22) and (25) into the equation in (23) and simplifying lead to the following quadratic equation:(27)aih∗2+bih∗+c=0,where(28)a=1+γα+α+γ+ρβ1-δ2,b=1+γα+α+γ+ρβ1-δδα+γ+ρβν1-δ2-R02+ρ,c=δα+γ+ρβν1-δ21-R0+1-α22.From (27), it can easily be seen that a>0. Further, if R0>1, then c<0. Thus, the number of possible positive real roots of (27) can depend on the signs of b. This can be analyzed using the Descartes rule of signs on the quadratic polynomial (27). The different possibilities for the roots of (27) are tabulated in Table 1.Table 1
Number of possible positive real roots of (27) for R0<1 and R0>1.
Cases a b c R 0 Number of sign changes Number of possible positive real roots 1 + + + <1 0 0 2 + + − >1 1 1 3 + − − >1 1 1Thus, the malaria model has a unique endemic equilibrium ifR0>1 and whenever cases 2 and 3 are satisfied. Hence, the endemic equilibrium then becomes(29)nh∗=1+γα+α+γ+ρβ1-δih∗+δα+γ+ρβν1-δ,sh∗=δα+γ+ρβν1-δ+α+γ+ρβ1-δih∗,nv∗=1-δ,iv∗=1-δνih∗δ+νih∗,and ih∗ is the unique positive root of (27).Theorem 5.
IfR0>1, then the endemic equilibrium E2 of the malaria model (9) is globally asymptotically stable in Ω.Proof.
We shall propose the Lyapunov function(30)Vnh,sh,ih,nv,iv=a1nh-nh∗-nh∗ln⁡nhnh∗+a2sh-sh∗-sh∗ln⁡shsh∗+a3ih-ih∗-ih∗ln⁡ihih∗+a4nv-nv∗-nv∗ln⁡nvnv∗+a5iv-iv∗-iv∗ln⁡iviv∗,where(31)a1=a2=a3=a4=1,a5=εν.The Lyapunov function V is continuous for all nh,sh,ih,nv,iv>0. The time derivative of the function V along the solutions of system (9) becomes(32)dVdt=a11-nh∗nhdnhdt+a21-sh∗shdshdt+a31-ih∗ihdihdt+a41-nv∗nvdnvdt+a51-iv∗ivdivdt=a11-nh∗nhnh1-nh-αnh-ρih+a21-sh∗shnh1-nh-βshiv-αsh+a31-ih∗ihβshiv-α+γ+ρih+a41-nv∗nvnv1-nv-δnvε+a51-iv∗ivνnv-ivih-δivε.From the equilibrium point of the malaria model (9), we have the following relations:(33)α=nh∗1-nh∗+ρih∗nh∗,α+γ+ρ=βsh∗iv∗ih∗,δ=1-nv∗,ν=δiv∗nv∗-iv∗ih∗,nh∗1-nh∗=βsh∗iv∗+αsh∗.By adding and subtracting βsh∗iv∗,βsh∗iv∗(nhiv∗/nh∗iv) and using (31) and (33) in (32), after intensive simplification, we have(34)dVdt=nh1-nh2-nh∗nh-sh∗sh+αsh∗2-nhnh∗-shsh∗+βsh∗iv∗3-ihih∗-nhiv∗nh∗iv+shsh∗ih∗ihiviv∗+βsh∗iv∗1-iv∗iviviv∗-nhnh∗+ρih∗nh∗nh-1nhnh∗+ihih∗+nv∗iv∗nvihnv∗ih∗-iviv∗1-iv∗iv-1εnv-nv∗2-ih∗iv∗1-ihih∗1-iviv∗.Since the arithmetic mean is greater than or equal to the geometric mean, then we have(35)2-nh∗nh-sh∗sh≤0,2-nhnh∗-shsh∗≤0,3-ihih∗-nhiv∗nh∗iv+shsh∗ih∗ihiviv∗≤0.Also,(36)1-iv∗iviviv∗-nhnh∗≤0,if(37)iviv∗≤nhnh∗wheniv≥iv∗,iviv∗≥nhnh∗wheniv≤iv∗,nh∗nh-1nhnh∗+ihih∗≤0,ifnh≥nh∗.
Furthermore,(38)nvihnv∗ih∗-iviv∗1-iv∗iv≤0,if (39)nvihnv∗ih∗≤iviv∗wheniv∗≤iv,nvihnv∗ih∗≥iviv∗wheniv∗≥iv.Hence, it follows from (34), (35), (36), and (38) that dV/dt≤0 in Ω. Thus, the equality dV/dt=0 holds only when nh=nh∗, sh=sh∗, ih=ih∗, nv=nv∗, and iv=iv∗. Therefore, the largest compact invariant set in {nh,sh,ih,nv,iv∈Ω:V˙=0} is the singleton {E2}, where E2 is the endemic equilibrium. From the LaSalle’s invariant principle [30], the unique equilibrium E2 of system (9) is globally asymptomatically stable for R0>1.


