Malaria is a life-threatening disease spread to humans by certain types of mosquitoes. According to scientific studies of National Institutes of Health (NIH), Malaria is the fifth most deadly infectious disease [1] . It is mostly found in tropical countries. One of the deadliest forms of malaria is cerebral malaria, where the patient experiences severe neurological symptoms, such as seizures and coma, the condition is lethal without treatment. Symptoms can range from mild to life-threatening. Mild symptoms include fever, chills, and headache. Severe symptoms include fatigue, confusion, seizures, muscle aches, nausea, vomiting, diarrhea and difficulty breathing [2] . Infants, children under 5 years, pregnant women, travellers, and people with HIV or AIDS are at higher risk of severe infection. Malaria can be prevented by avoiding mosquito bites and through medication. Environmental factors are more significant than genetics in determining a person immunity, according to scientific studies of National Institutes of Health (NIH) [1] . Treatments can prevent mild cases from worsening as mentioned by the World Health Organization (WHO) [2] , where the NIH clinical trials [3] - [5] have found that one dose of an antibody-drug safely protected healthy, non-pregnant adults from malaria infection during an intense six-month malaria season in Mali, Africa. The antibody was up to 88.2% effective at preventing infection over a 24-week period, demonstrating for the first time that a monoclonal antibody can prevent malaria infection in an endemic region. According to the World Malaria Report (WHO) in 2023 [2] , there were 249 million cases of malaria in 2022 compared to 244 million cases in 2021. The estimated number of malaria deaths stood at 608,000 in 2022 compared to 610,000 in 2021. The WHO African Region continues to bear a disproportionately high share of the global malaria burden. In 2022, the region was home to about 94% of all malaria cases and 95% of deaths. Children under 5 years of age accounted for about 78% of all malaria deaths in the region. Four African countries accounted for just over half of all malaria deaths worldwide: Nigeria (26.8%), the Democratic Republic of the Congo (12.3%), Uganda (5.1%), and Mozambique (4.2%) [2] .

The Centers for Disease Control and Prevention reported that a number of Malaria cases were transmitted within the USA in Texas and Florida, and became very important to understand the interactions of between climate, mosquitoes and pathogens mosquitoes transmit to humans, and this was the importance of mathematical modelling. For the last 20 years, researchers focused on understanding the impacts of environmental change on the ecology of insect vectors and the diseases they transmit, combining field studies and modelling approaches, statistically and mathematically to explain the interactions. This approach advances our capacity to engage in more relational science, where factors are not understood as fixed and independent forces, but as dynamic and interdependent. Relations between variables can be expressed by nonlinear relationship that can be easily grasped by machine learning algorithms and other data science tools. Computers have made it easier to deal with larger datasets. The automation of data assimilation can lead to predicting where and when to expect the transmission of some vector-borne diseases. Mathematical models also show how the stability of natural systems can collapse following small changes in the environment, and that has clear implications about why we need to worry as climate change continues its current course [6] . Mathematical modelling has been used by epidemiologists as a tool to predict the occurrence of epidemics of infectious diseases, and also as a tool for guiding research for eradication of malaria at the present time [7] .

The route of mathematical modelling was explained in [8] and gave an insight of Malaria modelling by mathematical tools. Mathematical models have been used to provide an explicit framework for understanding malaria transmission dynamics in human population for over 100 years.

The SIR-SI model represents an advancement over Ross’s SIS [9] model by incorporating the relationship between human hosts and vector (mosquito) compartments. It categorizes humans into three groups: susceptible, infectious, and recovered, while vectors are divided into susceptible and infectious categories. Importantly, unlike its predecessor, individuals in this model do not revert to susceptibility once they acquire immunity. The model discussed in [10] addresses the impact of environmental immunity on the mathematical modelling of malaria transmission between vector and host populations. However, it lacks some of the mathematical analysis, and therefore, the primary objective of this paper is to conduct a comprehensive mathematical analysis.

Our study aims to explore the fundamental qualitative properties of the proposed model, including the investigation of the global stability. To validate our theoretical findings, we utilized numerical simulations. Finally, we provide a thorough discussion of the results obtained, offering insights into the dynamics of malaria transmission within the context of environmental immunity.

