# Malaria Transmission Model with Transmission-Blocking Drugs and a Time Delay

Authors: Huike Wu; Zhixing Hu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

A class of transmission-blocking drugs (TBDs) that block the transmission of parasites between humans and mosquitoes has recently been shown to be effective in controlling malaria transmission. In this paper, we develop a time-delay differential equation model for malaria using TBDs intervention, in which the human population consists of a treated class and a successfully treated class. In classifying the positive equilibria, the control reproduction numberRT was obtained and the forward and backward branching cases were explored. Then, by constructing a Lyapunov function, the disease-free equilibrium is globally asymptotically stable under certain conditions. In addition, when RT>1, the model exhibits Hopf bifurcation, the positive equilibrium becomes unstable from stable, and the model exhibits a periodic solution due to the change of time delay. On the other hand, it is concluded that the use of TBDs has a positive effect on disease control when the treatment rate and the efficacy of TBDs meet certain conditions. Finally, numerical simulation was used to observe the effect of treatment rate and the efficacy of TBDs on RT, and it was found that the increase in the efficacy of TBDs had a more pronounced effect on disease control compared to treatment rate.


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## Body

## 1. Introduction

Malaria is an insect-borne disease caused byPlasmodium infection through the bite of Anopheles mosquitoes or by importing the blood of a person with Plasmodium and is a major infectious disease that seriously endangers people’s health and life safety. The first symptoms of malaria usually appear 10–15 days after people are bitten by mosquitoes that have been infected with the virus. The disease mainly manifests as periodic regular attacks with generalized chills, fever, and excessive sweating, and after many long-term attacks, it can cause anemia and splenomegaly. Plasmodium falciparum can lead to the most serious diseases (see [1]). WHO once ranked malaria as one of the top three global public health problems, along with AIDS and Tuberculosis. Data from 1980 to 2010 show that malaria infection rates have declined, but mortality remains a concern. To date, malaria remains a serious global epidemic, with about 40% of the world’s population living in malaria-endemic regions, 90% of which are on the African continent. More than 2 million people die from malaria each year, especially children under five years of age, pregnant women, and those suffering from other diseases (see [2]). The tropics and subtropics are the most malaria-prone regions, forming a “malaria belt” around the equator that includes Latin America, sub-Saharan Africa, South Asia, and Southeast Asia. WHO data show that, in 2017, there were an estimated 219 million cases of malaria in 87 countries, and in 2019, there were an estimated 229 million malaria cases worldwide, with an estimated 409,000 malaria deaths.Early diagnosis and treatment of malaria can reduce disease and avert deaths, and since 2000, the expansion of vector control interventions has led to progress in malaria control, particularly in sub-Saharan Africa (see [3]). The best available treatments, particularly for falciparum malaria, are artemisinin-based combination therapies. However, these gains are jeopardized by the emergence of resistance to antimalarial drugs in Anopheles mosquitoes. Furthermore, some studies have shown an increase in resistance to artemisinin combination therapies paired with drugs (see [4, 5]). Rather than relying on vector control to prevent mosquito bites, the transmission of gametocytes (the intraerythrocytic sexual stages) to the mosquito is potentially more amenable to direct intervention because it is easily targetable within the human blood compartment (see [6]). Most antimalarial drugs are inactive against the sexual stage of P. falciparum, and for effective long-term malaria control, measures are needed to stop the transmission of P. falciparum between humans and mosquitoes. Several studies have shown that a potential approach to directly block parasite transmission is to target Plasmodium using transmission-blocking interventions (TBIs), which can be broadly categorized as transmission-blocking drugs (TBDs) or transmission-blocking vaccines (TBVs) that target parasitic stages (see [7]). These drugs can be divided into two categories, one targeting the parasitic stage exposed only in mosquitoes and the other targeting the parasitic stage exposed only in humans. For TBDs that target parasites in humans, drugs can kill the asexual phase of the parasite, stopping/reducing development to gametes, drugs target immature and mature gametes in humans directly, or drugs provide chemotherapeutic prophylaxis by direct action on Sporozoa, thus stopping infection. From these aspects, there are many scholars working on transmission-blocking drugs that can block the different stages of the parasite within the host. Antimalarial drugs with transmission-blocking activity have been prioritized, and several are in various stages of clinical development (see [8]).Exploring the dynamics of infectious diseases through mathematical modeling can be used to guide disease control. A number of mathematicians have explored the dynamics of malaria transmission between humans and mosquitoes by modeling differential equations. Following the studies of Ross (see [9]) and Macdonald (see [10]), a series of differential equation models for malaria have been proposed one after another from different perspectives (see [11–16]). In recent years, Ngwa proposed a deterministic differential equation model for malaria epidemics (see [17]), in which human and mosquito populations have SEIR and SEI structures, respectively, and most subsequent studies have followed this structure. Chitnis proposed a mathematical model for the transmission of malaria in populations and mosquitoes with ordinary differential equations (see [18]), and numerical simulations showed that, for larger disease lethality, subcritical (backward) bifurcation may occur when the basic reproduction number is equal to 1. In 2008, Chitnis identified important parameters of malaria transmission by sensitivity analysis of the mathematical model (see [19]). Ruan considered the incubation period of the parasite in humans and mosquitoes and proposed a delayed Ross-Macdonald model (see [20]), which demonstrated that the basic reproduction number is a decreasing function of two time delays and that the prevalence of infection can be reduced by extending the incubation period in humans or mosquitoes. In the context of some progress in new antimalarial drug research, a regional deterministic model assessed the effectiveness of a malaria transmission-blocking vaccine targeting the parasitic stage of mosquitoes (see [21]). In addition, a mathematical model assessed the role of gametocytes (the infectious stage of the malaria parasite) in the dynamics of malaria transmission (see [22]), which was extended to include some hypothetical therapeutic features of an imperfect vaccine.Mathematical models of drug in-host kinetics are often used to guide drug development, and these models often focus on assessing efficacy and response duration to guide patient treatment (see [23]). Based on this motivation, in this paper, we refer to the modeling ideas from the literature (see [24]) and add TBDs targeting parasite blockade in humans to the malaria model, assuming that a fraction of human infected patients are treated with TBDs and that a fraction of patients successfully block parasite transmission to mosquitoes after treatment with TBDs. In addition, usually, people develop the first symptoms of malaria after being infected with the virus for a period of time (see [25, 26]). Being undetected during the incubation period leads to an increased risk of malaria infection in humans by not treating the parasite promptly or using TBDs to block transmission to mosquitoes in a timely manner. Based on these two considerations, this paper develops a differential equation model considering TBDs and a time delay and is dedicated to analyzing the effect of the incubation period of Plasmodium in humans on the stability of the model’s equilibrium, as well as the effect of treatment with TBDs on malaria control in humans, and to assessing the range of treatment rate and the efficacy of TBDs that can contribute most to malaria control under certain conditions.