### 3.3. Sensitivity Analysis

To understand the relative importance of parameters which are responsible for transmission and prevalence of malaria disease, described in the model (9), we perform a sensitivity analysis. Sensitivity indices help us to measure the relative change in a state variable while a parameter changes. The normalized sensitivity index of a variable to a parameter is the ratio of the relative change in the variable to the relative change in the parameter [27]. We calculate the sensitivity indices of R0 to assess which parameter has a great impact on R0 and hence the greatest effect in determining whether the disease dies out or persists with population.LetP be the generic parameter of model (9). We, now only, derive the normalized sensitivity index of R0 to each of the parameters involved in R0, defined by the ratio of the relative change in R0 to the relative change in the parameter P; that is, (40)ΠPR0=PR0∂R0∂P.This index shows how sensitive R0 is to a change in the parameter P. We notice that (41)ΠβR0=βR0∂R0∂β=βR0ν1-α1-δδα+γ+ρ∂β∂β=12.This indicates that ΠβR0 does not depend on any parameter value. Similarly, for the other parameters, we have (42)ΠνR0=12,ΠαR0=-βνα1-δ1+γ+ρ2R02δα+γ+ρ2ΠγR0=-γ2α+γ+ρ,ΠρR0=-ρ2α+γ+ρ,ΠδR0=-βν1-α2R02δα+γ+ρ2.We evaluate the above sensitivity indices, in Table3, using the parameter values in Table 2. The basic reproduction number, R0, is most sensitive to the contact rates of human to vector and vector to human, with πβR0=0.5000 and πνR0=0.5000 as it can be seen in Table 3. This shows that any increase (decrease) by 10% in β or ν will increase (decrease) by 5% in R0. The other parameters with highest sensitivity indices are δ, with πδR0=-0.6266, and ρ, with πρR0=-0.3812. Increasing (decreasing) δ by 10% will decrease (increase) in R0 by 6.266% and increasing (decreasing) ρ by 10% will decrease (increase) in R0 by 3.812% or vice versa. The rest of parameters, γ and α, have less significant effect in R0.Table 2
Some parameter values of the malaria model; all parameters are nondimensional.
Parameter Baseline value Source β 0.0220 [31] ν 0.4800 [31] γ 0.1096 [32] δ 0.2538 [31] α 0.1455 [31] ρ 0.8182 [31]Table 3
Sensitivity indices ofR0 to parameters for the malaria model.
Parameter Sensitivity index β +0.5000 ν +0.5000 γ - 0.0511 δ - 0.6266 α - 0.1528 ρ - 0.3812In conclusion, the vector death rate, the human induced death rate, and the contact rates are important parameters in the model which have a significant impact on prevalence and transmission of the malaria disease; these parameters are able to control so that an intensive effort/work has to be done to eradicate the malaria disease from the population. Furthermore, one can understand from the sensitivity indices that vector control is the most effective control strategy.