In this section, we study the global stability of the SIR-SI model ( [10] and references therein). The model comprises five compartments: the number of susceptible healthy human to the disease, $S_h$, the number of the infected human $I_h$, the number of the recovered humans $R_h$, the number of susceptible mosquitoes (or vector) $S_v$, and the number of infected mosquitoes (or vector) $I_v$. The environmental immunity parameter was incorporated into the infectious and recovered classes to examine its impact on the basic reproduction number and the equilibrium points. The schematic diagram of the model is illustrated in Figure 1 . The outlines of the adopted model are [10] :

A₁. The model considers the interactions between five compartments: $S_h$, $I_h$, $R_h$, $S_v$, $I_v$. Total human population at time t is denoted by $N_h\left(t\right)$ while the total vector (mosquito) population at t is $N_v\left(t\right)$.

A₂. The parameters of birth/migration and death-rates of the susceptible humans are ${\mu }_h$ and ${\alpha }_h$, respectively.

A₃. The parameters of birth/migration and death-rates of the susceptible vectors are ${\mu }_v$ and ${\alpha }_v$, respectively.

A₄. The rates of $I_h$ are: contact/transmission ${\beta }_h$, recovery ${\gamma }_h$, disease induced death ${\rho }_h$, environmental immunity due to nutrition and supplement $m_h$, and which infected humans enter the susceptible class at rate ${\tau }_h$.

A₅. The infected vectors’ contact/transmission rate is ${\beta }_v$.

Based on Assumptions $A_1$ to $A_5$, the model was formulated as:

$\text{d}{S}_h/\text{d}t={S^{\prime }}_h\left(t\right)={\mu }_h{N}_h\left(t\right)-{\beta }_h{S}_h\left(t\right)I_v\left(t\right)-{\alpha }_h{S}_h\left(t\right)+{\tau }_h{I}_h\left(t\right)$(1a)

$\text{d}{I}_h/\text{d}t={I^{\prime }}_h\left(t\right)={\beta }_h{S}_h\left(t\right)I_v\left(t\right)-\left({\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h\right)I_h\left(t\right)$(1b)

$\text{d}{R}_h/\text{d}t={R^{\prime }}_h\left(t\right)=\left({\gamma }_h+m_h\right)I_h\left(t\right)-{\alpha }_h{R}_h\left(t\right)$(1c)

$\text{d}{S}_v/\text{d}t={S^{\prime }}_v\left(t\right)={\mu }_v{N}_v\left(t\right)-{\beta }_v{S}_v\left(t\right)I_h\left(t\right)-{\alpha }_v{S}_v\left(t\right)$(1d)

$\text{d}{I}_v/\text{d}t={I^{\prime }}_v\left(t\right)={\beta }_v{S}_v\left(t\right)I_h\left(t\right)-{\alpha }_v{I}_v\left(t\right)$(1e)

In this section, we study the basic qualitative properties of system (1a)-(1e) including the non-negativity, boundness, and stability. To prove that the solution of the system (1a)-(1e) is non-negative and bounded; we apply the following lemma.

Lemma 1. The solutions of system (1a)-(1e) are nonnegative and bounded.

Proof. We have that

${{S^{\prime }}_h|}_{S_h=0}={\mu }_h{N}_h+{\tau }_h{I}_h\ge 0\text{for}\text{all}{I}_h\ge 0,{{I^{\prime }}_h|}_{I_h=0}={\beta }_h{S}_h{I}_v\ge 0\text{for}\text{all}{S}_h,I_v\ge 0,$

${{R^{\prime }}_h|}_{R_h=0}=\left({\gamma }_h+m_h\right)I_h\ge 0\text{for}\text{all}{I}_h\ge 0,{{S^{\prime }}_v|}_{S_v=0}={\mu }_v{N}_v>0,$

${{I^{\prime }}_v|}_{I_v=0}={\beta }_v{S}_v{I}_h\ge 0\text{for}\text{all}{S}_v,I_h\ge 0.$

This guarantees that: $\left(S_h\left(t\right)\mathrm{,}{I}_h\left(t\right)\mathrm{,}{R}_h\left(t\right)\mathrm{,}{S}_v\left(t\right)\mathrm{,}{I}_v\left(t\right)\right)\in {ℝ}_{\ge 0}^5$ for all $t\ge 0$ when $\left(S_h\left(0\right)\mathrm{,}{I}_h\left(0\right)\mathrm{,}{R}_h\left(0\right)\mathrm{,}{S}_v\left(0\right)\mathrm{,}{I}_v\left(0\right)\right)\in {ℝ}_{\ge 0}^5$. This proves the non-negativity of the solution.