## 2. Model Description

In this model, the total human populationNh is divided into susceptible individuals Sh, infected individuals Ih, patients treated with TBDs Th, successful treatment with TBDs and not transmitting gamete cells to mosquito individuals Ph, and recovered individuals but partially lost their immunity Rh. The total mosquito population Nv is divided into susceptible mosquitoes Sv and infected mosquitoes Iv. At any time t, the human population has the expression Nht=Sht+Iht+Tht+Pht+Rht and the mosquito population has the expression Nvt=Svt+Ivt. The period from the time of human being infected with Plasmodium to the time of onset is recorded as the time delay τ. The model considered in this paper is(1)dShtdt=Λh+γhRht+υPht−rρvIvt−τNht−τSht−τe−dhτ−dhSht,dIhtdt=rρvIvt−τNht−τSht−τe−dhτ+κhψ2Tht−dh+δh+θh+αhIht,dThtdt=αhIht−ψ1Tht−κhψ2Tht−1−κhψ2Tht−dhTht,dPhtdt=ψ1Tht−υPht−dhPht,dRhtdt=θhIht+1−κhψ2Tht−γhRht−dhRht,dSvtdt=Λv−rρhIht+ξtTht+ξrRhtNhtSvt−dvSvt,dIvtdt=rρhIht+ξtTht+ξrRhtNhtSvt−dvIvt,where Λh and Λv are the input of susceptible humans and susceptible mosquitoes, respectively, and dh and dv are the natural mortality rates of humans and mosquitoes, respectively. The exponential term e−dhτ represents the human survival rate during the incubation period. γh is the immune loss rate after human recovery, and υ represents the rate of loss protection and transfer to the susceptible class. ψ1 is the percentage of patients who were successfully protected after TBDs treatment, and ψ2 is the percentage of patients who were not protected after TBDs treatment. Among the patients who were not protected after TBDs treatment, κhψ2 is the rate of being moved to infection class, and 1−κhψ2 is the rate of being moved to the recovery class. ψ1 and ψ2 are functions of pe (pe is the efficacy of TBDs), ψ1 is proportional to pe, and ψ2 is inversely proportional to pe, respectively, expressed as simple functions ψ1=ηhpe and ψ2=μh1−pe. ηh is the effective rate of TBDs treatment. μh is the rate of ineffective TBDs treatment. αh represents the rate of infected patients receiving TBDs treatment (abbreviated as treatment rate), δh is the human mortality rate due to disease, and θh is the human recovery rate after infection. fh=rρvIv/Nh is the human infection rate, where r is the average number of times a mosquito bites a human and ρv is the probability of human infection for each bite of mosquito. fv=rρhIh+ξtTh+ξrRh/Nh represents the infection rate of mosquitoes, and ρh represents the probability of mosquito infection per bite. ξt and ξr represent the relative infection rates respectively, and 0<ξr<ξt<1. Let β1h=dh+δh+θh+αh, assuming that the input of the susceptible population is greater than the natural human mortality rate Λh>dh; otherwise, the human will go extinct.LetC=C−τ,0,R+7 be a Banach space of continuous functions mapping the interval −τ,0 into R+7 with upper bound norm. According to the biological significance, the initial conditions of system (1) are given(2)Shθ=φ1θ,Ihθ=φ2θ,Thθ=φ3θ,Phθ=φ4θ,Rhθ=φ5θ,Ivθ=φ6θ,Nvθ=φ7θ,θ∈−τ,0.Among them(3)φ=φ1,φ2,φ3,φ4,φ5,φ6,φ7T∈C+,where(4)C+=φ=φ1,φ2,φ3,φ4,φ5,φ6,φ7T∈C:φi≥0,i=1,2,…,7.According to the basic theory of functional differential equation, system (1) has a unique solution which satisfies the initial conditions.According to the first five equations of system (1), for t≥0, there is(5)N˙h=S˙h+I˙h+T˙h+P˙h+R˙h=Λh−dhNh−δhIh.Thus,limt⟶+∞supNht≤Λh/dh≡Nhmax. Similarly, from the last two expressions of system (1), there is limt⟶+∞supNvt≤Λv/dv≡Nvmax. Therefore, the solution of system (1) is ultimately uniformly bounded.Let(6)Ω=Nh,Nv|0<Nh≤Nhmax,0<Nv≤Nvmax,and then all trajectories in the first quadrant of system (1) will enter or stay in Ω. Therefore, all variables on Ω are nonnegative and ultimately uniformly bounded, and the initial value problem of system (1) satisfies the biological significance. Next, the dynamic properties of system (1) are considered on Ω.In the third section, the control reproduction number and the classification of the equilibrium are calculated, which indicates that the system has a backward bifurcation. In the fourth section, the local stability and global stability of the disease-free equilibrium are discussed. In the fifth section, the influence of TBDs on the propagation of the model is discussed. In the sixth section, the local stability of the endemic equilibrium and the existence of Hopf bifurcation are analyzed. Finally, numerical simulation is used to verify the conclusions obtained in this paper.