## 3.1. Stability of Disease-Free Equilibrium and Reproduction Number

Our model (9) admits two disease-free equilibria, namely, E0=(0,0,0,0,0) and E1=(1-α,1-α,0,1-δ,0), where α and δ lie between 0 and 1, and a unique endemic equilibrium, E2=(nh∗,sh∗,ih∗,nv∗,iv∗). The equilibria of the system (9) are obtained by setting the right side equal to zero.The basic reproduction number,R0, is the single most important parameter in epidemiological modeling. It measures the average number of the secondary infections caused by a single infective in an entirely susceptible population during its whole infectious period [26]. To derive the basic reproduction number R0 of model (9), we use the next generation matrix approach described in [27–29]. The infected compartments of system (9) are ih and iv. Following [29], the new infection matrix F and the transition matrix V are given, respectively, by(11)F=0β1-αν1-δ0,V=α+ρ+γ00δ.Hence, the basic reproduction number,R0, is the dominant eigenvalue of the next generation matrix FV-1 and becomes(12)R0=ρFV-1,=β1-αα+ρ+γν1-δδ=R0vh×R0hv.From (12), it is noted that the reproduction number depends on the product of the number of humans that one mosquito infects through its infectious lifetime, R0vh, and the number of mosquitoes that one human infects through its infectious lifetime, R0hv.Theorem 2.
The disease-free equilibrium point,E0, is locally asymptotically stable if all eigenvalues of the characteristic equation of the variational matrix lie below zero.Proof.
At the equilibrium point,E0, the variational matrix is given by (13)JE0=1-α0-ρ001-α00000-α+ρ+γ000001-δ00000-δ.The characteristic equation may be written as det⁡J(E0)-λI=0. It implies (14)1-α-λ-α-λ-α-ρ-γ-λ1-δ-λ-δ-λ=0.Clearly, we have (15)λ1=1-α>0,asα<1,λ2=-α<0,λ3=-α+ρ+γ<0,λ4=1-δ>0asδ<1,λ5=-δ<0.Thus, this shows that the solution, E0, is unstable since λ1 and λ4 lie above zero.Theorem 3.
The disease-free equilibrium point,E1, is locally asymptotically stable if R0<1.Proof.
The variational matrix at the equilibrium point,E1, becomes (16)JE1=α-10-ρ002α-1-α00βα-100-α+ρ+γ0β1-α000δ-1000ν1-δ0-δ.Thus, the characteristic equation of the variational matrix is given by(17)α-1-λ-α-λδ-1-λλ2+a1λ+a0=0,where(18)a0=δα+ρ+γ1-R02,a1=δ+α+γ+ρ.The characteristic polynomial in (17) has roots λ1=α-1, λ2=δ-1, λ3=-α, with negative real parts since 0<α,δ<1. By Routh-Hurwitz criterion [28], the other roots λ4 and λ5 have negative real parts if both a0 and a1 lie above zero. From the second equation of (18), a1>0, and from the first equation of (18), a0>0, when R0<1. Hence, the disease-free equilibrium point, E1, is locally asymptotically stable.Theorem 4.
The disease-free equilibrium point,E1, is globally asymptotically stable in Ω if R0≤1; otherwise it is unstable.Proof.
Consider the Lyapunov function(19)Vih,iv=aih+biv,where (20)a=ν1-δδα+γ+ρ,b=εδ.The time derivative of the function V along the solutions of (9) becomes (21)V˙ih,iv=adihdt+bdivdt=aβshiv-α+γ+ρih+bνnv-ivih-δivε=aβ1-α-bδεiv+bνε1-δ-iv-aα+γ+ρih=βν1-α1-δδα+γ+ρ-1iv-νδihiv=R02-1iv-νδihiv≤0.Thus V˙≤0 if R0≤1 and the equality V˙=0 holds if and only if ih=iv=0. Therefore, the largest compact invariant set in {(ih,iv)∈Ω:V˙=0} is the singleton {E1}, where E1 is the disease-free equilibrium. LaSalle’s Invariant Principle [30] implies that E1 is globally asymptotically stable in Ω.