Next, let:

$\Psi \left(t\right)=S_h\left(t\right)+I_h\left(t\right)+R_h\left(t\right)+S_v\left(t\right)+I_v\left(t\right)\mathrm{,}$

then

$\begin{array}{c}{\Psi }^{\prime }={\mu }_h{N}_h+{\mu }_v{N}_v-{\alpha }_h{S}_h-\left({\alpha }_h+{\rho }_h\right)I_h-{\alpha }_h{R}_h-{\alpha }_v{S}_v-{\alpha }_v{I}_v\mathrm{,}\\ \le {\mu }_h{N}_h+{\mu }_v{N}_v-\varphi \left[S_h+I_h+R_h+S_v+I_v\right]={\mu }_h{N}_h+{\mu }_v{N}_v-\varphi \Psi \mathrm{,}\end{array}$

where $\varphi =\min \left\{{\alpha }_h\mathrm{,}{\alpha }_h+{\rho }_h\mathrm{,}{\alpha }_v\right\}$, thus, $0\le \Psi \left(t\right)\le {\Delta }_1$ if $\Psi \left(0\right)\le {\Delta }_1$ for $t\ge 0$, where ${\Delta }_1=\left({\mu }_h{N}_h+{\mu }_v{N}_v\right)/\varphi$. Since $S_h\mathrm{,}{I}_h\mathrm{,}{R}_h\mathrm{,}{S}_v$, and $I_v$ are all nonnegative, then $0\le S_h\mathrm{,}{I}_h\mathrm{,}{R}_h\mathrm{,}{S}_v\mathrm{,}{I}_v\le {\Delta }_1$, if $S_h\left(0\right)+I_h\left(0\right)+R_h\left(0\right)+S_v\left(0\right)+I_v\left(0\right)\le {\Delta }_1$. This proves the boundedness of the solutions.

The model has two equilibrium points. To compute the disease-free equilibrium, the change with time was set to zero. The resulting system of nonlinear algebraic equations, then, was solved symbolically, using Matlab (see Appendices for Matlab-coding):

${\mu }_h{N}_h-{\beta }_h{S}_h{I}_v-{\alpha }_h{S}_h+{\tau }_h{I}_h=0$(2a)

${\beta }_h{S}_h{I}_v-{\alpha }_h{I}_h-{\gamma }_h{I}_h-{\rho }_h{I}_h-m_h{I}_h-{\tau }_h{I}_h=0$(2b)

${\gamma }_h{I}_h-{\alpha }_h{R}_h+m_h{I}_h=0$(2c)

${\mu }_v{N}_v-{\beta }_v{S}_v{I}_h-{\alpha }_v{S}_v=0$(2d)

${\beta }_v{S}_v{I}_h-{\alpha }_v{I}_v=0$(2e)

Stability analysis is at the heart of dynamical analysis. Only stable solutions can be noticed experimentally. Therefore, in this section we examine the global asymptotic stability of all equilibria by establishing suitable Lyapunov functions [12] and applying the Lyapunov-LaSalle asymptotic stability theorem (L-LAST) [13] - [15] . Let a function $L_j\left(S_h\mathrm{,}{I}_h\mathrm{,}{S}_v\mathrm{,}{I}_v\right)$ and ${\widetilde{ℳ}}_j$ be the largest invariant subset of

${ℳ}_j=\left\{\left(S_h\mathrm{,}{I}_h\mathrm{,}{R}_h\mathrm{,}{S}_v\mathrm{,}{I}_v\right)\mathrm{<mo>\; :\; </mo>}\frac{\text{d}{L}_j}{\text{d}t}=0\right\}\text{,}j=\mathrm{0,1.}$

Define a function

$Ϝ\left(\upsilon \right)=\upsilon -1-\ln v\text{.}$

Theorem 1. If ${ℛ}_0\le 1$, the disease-free equilibrium ${\mathcal{D}}_0$ is globally asymptotically stable (G.A.S).