## 3. The Existence of Equilibrium

In this section, the equilibria of the model are classified and the control reproduction number of the model is determined.When there is no infection, the system has a stable state. Supposing that the disease-free equilibrium isE0=Sh0,Ih0,Th0,Ph0,Rh0,Sv0,Iv0, it is easy to obtain E0=Λh/dh,0,0,0,0,Λv/dv,0.When infection exists, the endemic equilibriumE∗=Sh∗,Ih∗,Th∗,Ph∗,Rh∗,Sv∗,Iv∗ of system (1) satisfies the following equations:(7)Λh+γhRh∗+υPh∗−fh∗Sh∗e−dhτ−dhSh∗=0,fh∗Sh∗e−dhτ+κhψ2Th∗−β1hIh∗=0,αhIh∗−ψ1Th∗−κhψ2Th∗−1−κhψ2Th∗−dhTh∗=0,ψ1Th∗−υPh∗−dhPh∗=0,θhIh∗+1−κhψ2Th∗−γhRh∗−dhRh∗=0,Λv−fv∗Sv∗−dvSv∗=0,fv∗Sv∗−dvIv∗=0.From the first five equations, it follows that(8)Sh∗=K2Ih∗T1fh∗e−dhτ,Th∗=αhIh∗T1,Ph∗=αhψ1Ih∗T1T2,Rh∗=θhT1+1−κhαhψ2T1T3Ih∗,Ih∗=a1fha2fh+a3,where(9)T1=ψ1+ψ2+dh,T2=υ+dh,T3=γh+dh,K1=T1T3+ξtαhT3+ξrK5,K2=β1hT1−κhαhψ2>0,K3=κhψ2θh+1−κhψ2β1h,K4=rρvΛvdhΛhdv,K5=θhT1+αhψ21−κh,a1=ΛhT1T2T3e−dhτ>0,a2=T2T3K2e−dhτ−T2γhK5e−dhτ−υαhψ1T3e−dhτ>0,a3=T2T3dhK2>0.From the model, there are(10)Nh∗=Λh−δhIh∗dh,Nv∗=Λvdv.Substituting the expressions ofIh∗, Th∗,Rh∗, and Nh∗ in (8) and (10) into fv=rρhIh+ξtTh+ξrRh/Nh, then(11)fv∗=rρhdha1K1fh∗T1T3b1fh∗+T1T3a3Λh,where b1=a2Λh−a1δh>0 and K1=T1T3+ξtαhT3+ξrK5>0.It can be obtained from the last two equations of (7) that(12)Sv∗=Λvfv∗+dv,Iv∗=Λvfv∗dvfv∗+dv.Substituting the expressions ofIv∗ and Nh∗ in (12) and (10) into fh=rρvIv/Nh, then(13)fh∗=rρvΛvdha2fh∗+a3fv∗dvfv∗+dvb1fh∗+a3Λh.From (11) and (13), the quadratic equation of one variable about fh∗ is obtained as follows:(14)Ffh∗=l1fh∗2+l2fh∗+l3=0,where(15)l1=rρhdhdva1b1K1+T1T3dv2b12,l2=Λhdhdv2a1a2K2T3e−dhτe−dhτrρhdha1a3K1+2T1T3dvb1a3dhdva1a2K2T3−RT,l3=T1T3dv2Λh2a321−RT,RT=r2ρhρvΛvdhK1e−dhτΛhdv2K2T3.RT is defined as the control reproduction number of the model.It follows that the number of positive equilibria of system (1) depends on the number of positive roots of equation (14). For equation (14), there are F0=T1T3dv2Λh2a321−RT and l1>0. When RT>1, there are F0<0 and limfh∗⟶+∞Ffh∗>0. Thus, when RT>1, equation (14) must have a unique positive root, denoted as fh∗.Let(16)M=e−dhτrρhdha1a3K1+2T1T3dvb1a3dhdva1a2K2T3,N=Λhdhdv2a1a2K2T3e−dhτ,Q=T1T3dv2Λh2a32,and then M>0, N>0, and Q>0.WhenRT=1, there are l3=0 and Ffh∗=l1fh∗2+l2fh∗. Thus, when l2<0, namely, M<1, equation (14) has a unique positive root.WhenRT<1, there is F0>0. If M>1, equation (14) has no positive roots; positive roots of equation (14) are possible only when M<1. Further, for equation (14) to have positive roots, the condition M<RT needs to be satisfied.The discriminant ofFfh∗=0 is Δ=l22−4l1l3; rearrange it as a function of RT and call it ΔRT.Let(17)R∗≜MN2−2l1Q+2l12Q2+l1N2Q1−MN2.By simple computations, it hasΔ0>0, ΔM<0, ΔR∗=0, and Δ1>0. Thus, 0<M<R∗<1.IfR∗<RT<1, equation (14) has two positive roots, denoted as fh∗ and fh∗∗. If RT=R∗, equation (14) has a positive root, denoted as fh′. If 0<RT<R∗, equation (14) has no positive root. Based on the above analysis, the following theorem can be obtained.Theorem 1.
System (1) always has a disease-free equilibrium E0=Λh/dh,0,0,0,0,Λv/dv,0. In addition, system (1) also has the following endemic equilibrium:(i)
IfRT>1, system (1) has a unique positive equilibrium E∗=Sh∗,Ih∗,Th∗,Ph∗,Rh∗,Sv∗,Iv∗(ii)
IfM<1, there are three situations:(1)
IfRT=1, system (1) has a unique positive equilibrium E∗=Sh∗,Ih∗,Th∗,Ph∗,Rh∗,Sv∗,Iv∗(2)
IfR∗<RT<1, system (1) has two positive equilibria E∗=Sh∗,Ih∗,Th∗,Ph∗,Rh∗,Sv∗,Iv∗ and E∗∗=Sh∗∗,Ih∗∗,Th∗∗,Ph∗∗,Rh∗∗,Sv∗∗,Tv∗∗(3)
IfRT=R∗, system (1) has a unique positive equilibrium E′=Sh′,Ih′,Th′,Ph′,Rh′,Sv′,Iv′
The positive roots of equation (14) are expressed as(18)fh∗=−l2+Δ2l1,fh∗∗=−l2−Δ2l1,fh′=−l22l1.


## 4. Stability of the Disease-Free Equilibrium

In this section, the local stability of the disease-free equilibrium is discussed by using the characteristic equation of the linear approximation equation of system (1). In addition, the global stability of the disease-free equilibrium is obtained by constructing the Lyapunov function when the time delay is zero.


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

For the local stability of the disease-free equilibriumE0, the following conclusions are obtained.Theorem 2.
IfRT<1, the disease-free equilibrium E0 is locally asymptotically stable for any τ≥0; if RT>1, E0 is unstable for any τ≥0.Proof.
It is shown that the characteristic equation of system (1) at E0 is(19)L0λ,τ=λ4+A3λ3+A2λ2+A1λ+A0+e−λτB2λ2+B1λ+B0=0,where(20)A3=T1+T3+dv+β1h,A2=K2+dvT3+T3+dv+T1+β1h,A1=dvT3T1+β1h+T3+dvK2,A0=dvT3K2,B2=−rρvK4e−dhτ,B1=−rρvK4T1+T3+θhξr+αhξte−dhτ,B0=−rρvK4K1e−dhτ.
Whenτ=0, equation (19) is rewritten as(21)L1λ=λ4+m3λ3+m2λ2+m1λ+m0=0,where m3=A3, m2=A2+B2, m1=A1+B1, and m0=A0+B0.
WhenRT<1, there are(22)m3=A3>0,m2=A2+B2>K2+dvT1+T3+T3T1+β1h>0,m1=A1+B1>dvT1T3+T3K2>0,m0=A0+B0=dvT3K21−RT>0.
Next, consider the signs ofΔ2=m3m2−m1 and Δ3=m1m3m2−m1−m0m32.(23)Δ2=m3m2−m1>a3dvT1+T3T1+β1h+T1+β1hK2+dv+T3dvT3+rρvK4T1+T3+θhξr+αhξt>0,Δ3=m1m3m2−m1−m0m32>T3K2+dvT1T3K2T1+β1h+dvT3dv+T3+T32K2a3T1+β1h+dvT12T3a3dv+T3+rρvK4a3T1T3T1+T3>0.
According to the Routh–Hurwitz criterion, whenτ=0 and RT<1, the disease-free equilibrium E0 is locally asymptotically stable. Next, the local stability of disease-free equilibrium E0 is discussed when τ>0. Suppose that, for some τ>0, equation (19) has a pure imaginary root λ=iω, ω>0. Substituting λ=iω into equation (19), then equation (19) becomes(24)ω4−A3ω3i−A2ω2+A1ωi+A0+cosωτ−isinωτ−B2ω2+B1ωi+B0=0.
Separating the real part and the imaginary part gives(25)−B2ω2+B0cosωτ+B1ωsinωτ=−ω4+A2ω2−A0,B1ωcosωτ−−B2ω2+B0sinωτ=A3ω3−A1ω.
Thenω must be the root of the following equation:(26)ω8+c3ω6+c2ω4+c1ω2+c0=0.
Letz=ω2; then equation (26) becomes the following equation:(27)z4+c3z3+c2z2+c1z+c0=0,where(28)c3=A32−2A2,c2=A22+2A0−2A1A3−B22,c1=A12−2A0A2−B12+2B0B2,c0=A02−B02.
WhenRT<1, there are(29)c0=dv2T32K221−RT2>0,c1>dv2T12T32+k22T32+2r2ρv2K42ξrαhψ21−κh+2dv2T32κhαhψ2>0,c2>K22+dv2T32+T12+T32T12+β1h2>0,c3>T12+T32+β1h2+dv2>0.
Therefore, whenRT<1, equation (27) has no positive root.
According to the above analysis, whenRT<1, all the roots of the characteristic equation (19) have negative real parts, and the disease-free equilibrium E0 is locally asymptotically stable. When RT>1, there are L00,τ=dvT3K21−RT<0 and limλ⟶+∞L0λ,τ>0, the characteristic equation (19) has at least one positive real root, and then the disease-free equilibrium E0 is unstable.