## 3.2. Stability of Endemic Equilibrium

To find the endemic equilibriumE2, we shall keep the assumptions about nv∗ and iv∗ from the singular perturbation equations (see the fourth and fifth equations of system (9)) and then focus on the first three equations of system (9). We consider the equilibrium solutions, using ih∗ as the basic quantity. From the third equation of (9), we have an expression for sh∗:(22)sh∗=α+γ+ρβν1-δδ+νih∗.Addition of the second and third equations of system (9) gives(23)nh∗-γαih∗-ih∗-sh∗=0.Now, let us consider the first equation of (9) which connects nh∗ and ih∗. That is,(24)nh∗1-nh∗-αnh∗-ρih∗=0.Equation (24) has only real and positive solutions for nh∗ if ρih∗<(1-α)/22, since the maximum value of nh∗1-nh∗-αnh∗ is (1-α)2/4. If ρih∗<(1-α)2/4, then (24) has two solutions:(25)nh∗+,nh∗-=1-α2±1-α24-ρih∗,for each value of ρih∗.One might wonder what happens when the inequality is violated andρih∗>(1-α)/22. The solution of (24) will then be majorized by the equation (26)dydt=y1-y-αy-1-α22whenρih∗=1-α22,which has only one unstable equilibrium point at y=(1-α)/2 and otherwise tends to 0. Moreover, it is clear that nh∗ needs to be larger or equal to ih∗. Let us, therefore, consider nh∗+ and nh∗- in the light of this restriction. The various situations are best described in terms of the graph shown in Figure 2.Figure 2
Graph illustrating the behaviour of the solutions of the equation betweennh∗ and ih∗.The parabola is the locus of the solutions fornh∗+ and nh∗- when 0≤ρih∗≤1-α2/4. The line, ih∗=(1/ρ)ρih∗, defines the limit of the region for solutions fulfilling ih∗≤nh∗. In the graph, the upper solution (upper red dot) is acceptable since nh∗+>ih∗, whereas the lower solution is outside the region and hence unacceptable. Further inspection of the graph shows the following.(1)
If1-α≤ρ, both solutions, nh∗- and nh∗+, are larger than ih∗ (consider the line y2).(2)
Ifρ<1-α/2, only nh∗+ is larger than ih∗ when ih∗ is positive (consider the line y1).(3)
For1-α/2<ρ<1-α, one or both of the solutions are acceptable.The main conclusion is that there are acceptable solutions with respect to size for allih∗ where 0≤ρih∗≤1-α2/4.Assume thatρih∗<1-α2/4, where only nh∗+ is acceptable with respect to size. Then substituting expressions (22) and (25) into the equation in (23) and simplifying lead to the following quadratic equation:(27)aih∗2+bih∗+c=0,where(28)a=1+γα+α+γ+ρβ1-δ2,b=1+γα+α+γ+ρβ1-δδα+γ+ρβν1-δ2-R02+ρ,c=δα+γ+ρβν1-δ21-R0+1-α22.From (27), it can easily be seen that a>0. Further, if R0>1, then c<0. Thus, the number of possible positive real roots of (27) can depend on the signs of b. This can be analyzed using the Descartes rule of signs on the quadratic polynomial (27). The different possibilities for the roots of (27) are tabulated in Table 1.Table 1