Proof. Define

$\begin{array}{c}{L}_0=S_h^0Ϝ\left(\frac{S_h}{S_h^0}\right)+I_h+\frac{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}{{\beta }_v{S}_v^0}{S}_v^0Ϝ\left(\frac{S_v}{S_v^0}\right)\\ +\frac{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}{{\beta }_v{S}_v^0}{I}_v\mathrm{,}\end{array}$

note that, $L_0>0$ for all $S_h,I_h,S_v,I_v>0$, and $L_0\left(S_h^0\mathrm{,0,}{S}_v^0\mathrm{,0}\right)=0$. By differentiating $L_0$ along the solutions of system of OD Equations (1a)-(1e):

$\begin{array}{c}\frac{\text{d}{L}_0}{\text{d}t}=\left(1-\frac{S_h^0}{S_h}\right)\left({\mu }_h{N}_h-{\beta }_h{S}_h{I}_v-{\alpha }_h{S}_h+{\tau }_h{I}_h\right)\\ +{\beta }_h{S}_h{I}_v-\left({\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h\right)I_h\\ +\frac{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}{{\beta }_v{S}_v^0}\left(1-\frac{S_v^0}{S_v}\right)\left[{\mu }_v{N}_v-{\beta }_v{S}_v{I}_h-{\alpha }_v{S}_v\right]\\ +\frac{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}{{\beta }_v{S}_v^0}{\beta }_v{S}_v{I}_h-\frac{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}{{\beta }_v{S}_v^0}{\alpha }_v{I}_v\\ =\left(1-\frac{S_h^0}{S_h}\right)\left({\mu }_h{N}_h-{\alpha }_h{S}_h\right)+{\tau }_h{I}_h\left(1-\frac{S_h^0}{S_h}\right)\\ +\frac{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}{{\beta }_v{S}_v^0}\left(1-\frac{S_v^0}{S_v}\right)\left({\mu }_v{N}_v-{\alpha }_v{S}_v\right)\\ +\left({\beta }_h{S}_h^0-\frac{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}{{\beta }_v{S}_v^0}{\alpha }_v\right)I_v\\ =\left(1-\frac{S_h^0}{S_h}\right)\left({\mu }_h{N}_h-{\alpha }_h{S}_h\right)+{\tau }_h{I}_h\left(1-\frac{S_h^0}{S_h}\right)\\ +\frac{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}{{\beta }_v{S}_v^0}\left(1-\frac{S_v^0}{S_v}\right)\left({\mu }_v{N}_v-{\alpha }_v{S}_v\right)\\ +\frac{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}{{\beta }_v{S}_v^0}{\alpha }_v\left(\frac{{\beta }_h{S}_h^0}{{\alpha }_v}\frac{{\beta }_v{S}_v^0}{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}-1\right)I_v\mathrm{.}\end{array}$

Using the equilibrium condition: ${\mu }_h{N}_h={\alpha }_h{S}_{h_0}$, and ${\mu }_v{N}_v={\alpha }_v{S}_{v_0}$ we get:

$\begin{array}{c}\frac{\text{d}{L}_0}{\text{d}t}=-{\alpha }_h\frac{{(S_h-S_h^0)}^2}{S_h}+{\tau }_h{I}_h\left(1-\frac{S_h^0}{S_h}\right)\\ -{\alpha }_v\frac{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}{{\beta }_v{S}_v^0}\frac{{\left(S_v-S_v^0\right)}^2}{S_v}\\ +\frac{{\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h}{{\beta }_v{S}_v^0}{\alpha }_v\left({ℛ}_0^2-1\right)I_v.\end{array}$

Since ${ℛ}_0^2<{ℛ}_0<1$ and $S_h^0/S_h\ge 1$, then ${L^{\prime }}_0\le 0$ for all $S_h,I_h,S_v,I_v>0$. In addition ${L^{\prime }}_0=0$ when $S_h=S_h^0$, $S_v=S_v^0$ and $I_h=R_h=I_v=0$. The solution of system (1a)-(1e) tends to $ℳ=\left\{{\mathcal{D}}_0\right\}$. Therefore, by applying Lyapunov-LaSalle asymptotic stability theorem (L-LAST) [13] - [15] , we obtain that the disease-free equilibrium ${\mathcal{D}}_0$ is G.A.S.