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

From the above discussion, the disease-free equilibriumE0 is locally asymptotically stable when RT<1. In this section, the global stability of disease-free equilibrium E0 is considered at τ=0.Let(30)R˜=dh2K1ΛhK2T3,Γ=Sh,Ih,Th,Ph,Rh,Sv,Iv∈R+7:0<Sh+Ih+Th+Ph+Rh≤Λhdh;0<Sv+Iv≤Λvdv.Theorem 3.
IfRT<R˜, the disease-free equilibrium E0 is globally asymptotically stable in Γ at τ=0.Proof.
It can be seen from the above section that the disease-free equilibriumE0 is locally asymptotically stable. Next, only the global attraction of the disease-free equilibrium E0 is required to be proved.
By calculation, there is(31)R˜=dh2K1ΛhK2T3≤dh2K1ΛhdhK1+K6<1,where K6=ΛhδhT1T3+θhT1γh+αhψ1T3+αhψ21−κhγh>0.
Constructing the Lyapunov function onΓ,(32)V=rρhΛvdhdvIh+rρhΛvdhdvTh+rρhΛvdhdvRh+Iv.
Vt is positive and bounded according to the nonnegative uniform boundedness of the solution. The total derivative of Vt along the solution xt of system (1) is(33)V˙xt=rρhΛvdhdvI˙h+rρhΛvdhdvT˙h+rρhΛvdhdvR˙h+I˙v=rρhΛvdhdvrρvIvNhShe−dhτ+κhψ2Th−β1hIh+αhIh−ψ1Th−ψ2Th−dhTh+rρhΛvdhdvθhIh+1−κhψ2Th−γhRh−dhRh+rρhIh+ξtTh+ξrRhNhSv−dvIv≤rρhΛvdhdvrρvIve−dhτ−dh+δhIh−ψ1Th−dhTh−γhRh−dhRh+rρhΛvdvIh+Th+Rh−dvIv≤dvRTR˜−1Iv−rρhΛvdhdvδhIh+ψ1Th+γhRh.
WhenRT<R˜, there is V˙xt≤0, if and only if Iv=0, Ih=0, Th=0, and Rh=0, and there is V˙xt=0. Define the set(34)Σ=Sh,Ih,Th,Ph,Rh,Sv,Iv∈Γ|dVdt=0.
LetM be the largest invariant set in Σ. From system (1) and the invariant property of M, M=E0 can be obtained. According to the Lyapunov–LaSalle principle, the solutions of system (1) with initial conditions in Γ all approach the disease-free equilibrium E0 at t⟶+∞. That is, when RT<R˜, there is Sht,Iht,Tht,Pht,Rht,Svt,Ivt⟶Λh/dh,0,0,0,0,Λv/dv,0; the disease-free equilibrium E0 is globally asymptotically stable.


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

For the local stability of the disease-free equilibriumE0, the following conclusions are obtained.Theorem 2.
IfRT<1, the disease-free equilibrium E0 is locally asymptotically stable for any τ≥0; if RT>1, E0 is unstable for any τ≥0.Proof.
It is shown that the characteristic equation of system (1) at E0 is(19)L0λ,τ=λ4+A3λ3+A2λ2+A1λ+A0+e−λτB2λ2+B1λ+B0=0,where(20)A3=T1+T3+dv+β1h,A2=K2+dvT3+T3+dv+T1+β1h,A1=dvT3T1+β1h+T3+dvK2,A0=dvT3K2,B2=−rρvK4e−dhτ,B1=−rρvK4T1+T3+θhξr+αhξte−dhτ,B0=−rρvK4K1e−dhτ.
Whenτ=0, equation (19) is rewritten as(21)L1λ=λ4+m3λ3+m2λ2+m1λ+m0=0,where m3=A3, m2=A2+B2, m1=A1+B1, and m0=A0+B0.
WhenRT<1, there are(22)m3=A3>0,m2=A2+B2>K2+dvT1+T3+T3T1+β1h>0,m1=A1+B1>dvT1T3+T3K2>0,m0=A0+B0=dvT3K21−RT>0.
Next, consider the signs ofΔ2=m3m2−m1 and Δ3=m1m3m2−m1−m0m32.(23)Δ2=m3m2−m1>a3dvT1+T3T1+β1h+T1+β1hK2+dv+T3dvT3+rρvK4T1+T3+θhξr+αhξt>0,Δ3=m1m3m2−m1−m0m32>T3K2+dvT1T3K2T1+β1h+dvT3dv+T3+T32K2a3T1+β1h+dvT12T3a3dv+T3+rρvK4a3T1T3T1+T3>0.
According to the Routh–Hurwitz criterion, whenτ=0 and RT<1, the disease-free equilibrium E0 is locally asymptotically stable. Next, the local stability of disease-free equilibrium E0 is discussed when τ>0. Suppose that, for some τ>0, equation (19) has a pure imaginary root λ=iω, ω>0. Substituting λ=iω into equation (19), then equation (19) becomes(24)ω4−A3ω3i−A2ω2+A1ωi+A0+cosωτ−isinωτ−B2ω2+B1ωi+B0=0.
Separating the real part and the imaginary part gives(25)−B2ω2+B0cosωτ+B1ωsinωτ=−ω4+A2ω2−A0,B1ωcosωτ−−B2ω2+B0sinωτ=A3ω3−A1ω.
Thenω must be the root of the following equation:(26)ω8+c3ω6+c2ω4+c1ω2+c0=0.
Letz=ω2; then equation (26) becomes the following equation:(27)z4+c3z3+c2z2+c1z+c0=0,where(28)c3=A32−2A2,c2=A22+2A0−2A1A3−B22,c1=A12−2A0A2−B12+2B0B2,c0=A02−B02.
WhenRT<1, there are(29)c0=dv2T32K221−RT2>0,c1>dv2T12T32+k22T32+2r2ρv2K42ξrαhψ21−κh+2dv2T32κhαhψ2>0,c2>K22+dv2T32+T12+T32T12+β1h2>0,c3>T12+T32+β1h2+dv2>0.
Therefore, whenRT<1, equation (27) has no positive root.
According to the above analysis, whenRT<1, all the roots of the characteristic equation (19) have negative real parts, and the disease-free equilibrium E0 is locally asymptotically stable. When RT>1, there are L00,τ=dvT3K21−RT<0 and limλ⟶+∞L0λ,τ>0, the characteristic equation (19) has at least one positive real root, and then the disease-free equilibrium E0 is unstable.