Number of possible positive real roots of (27) for R0<1 and R0>1.
Cases a b c R 0 Number of sign changes Number of possible positive real roots 1 + + + <1 0 0 2 + + − >1 1 1 3 + − − >1 1 1Thus, the malaria model has a unique endemic equilibrium ifR0>1 and whenever cases 2 and 3 are satisfied. Hence, the endemic equilibrium then becomes(29)nh∗=1+γα+α+γ+ρβ1-δih∗+δα+γ+ρβν1-δ,sh∗=δα+γ+ρβν1-δ+α+γ+ρβ1-δih∗,nv∗=1-δ,iv∗=1-δνih∗δ+νih∗,and ih∗ is the unique positive root of (27).Theorem 5.
IfR0>1, then the endemic equilibrium E2 of the malaria model (9) is globally asymptotically stable in Ω.Proof.
We shall propose the Lyapunov function(30)Vnh,sh,ih,nv,iv=a1nh-nh∗-nh∗ln⁡nhnh∗+a2sh-sh∗-sh∗ln⁡shsh∗+a3ih-ih∗-ih∗ln⁡ihih∗+a4nv-nv∗-nv∗ln⁡nvnv∗+a5iv-iv∗-iv∗ln⁡iviv∗,where(31)a1=a2=a3=a4=1,a5=εν.The Lyapunov function V is continuous for all nh,sh,ih,nv,iv>0. The time derivative of the function V along the solutions of system (9) becomes(32)dVdt=a11-nh∗nhdnhdt+a21-sh∗shdshdt+a31-ih∗ihdihdt+a41-nv∗nvdnvdt+a51-iv∗ivdivdt=a11-nh∗nhnh1-nh-αnh-ρih+a21-sh∗shnh1-nh-βshiv-αsh+a31-ih∗ihβshiv-α+γ+ρih+a41-nv∗nvnv1-nv-δnvε+a51-iv∗ivνnv-ivih-δivε.From the equilibrium point of the malaria model (9), we have the following relations:(33)α=nh∗1-nh∗+ρih∗nh∗,α+γ+ρ=βsh∗iv∗ih∗,δ=1-nv∗,ν=δiv∗nv∗-iv∗ih∗,nh∗1-nh∗=βsh∗iv∗+αsh∗.By adding and subtracting βsh∗iv∗,βsh∗iv∗(nhiv∗/nh∗iv) and using (31) and (33) in (32), after intensive simplification, we have(34)dVdt=nh1-nh2-nh∗nh-sh∗sh+αsh∗2-nhnh∗-shsh∗+βsh∗iv∗3-ihih∗-nhiv∗nh∗iv+shsh∗ih∗ihiviv∗+βsh∗iv∗1-iv∗iviviv∗-nhnh∗+ρih∗nh∗nh-1nhnh∗+ihih∗+nv∗iv∗nvihnv∗ih∗-iviv∗1-iv∗iv-1εnv-nv∗2-ih∗iv∗1-ihih∗1-iviv∗.Since the arithmetic mean is greater than or equal to the geometric mean, then we have(35)2-nh∗nh-sh∗sh≤0,2-nhnh∗-shsh∗≤0,3-ihih∗-nhiv∗nh∗iv+shsh∗ih∗ihiviv∗≤0.Also,(36)1-iv∗iviviv∗-nhnh∗≤0,if(37)iviv∗≤nhnh∗wheniv≥iv∗,iviv∗≥nhnh∗wheniv≤iv∗,nh∗nh-1nhnh∗+ihih∗≤0,ifnh≥nh∗.
Furthermore,(38)nvihnv∗ih∗-iviv∗1-iv∗iv≤0,if (39)nvihnv∗ih∗≤iviv∗wheniv∗≤iv,nvihnv∗ih∗≥iviv∗wheniv∗≥iv.Hence, it follows from (34), (35), (36), and (38) that dV/dt≤0 in Ω. Thus, the equality dV/dt=0 holds only when nh=nh∗, sh=sh∗, ih=ih∗, nv=nv∗, and iv=iv∗. Therefore, the largest compact invariant set in {nh,sh,ih,nv,iv∈Ω:V˙=0} is the singleton {E2}, where E2 is the endemic equilibrium. From the LaSalle’s invariant principle [30], the unique equilibrium E2 of system (9) is globally asymptomatically stable for R0>1.