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

From the above discussion, the disease-free equilibriumE0 is locally asymptotically stable when RT<1. In this section, the global stability of disease-free equilibrium E0 is considered at τ=0.Let(30)R˜=dh2K1ΛhK2T3,Γ=Sh,Ih,Th,Ph,Rh,Sv,Iv∈R+7:0<Sh+Ih+Th+Ph+Rh≤Λhdh;0<Sv+Iv≤Λvdv.Theorem 3.
IfRT<R˜, the disease-free equilibrium E0 is globally asymptotically stable in Γ at τ=0.Proof.
It can be seen from the above section that the disease-free equilibriumE0 is locally asymptotically stable. Next, only the global attraction of the disease-free equilibrium E0 is required to be proved.
By calculation, there is(31)R˜=dh2K1ΛhK2T3≤dh2K1ΛhdhK1+K6<1,where K6=ΛhδhT1T3+θhT1γh+αhψ1T3+αhψ21−κhγh>0.
Constructing the Lyapunov function onΓ,(32)V=rρhΛvdhdvIh+rρhΛvdhdvTh+rρhΛvdhdvRh+Iv.
Vt is positive and bounded according to the nonnegative uniform boundedness of the solution. The total derivative of Vt along the solution xt of system (1) is(33)V˙xt=rρhΛvdhdvI˙h+rρhΛvdhdvT˙h+rρhΛvdhdvR˙h+I˙v=rρhΛvdhdvrρvIvNhShe−dhτ+κhψ2Th−β1hIh+αhIh−ψ1Th−ψ2Th−dhTh+rρhΛvdhdvθhIh+1−κhψ2Th−γhRh−dhRh+rρhIh+ξtTh+ξrRhNhSv−dvIv≤rρhΛvdhdvrρvIve−dhτ−dh+δhIh−ψ1Th−dhTh−γhRh−dhRh+rρhΛvdvIh+Th+Rh−dvIv≤dvRTR˜−1Iv−rρhΛvdhdvδhIh+ψ1Th+γhRh.
WhenRT<R˜, there is V˙xt≤0, if and only if Iv=0, Ih=0, Th=0, and Rh=0, and there is V˙xt=0. Define the set(34)Σ=Sh,Ih,Th,Ph,Rh,Sv,Iv∈Γ|dVdt=0.
LetM be the largest invariant set in Σ. From system (1) and the invariant property of M, M=E0 can be obtained. According to the Lyapunov–LaSalle principle, the solutions of system (1) with initial conditions in Γ all approach the disease-free equilibrium E0 at t⟶+∞. That is, when RT<R˜, there is Sht,Iht,Tht,Pht,Rht,Svt,Ivt⟶Λh/dh,0,0,0,0,Λv/dv,0; the disease-free equilibrium E0 is globally asymptotically stable.


## 5. Assessment of TBDs

In order to effectively control malaria for a long time, it is necessary to know what conditions the parameters meet when the use of TBDs plays a positive role in disease control. In this section, the impact of the use of TBDs on malaria transmission is analyzed. Firstly, the treatment rateαh was taken as the main parameter to consider the influence of αh on the control reproduction number RT.Let(35)c0=r2ρhρvΛvdhe−dhτ,c1=T1T3+ξrθh,c2=T1dh+δh+θh,d0=T3Λhdv2,d1=ξtT3+ξrψ21−κh,d2=T1−κhψ2.The derivative ofRT with respect to αh is(36)dRTdαh=c0c2d1−c1d2d0c2+d2αh2,if and only if c2d1−c1d2<0, and there is dRT/dαh<0. In general, when the control reproduction number is less than one, the disease can finally disappear, so there is the following theorem.Theorem 4.
For system (1),(1)
Whenc2d1/c1d2<1, the TBDs have a positive effect on reducing the burden of disease(2)
Whenc2d1/c1d2=1, the TBDs have no effect on reducing the burden of disease(3)
Whenc2d1/c1d2>1, the TBDs have a negative effect on reducing the burden of diseaseIt is generally believed that the more the people who receive the TBDs treatment, the more the effectiveness on controlling malaria. However, humans treated with TBDs remain infectious; the increase in the number ofTh does not control the disease well. It is worth noting the percentage of humans treated with TBDs who were successfully protected. Next, taking the efficacy of TBDs pe as the main parameter, to observe which conditions the parameters meet, the TBDs have a positive effect on reducing the burden of disease.The expressionc2d1−c1d2 is rewritten as the expression T1E1+E2pe about the efficacy of TBDs pe, where(37)E1=dh+δh+θhξrT3−dhT3−dhθhξr−1−κhT3μh+dh+δhξr1−κhμh,E2=1−κhT3μh−T3+θhξrηh−dh+δhξr1−κhμh.According to Theorem4 and equation (36), the following theorem can be obtained.Theorem 5.
For system (1),(1)
WhenE1>0 and E2>0, for any pe∈0,1, the TBDs have a negative effect on reducing the burden of disease(2)
WhenE1>0 and E2<0, if pe∈min−E1/E2,1,1, the TBDs have a positive effect on reducing the burden of disease(3)
WhenE1<0 and E2>0, if pe∈0,min−E1/E2,1, the TBDs have a positive effect on reducing the burden of disease(4)
WhenE1<0 and E2<0, for any pe∈0,1, the TBDs have a positive effect on reducing the burden of disease


## 6. Stability of Positive Equilibrium and Hopf Bifurcation

### 6.1. Local Stability of Positive Equilibrium

WhenRT>1, there is a unique positive equilibrium in system (1), and the characteristic equation of system (1) at E∗ is(38)L2λ,τ=λ7+F6λ6+F5λ5+F4λ4+F3λ3+F2λ2+F1λ+F0+e−λτJ6λ6+J5λ5+J4λ4+J3λ3+J2λ2+J1λ+J0=0.The coefficients of equation (38) are given in Appendix.Whenτ=0, the characteristic equation is written as(39)L3λ=λ7+H6λ6+H5λ5+H4λ4+H3λ3+H2λ2+H1λ+H0=0,where Hi=Fi+Ji,i=1,2,…,6.Define(40)Δ1=H6,Δ2=H61H4H5,Δ3=H610H4H5H6H2H3H4,Δ4=H6100H4H5H61H2H3H4H5H0H1H2H3,Δ5=H61000H4H5H610H2H3H4H5H6H0H1H2H3H400H0H1H2,Δ6=H610000H4H5H6100H2H3H4H5H61H0H1H2H3H4H500H0H1H2H30000H0H1,Δ7=H0Δ6.According to the Routh–Hurwitz criterion, ifΔi>0,i=1,2,…,7, then all the roots of equation (39) have negative real parts.Theorem 6.
WhenRT>1 and τ=0, if Δi>0,i=1,2,…,7, then the positive equilibrium E∗ is locally asymptotically stable.In the classification of equilibria in section three, the occurrence of backward bifurcation means that there are two positive equilibria in the system atR∗<RT<1. The stability analysis of these two positive equilibria will be explained in numerical simulation.


### 6.2. Existence of Hopf Bifurcation

Suppose that equation (38) has a pure imaginary root λ=iωω>0. λ=0 is not the root of equation (38) since A0+B0≠0. Substituting λ=iω into equation (38) and separating real and imaginary parts, it follows that(41)p1cosωτ+p2sinωτ=F6ω6−F4ω4+F2ω2−F0,p2cosωτ−p1sinωτ=ω7−F5ω5+F3ω3−F1ω,where p1=−J6ω6+J4ω4−J2ω2+J0 and p2=J5ω5−J3ω3+J1ω.Thenω satisfies the equation(42)ω14+H6ω12+H5ω10+H4ω8+H3ω6+H2ω4+H1ω2+H0=0,where(43)H6=F62−J62−2F5,H5=F52+2F3−2F4F6+2J4J6−J52,H4=F42+2F2F6−2F1−2F3F5−J42−2J2J6+2J3J5,H3=F32+2F1F5−2F0F6−2F2F4−J32+2J0J6+2J2J4−2J1J5,H2=F22+2F0F4−2F1F3−J22−2J0J4+2J1J3,H1=F12−2F0F2−J12+2J0J2,H0=F02−J02.Letz=ω2, and define(44)Gz=z7+H6z6+H5z5+H4z4+H3z3+H2z2+H1z+H0=0.Therefore, if equation (38) has a pure imaginary root, then equation Gz=0 has a positive real root. Assume that equation (44) has seven positive real roots, denoted as zk,k=1,2,…,7.It can be obtained by calculation that(45)coszkτ=Uk,sinzkτ=Vk,where(46)Uk≜p3F6zk3−F4zk2+F2zk−F0+zkp4zk3−F5zk2+F3zk−F1p32+zkp42,Vk≜zkp4F6zk3−F4zk2+F2zk−F0−zkp3zk3−F5zk2+F3zk−F1p32+zkp42,p3=−J6zk3+J4zk2−J2zk+J0,p4=J5zk2−J3zk+J1.Then(47)τkj=1ykarccosUk+2jπ,Vk≥0,1yk2π−arccosUk+2jπ,Vk<0,where k=1,2,…,7 and j=0,1,2,…. Therefore, the characteristic equation L2λ,τkj=0 has a pair of pure imaginary roots ±izk.Let(48)τ0=τk00=minτkj|k=1,2,…,7;j=0,1,2,…,ω0=ωk0.Assuming thatλτ=ατ+iωτ is the root of equation (38), then ατ0=0 and ωτ0=ω0.Theorem 7.
Suppose that equation (44) has root z=ω02 and G′z≠0; then dReλ/dτ|τ=τ0 and G′z have the same sign.Proof.
Firstly, the derivative ofλ with respect to τ in the characteristic equation (38) is calculated as follows:(49)dλdτ−1=−7λ6+6F6λ5+5F5λ4+4F4λ3+3F3λ2+2F2λ+F1λλ7+F6λ6+F5λ5+F4λ4+F3λ3+F2λ2+F1λ+F0+6J6λ5+5J5λ4+4J4λ3+3J3λ2+2J2λ+J1λJ6λ6+J5λ5+J4λ4+J3λ3+J2λ2+J1λ+J0−τλ.
Substitutingλ=iω0 into equation (49), it follows that(50)Redλdτ−1|τ=τ0=G′ω02J6ω06−J4ω04−J2ω02−J02+J5ω05−J3ω03−J1ω02.
Therefore, it can be obtained that(51)signdReλdτ|τ=τ0=signRedλdτ−1|τ=τ0=signG′ω02.Based on Theorem7 and Hopf bifurcation theory, the following conclusions can be drawn.Theorem 8.
Suppose thatE∗ is locally asymptotically stable when τ=0 and RT>1; there are the following conclusions:(1)
If equation (44) has no positive real root, then the equilibrium E∗ is locally asymptotically stable for any τ≥0(2)
If equation (44) has a positive real root, when τ∈0,τ0, the equilibrium E∗ is locally asymptotically stable, while when τ>τ0, E∗ is unstable(3)
Ifzk0 is a single root of equation (44), then at τ=τ0, system (1) will undergo a Hopf bifurcationProof.
(1)
IfGz=0 has no positive real root, then the root of the characteristic equation on the left side of the imaginary axis is still there and does not cross the imaginary axis. Therefore, for any τ≥0, the equilibrium E∗ is locally asymptotically stable.(2)
From the definition ofτ0, it can be seen that there is no positive root of equation Gz=0 in the set τ∈0,τ0, and the root of its characteristic equation still does not cross the imaginary axis. Therefore, when τ∈0,τ0, the equilibrium E∗ is locally asymptotically stable. When τ>τ0, equation Gz=0 has a root with a positive real part; thus, E∗ is unstable.(3)
Ifzk0 is a single root of equation Gz=0, then G′zk0≠0 and dReλ/dτ|τ=τ0≠0. If dReλ/dτ|τ=τ0<0, then the root of the characteristic equation (38) at τ slightly less than τ0 has a positive real part, which contradicts the definition of τ0. Therefore, dReλ/dτ|τ=τ0>0 and G′ω02>0. If dReλ/dτ|τ=τ0>0, when the value of τ is slightly less than τ0, it can be seen from Theorem 6 that all the roots of (38) have negative real parts. When the value of τ is slightly greater than τ0, equation (38) has the root with positive real part, which means that system (1) has Hopf bifurcation.