## 3.3. Sensitivity Analysis

To understand the relative importance of parameters which are responsible for transmission and prevalence of malaria disease, described in the model (9), we perform a sensitivity analysis. Sensitivity indices help us to measure the relative change in a state variable while a parameter changes. The normalized sensitivity index of a variable to a parameter is the ratio of the relative change in the variable to the relative change in the parameter [27]. We calculate the sensitivity indices of R0 to assess which parameter has a great impact on R0 and hence the greatest effect in determining whether the disease dies out or persists with population.LetP be the generic parameter of model (9). We, now only, derive the normalized sensitivity index of R0 to each of the parameters involved in R0, defined by the ratio of the relative change in R0 to the relative change in the parameter P; that is, (40)ΠPR0=PR0∂R0∂P.This index shows how sensitive R0 is to a change in the parameter P. We notice that (41)ΠβR0=βR0∂R0∂β=βR0ν1-α1-δδα+γ+ρ∂β∂β=12.This indicates that ΠβR0 does not depend on any parameter value. Similarly, for the other parameters, we have (42)ΠνR0=12,ΠαR0=-βνα1-δ1+γ+ρ2R02δα+γ+ρ2ΠγR0=-γ2α+γ+ρ,ΠρR0=-ρ2α+γ+ρ,ΠδR0=-βν1-α2R02δα+γ+ρ2.We evaluate the above sensitivity indices, in Table3, using the parameter values in Table 2. The basic reproduction number, R0, is most sensitive to the contact rates of human to vector and vector to human, with πβR0=0.5000 and πνR0=0.5000 as it can be seen in Table 3. This shows that any increase (decrease) by 10% in β or ν will increase (decrease) by 5% in R0. The other parameters with highest sensitivity indices are δ, with πδR0=-0.6266, and ρ, with πρR0=-0.3812. Increasing (decreasing) δ by 10% will decrease (increase) in R0 by 6.266% and increasing (decreasing) ρ by 10% will decrease (increase) in R0 by 3.812% or vice versa. The rest of parameters, γ and α, have less significant effect in R0.Table 2
Some parameter values of the malaria model; all parameters are nondimensional.
Parameter Baseline value Source β 0.0220 [31] ν 0.4800 [31] γ 0.1096 [32] δ 0.2538 [31] α 0.1455 [31] ρ 0.8182 [31]Table 3
Sensitivity indices ofR0 to parameters for the malaria model.
Parameter Sensitivity index β +0.5000 ν +0.5000 γ - 0.0511 δ - 0.6266 α - 0.1528 ρ - 0.3812In conclusion, the vector death rate, the human induced death rate, and the contact rates are important parameters in the model which have a significant impact on prevalence and transmission of the malaria disease; these parameters are able to control so that an intensive effort/work has to be done to eradicate the malaria disease from the population. Furthermore, one can understand from the sensitivity indices that vector control is the most effective control strategy.