## 6.1. Local Stability of Positive Equilibrium

WhenRT>1, there is a unique positive equilibrium in system (1), and the characteristic equation of system (1) at E∗ is(38)L2λ,τ=λ7+F6λ6+F5λ5+F4λ4+F3λ3+F2λ2+F1λ+F0+e−λτJ6λ6+J5λ5+J4λ4+J3λ3+J2λ2+J1λ+J0=0.The coefficients of equation (38) are given in Appendix.Whenτ=0, the characteristic equation is written as(39)L3λ=λ7+H6λ6+H5λ5+H4λ4+H3λ3+H2λ2+H1λ+H0=0,where Hi=Fi+Ji,i=1,2,…,6.Define(40)Δ1=H6,Δ2=H61H4H5,Δ3=H610H4H5H6H2H3H4,Δ4=H6100H4H5H61H2H3H4H5H0H1H2H3,Δ5=H61000H4H5H610H2H3H4H5H6H0H1H2H3H400H0H1H2,Δ6=H610000H4H5H6100H2H3H4H5H61H0H1H2H3H4H500H0H1H2H30000H0H1,Δ7=H0Δ6.According to the Routh–Hurwitz criterion, ifΔi>0,i=1,2,…,7, then all the roots of equation (39) have negative real parts.Theorem 6.
WhenRT>1 and τ=0, if Δi>0,i=1,2,…,7, then the positive equilibrium E∗ is locally asymptotically stable.In the classification of equilibria in section three, the occurrence of backward bifurcation means that there are two positive equilibria in the system atR∗<RT<1. The stability analysis of these two positive equilibria will be explained in numerical simulation.


## 6.2. Existence of Hopf Bifurcation

Suppose that equation (38) has a pure imaginary root λ=iωω>0. λ=0 is not the root of equation (38) since A0+B0≠0. Substituting λ=iω into equation (38) and separating real and imaginary parts, it follows that(41)p1cosωτ+p2sinωτ=F6ω6−F4ω4+F2ω2−F0,p2cosωτ−p1sinωτ=ω7−F5ω5+F3ω3−F1ω,where p1=−J6ω6+J4ω4−J2ω2+J0 and p2=J5ω5−J3ω3+J1ω.Thenω satisfies the equation(42)ω14+H6ω12+H5ω10+H4ω8+H3ω6+H2ω4+H1ω2+H0=0,where(43)H6=F62−J62−2F5,H5=F52+2F3−2F4F6+2J4J6−J52,H4=F42+2F2F6−2F1−2F3F5−J42−2J2J6+2J3J5,H3=F32+2F1F5−2F0F6−2F2F4−J32+2J0J6+2J2J4−2J1J5,H2=F22+2F0F4−2F1F3−J22−2J0J4+2J1J3,H1=F12−2F0F2−J12+2J0J2,H0=F02−J02.Letz=ω2, and define(44)Gz=z7+H6z6+H5z5+H4z4+H3z3+H2z2+H1z+H0=0.Therefore, if equation (38) has a pure imaginary root, then equation Gz=0 has a positive real root. Assume that equation (44) has seven positive real roots, denoted as zk,k=1,2,…,7.It can be obtained by calculation that(45)coszkτ=Uk,sinzkτ=Vk,where(46)Uk≜p3F6zk3−F4zk2+F2zk−F0+zkp4zk3−F5zk2+F3zk−F1p32+zkp42,Vk≜zkp4F6zk3−F4zk2+F2zk−F0−zkp3zk3−F5zk2+F3zk−F1p32+zkp42,p3=−J6zk3+J4zk2−J2zk+J0,p4=J5zk2−J3zk+J1.Then(47)τkj=1ykarccosUk+2jπ,Vk≥0,1yk2π−arccosUk+2jπ,Vk<0,where k=1,2,…,7 and j=0,1,2,…. Therefore, the characteristic equation L2λ,τkj=0 has a pair of pure imaginary roots ±izk.Let(48)τ0=τk00=minτkj|k=1,2,…,7;j=0,1,2,…,ω0=ωk0.Assuming thatλτ=ατ+iωτ is the root of equation (38), then ατ0=0 and ωτ0=ω0.Theorem 7.
Suppose that equation (44) has root z=ω02 and G′z≠0; then dReλ/dτ|τ=τ0 and G′z have the same sign.Proof.
Firstly, the derivative ofλ with respect to τ in the characteristic equation (38) is calculated as follows:(49)dλdτ−1=−7λ6+6F6λ5+5F5λ4+4F4λ3+3F3λ2+2F2λ+F1λλ7+F6λ6+F5λ5+F4λ4+F3λ3+F2λ2+F1λ+F0+6J6λ5+5J5λ4+4J4λ3+3J3λ2+2J2λ+J1λJ6λ6+J5λ5+J4λ4+J3λ3+J2λ2+J1λ+J0−τλ.
Substitutingλ=iω0 into equation (49), it follows that(50)Redλdτ−1|τ=τ0=G′ω02J6ω06−J4ω04−J2ω02−J02+J5ω05−J3ω03−J1ω02.
Therefore, it can be obtained that(51)signdReλdτ|τ=τ0=signRedλdτ−1|τ=τ0=signG′ω02.Based on Theorem7 and Hopf bifurcation theory, the following conclusions can be drawn.Theorem 8.
Suppose thatE∗ is locally asymptotically stable when τ=0 and RT>1; there are the following conclusions:(1)
If equation (44) has no positive real root, then the equilibrium E∗ is locally asymptotically stable for any τ≥0(2)
If equation (44) has a positive real root, when τ∈0,τ0, the equilibrium E∗ is locally asymptotically stable, while when τ>τ0, E∗ is unstable(3)
Ifzk0 is a single root of equation (44), then at τ=τ0, system (1) will undergo a Hopf bifurcationProof.
(1)
IfGz=0 has no positive real root, then the root of the characteristic equation on the left side of the imaginary axis is still there and does not cross the imaginary axis. Therefore, for any τ≥0, the equilibrium E∗ is locally asymptotically stable.(2)
From the definition ofτ0, it can be seen that there is no positive root of equation Gz=0 in the set τ∈0,τ0, and the root of its characteristic equation still does not cross the imaginary axis. Therefore, when τ∈0,τ0, the equilibrium E∗ is locally asymptotically stable. When τ>τ0, equation Gz=0 has a root with a positive real part; thus, E∗ is unstable.(3)
Ifzk0 is a single root of equation Gz=0, then G′zk0≠0 and dReλ/dτ|τ=τ0≠0. If dReλ/dτ|τ=τ0<0, then the root of the characteristic equation (38) at τ slightly less than τ0 has a positive real part, which contradicts the definition of τ0. Therefore, dReλ/dτ|τ=τ0>0 and G′ω02>0. If dReλ/dτ|τ=τ0>0, when the value of τ is slightly less than τ0, it can be seen from Theorem 6 that all the roots of (38) have negative real parts. When the value of τ is slightly greater than τ0, equation (38) has the root with positive real part, which means that system (1) has Hopf bifurcation.