## 4. Application of the Model

In this section, we present more simulations illustrating the abrupt and periodic variations of the model. We fix a reasonable parameter values of the model for numerical simulations.We allow the mosquito sustainable level,Kv(t)=Kv0gv(t), noted in Section 3, to vary with respect to time. However, we first keep Kh fixed in order to investigate the impact of fast variation in Kv on the human populations. Periodic variations in Kv are shown in these plots for different periods. Plots of the abrupt changes in the sustainable populations, Kh=Kh0gh(t) (see in Section 3) and Kv, are located in Figures 3–7, whereas plots of periodic variations are shown in Figures 8 and 9.Figure 3
Numerical simulation for the fractions of human and vector population versus time with constant parameter values,ε=0.01, α=0.01, β=2, ρ=0.01, γ=0.1, δ=0.4, and ν=0.5, and initial values of the fractions of human host and mosquito vector nh0=1.0, sh0=0.99, ih0=0.01, nv0=0.8, iv0=0.5, and Kh=1. Steps down the vector population by 50%.Figure 4
Numerical simulation for the fractions of human and vector population versus time with constant parameter values the same as in Figure3 and steps up the vector population by 50%.Figure 5
Numerical simulation for the fractions of human and vector population versus time with constant parameter values the same as in Figure3 and steps up the human population by 50%.Figure 6
Numerical simulation for the fractions of human and vector population versus time with constant parameter values the same as in Figure3 and increasing the human population by 100%.Figure 7
Numerical simulation for the fractions of human and vector population versus time with constant parameter values the same as in Figure3 and ε=0.001, and steps down the human population by 50% in the form of a constant drop over a time interval of length 100.Figure 8
Numerical simulation for the fractions of human and vector population versus time with constant parameter values,ε=0.001, α=0.01, β=0.7, ρ=0.01, γ=0.1, δ=0.4, and ν=0.5, and initial values of the fractions of human host and mosquito vector, nh0=1.0, sh0=0.99, ih0=0.01, nv0=0.8, and iv0=0.5, and the sustainable levels Kh=1 and periodic variations in Kv given by Kv=1+0.5sin⁡(2πt/15).Figure 9
Periodic variations inKv with Kv=1+0.3sin⁡(πt/6), for Kh=1, and the other parameters and initial values remain fixed as in Figure 8 apart from β=0.5 over the time interval [0100].Abrupt changes in the human and mosquito populations may, for example, be due to intensive spraying of the mosquitoes some massive emigration (refugee camps) or immigration for the humans. In Figures3 and 4, steps down and up in vector sustainable population about 50% are plotted. These plots show that the transition occurs very fast and the system adjusts quickly to the new equilibrium. In Figures 5–7 increases about 50% and 100% and decrease about 50% in Kh are shown. Step change in humans needs caution since it may lead to unphysical solutions. In Figure 5, after a transient, the solution converges to the new equilibrium. An increase in Kh in Figure 6 may lead to an increase in the fraction of the human susceptible population for small time intervals, but not much dramatic change is shown. However, the slow change in human population after transient is shown in Figure 7. In the plots of periodic variation, one observes that the fast variation is quickly adapted by the vector population. This is because of the fast time scale for the mosquito population. Most of the figures show that the humans follow a slow variations relative to the vectors. Therefore, it is possible to say that fast variations in Kv do not imply large variation in human population. In general, for periods shorter than 1/μh, the human population do not in practice show the variations in the vector populations, but for long periods they do, but apparently weaker.


## 5. Conclusions

In this work, we developed and analyzed a logistic malaria model to study the global stability of both disease-free and endemic equilibrium points. Mathematically, we formulated a five-dimensional system of deterministic ordinary differential equations and defined the domain where the model is epidemiologically feasible and mathematically well-posed. The model used the next generation matrix approach to obtain an explicit formula for a reproduction number,R0, which is the expected number of secondary cases produced by a single infectious individual during its entire period of infectiousness in a fully susceptible populations.Qualitative analysis of the model determines stability analysis of the equilibrium points. Accordingly, we obtained two diseases-free equilibrium pointsE0 and E1. The equilibrium point, E0, is unstable and unphysical, while the equilibrium point, E1, becomes both locally and globally stable whenever R0<1 and R0≤1, respectively. We also have shown that the endemic equilibrium point, E2, is globally asymptotically stable if R0>1. Furthermore, sensitivity analysis of the model shows that the human induced death rate, the contact rates (human to mosquito or vice versa), and mosquito death rate have a significant effect on transmission and prevalence of the malaria disease. Moreover, numerical simulations are carried out in the application of the model to investigate how variations in the sustainable level of the vectors affect the human population. One can see from these simulations that fast variations in Kv do not lead to large variations in the human population.



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