## 7. Numerical Simulation

In this section, the conclusions of the paper will be verified by numerical simulation and some of the conclusions will be extended. When selecting parameters, we refer to the parameter range in reference (see [24]), as shown in Table 1.Table 1
Parameter value range.
ParameterRange of possible valuesBaseline value usedDimReferenceΛh2.7×10−5,1000100H×day−1EstimatedΛv[0.002, 2200]200V×day−1Estimateddh2.74×10−5,0.0330.012day−1See [24]dv[0.03302, 0.1]0.08day−1See [24]γh5×10−4,10.01day−1Estimatedv[0, 0.1]0.01day−1See [24]pe[0, 1]0.51See [24]αh[0, 1]0.75day−1See [24]δh1×10−5,5×10−20.08day−1See [24]θh3.5×10−5,0.20.2day−1See [24]r[0.1, 30]5H×V−1×day−1See [24]ρh[0.01, 1]0.0031See [24]ρv[0.072, 1]0.241See [24]ξt[0.02, 1]0.41See [24]ξr[0.005, 1]0.21See [24]ηh[0, 0.2]0.1day−1See [24]μh[0, 0.2]0.2day−1See [24]κh[0, 1]0.6day−1See [24]LetΛh=100, dh=0.012, dv=0.08, δh=0.08, Λv=200, r=5, ρh=0.003, θh=0.2, υ=0.01, αh=0.1, ξt=0.4, ξr=0.2, ηh=0.1, μh=0.2, pe=0.1, κh=0.6, and τ=3. When γh=0.01, system (1) exhibits forward bifurcation (Figure 1(a)). When γh=0.6, system (1) exhibits backward bifurcation (Figure 1(b)); if ρv=0.88, then R∗<RT<1; the system has a bistable phenomenon. The disease-free equilibrium E0 and the positive equilibrium E∗ are locally asymptotically stable, while the positive equilibrium E∗∗ is unstable (Figure 2).Figure 1
The bifurcation diagrams (a, b) show the change of the number of infected individualsIh with the control reproduction number RT.
(a)(b)Figure 2
For different initial values, some trajectories tend to disease-free equilibriumE0 and others to positive equilibrium E∗.
(a)(b)LetΛh=5, dh=0.012, dv=0.06, δh=0.08, γh=0.6, Λv=100, r=0.1, ρh=0.002, ρv=0.005, θh=0.2, υ=0.01, αh=0.1, ξt=0.4, ξr=0.2, ηh=0.1, μh=0.2, pe=0.1, κh=0.6, and τ=0; there is RT<R˜<1; the disease-free equilibrium E0 is globally asymptotically stable (Figure 3). Compared with Figure 2, the selected initial values all tend to the disease-free equilibrium E0.Figure 3
All trajectories tend to disease-free equilibriumE0.
(a)(b)In Figures4(a) and 4(b), the parameters γh=0.9, Λv=200, r=5, ρh=0.003, ρv=0.9, θh=0.2, υ=0.01, αh=0.1, ξt=0.4, ξr=0.2, ηh=0.1, μh=0.2, and κh=0.6 are shared, and pe is a variable. In Figure 4(a), let Λh=100, dh=0.015, dv=0.08, δh=0.08, and τ=3. In Figure 4(b), let Λh=80, dh=0.013, dv=0.09, δh=0.24, and τ=0.1. When parameters meet conditions E1=−0.03202<0, E2=−0.02078<0, and RT>1, according to Theorem 5 (1), with the increase of the efficacy of TBDs pe∈0,1, the number of infected humans Ih gradually decreased (Figure 4(a)). When parameters meet conditions E1=0.00133>0, E2=−0.0182<0, and R∗<RT<1, take an initial value that tends to the equilibrium E∗. According to Theorem 5 (3), when pe∈0,0.073, the improvement of the efficacy of TBDs had no significant effect on disease control, while when pe∈0.073,1, the improvement of the efficacy of TBDs has a significant effect on disease control, and the trajectory eventually tends to the disease-free equilibrium E0 (Figure 4(b)).Figure 4
Effects of increased efficacy of TBDspe on the number of malaria patients Ih.
(a)(b)Letdh=0.013, let αh and pe be variables, and other parameters are the same as those in Figure 4(a). It can be found that when the treatment rate αh is large enough and the efficacy of TBDs pe is low, the control reproduction number is still greater than 1; when pe is large enough, the control reproduction number is less than 1 (Figure 5).Figure 5
Diagram of the influence ofαh and pe on the control reproduction number RT.LetΛh=855, dh=0.01, dv=0.055, δh=0.02, γh=0.8, Λv=2100, r=5, ρh=0.003, ρv=0.8, θh=0.2, υ=0.2, αh=0.02, ξt=0.07, ξr=0.03, ηh=0.2, μh=0.16, pe=0.16, and κh=0.2; there is RT>1. In Figure 6, when τ=4.2, there is only one positive real root in equation (44), and the bifurcation threshold τ10=4.48 is obtained. In this case, τ<τ10, the positive equilibrium E∗ is locally asymptotically stable in Figure 6. In Figure 7, when τ=5, only one positive real root exists in equation (44), and the bifurcation threshold is τ10=4.58. In this case, τ>τ10, the positive equilibrium E∗ is unstable, and the system has a periodic solution. According to Theorem 8, at τ10, system (1) undergoes a Hopf bifurcation.Figure 6
Whenτ=4.2 is less than the threshold τ10=4.48, the positive equilibrium E∗ is locally asymptotically stable.Figure 7
Whenτ=5 is greater than the threshold τ10=4.58, E∗ is unstable and the system has a stable periodic solution.


## 8. Conclusion

In this paper, we establish a time-delay differential equation model with transmission-blocking drugs intervention, the effect of time delay on the stability of the equilibrium was obtained, and the effective range of treatment rate and the efficacy of TBDs under certain conditions was assessed. In the classification of equilibria, the model appears to have backward bifurcation. The bistable case suggests that disease may exist even if the control reproduction number is less than 1, and the existence of the disease depends on the initial conditions. The global stability of the disease-free equilibrium provides a reference for malaria eradication under certain conditions. In addition, when the parameters meet some conditions, the treatment rate and the efficacy of TBDs play a positive role in disease control. In contrast, the increase in the efficacy of TBDs was more significant for disease control. It is found through Figure4(b) that, in the bistable case, the trajectory that originally tends to the disease equilibrium starts to converge to the disease-free equilibrium after increasing the efficacy of TBDs pe, that is, making the initial value of disease extinction larger in range with increasing the efficacy of TBDs pe. Therefore, the development of the transmission-blocking drugs and the improvement of the efficacy of TBDs will play an important role in the long-term control of malaria. When considering the influence of time delay on the model, we find that when the control reproduction number is greater than 1, the model exhibits Hopf bifurcation. When the time delay is less than the bifurcation value, the positive equilibrium is locally asymptotically stable; when the time delay is slightly greater than the bifurcation value, the positive equilibrium is unstable and the system has a periodic solution. The appearance of periodic solution corresponds to the phenomenon of malaria periodic outbreaks in reality.



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*Source: 1339086-2021-09-11.xml*