Theorem 2. Let ${ℛ}_0>1$. If $I_h^{\mathrm{<mo>\; *\; </mo>}}/S_h^{\mathrm{<mo>\; *\; </mo>}}\le {\alpha }_h/{\tau }_h$. Then, the infected steady state ${\mathcal{D}}_1$ is (G.A.S.). In particular, is equivalent to:

$1<{ℛ}_0\le ℰ+1$

where,

$ℰ=\frac{{ℰ}_1{\alpha }_h\left({\alpha }_v{\mu }_v{N}_v{ℰ}_1+{\tau }_h\xi {\alpha }_h{\alpha }_v^3\left({ℛ}_0^2-1\right)\right)\left({ℰ}_2+{\alpha }_v{ℰ}_1\right)}{{\tau }_h\xi {\alpha }_h{\alpha }_v^2\left({ℰ}_1\left[{\beta }_h{\beta }_v{\mu }_v{N}_v+{\alpha }_h{\alpha }_v^2\right]+{\alpha }_h{\alpha }_v{ℰ}_2\left({ℛ}_0^2-1\right)\right)}\mathrm{,}$

${ℰ}_1=\xi {\alpha }_h{\alpha }_v{\beta }_v+{\beta }_h{\beta }_v{\mu }_v{N}_v\psi \mathrm{,}$

${ℰ}_2={\beta }_h\xi {\alpha }_h{\alpha }_v^2\mathrm{,}$

$\psi ={\alpha }_h+{\gamma }_h+{\rho }_h+m_h+{\tau }_h\mathrm{.}$

Proof. Define the global Lyapunov functional for Endemic equilibrium ${\mathcal{D}}_1$,

$\begin{array}{c}{L}_1=S_h^{\mathrm{<mo>\; *\; </mo>}}Ϝ\left(\frac{S_h}{S_h^{\mathrm{<mo>\; *\; </mo>}}}\right)+I_h^{\mathrm{<mo>\; *\; </mo>}}Ϝ\left(\frac{I_h}{I_h^{\mathrm{<mo>\; *\; </mo>}}}\right)+\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}{S}_v^{\mathrm{<mo>\; *\; </mo>}}Ϝ\left(\frac{S_v}{S_v^{\mathrm{<mo>\; *\; </mo>}}}\right)+\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}Ϝ\left(\frac{I_v}{I_v^{\mathrm{<mo>\; *\; </mo>}}}\right)\\ +\frac{{\tau }_h}{2\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}{\left[\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)+\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)\right]}^2\mathrm{.}\end{array}$

The time derivative of $L_1$ computed along solutions of the model is

$\begin{array}{c}\frac{\text{d}{L}_1}{\text{d}t}=\left(1-\frac{S_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h}\right){S^{\prime }}_h+\left(1-\frac{I_h^{\mathrm{<mo>\; *\; </mo>}}}{I_h}\right){I^{\prime }}_h+\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}\left(1-\frac{S_v^{\mathrm{<mo>\; *\; </mo>}}}{S_v}\right){S^{\prime }}_v+\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}\left(1-\frac{I_v^{\mathrm{<mo>\; *\; </mo>}}}{I_v}\right){I^{\prime }}_v\\ +\frac{{\tau }_h}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}\left[\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)+\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)\right]\left({S^{\prime }}_h+{I^{\prime }}_h\right)\mathrm{.}\end{array}$

$\begin{array}{c}\frac{\text{d}{L}_1}{\text{d}t}=\left(1-\frac{S_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h}\right)\left({\mu }_h{N}_h-{\beta }_h{S}_h{I}_v-{\alpha }_h{S}_h+{\tau }_h{I}_h\right)+\left(1-\frac{I_h^{\mathrm{<mo>\; *\; </mo>}}}{I_h}\right)\left({\beta }_h{S}_h{I}_v-\xi I_h\right)\\ +\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}\left(1-\frac{S_v^{\mathrm{<mo>\; *\; </mo>}}}{S_v}\right)\left({\mu }_v{h}_v-{\beta }_v{S}_v{I}_v-{\alpha }_v{S}_v\right)\\ +\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}\left(1-\frac{I_v^{\mathrm{<mo>\; *\; </mo>}}}{I_v}\right)\left({\beta }_v{S}_v{I}_h-{\alpha }_v{I}_v\right)+\frac{{\tau }_h}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}\\ \times \left[\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)+\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)\right]\left[{\mu }_h{N}_h-{\alpha }_h\left(S_h+I_h\right)-\left({\gamma }_h+{\rho }_h+m_h\right)I_h\right]\mathrm{.}\end{array}$

From the steady-states solution:

${\mu }_h{N}_h={\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}+{\alpha }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}-{\tau }_h{I}_h^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}$

${\mu }_h{N}_h={\alpha }_h\left(S_h^{\mathrm{<mo>\; *\; </mo>}}+I_h^{\mathrm{<mo>\; *\; </mo>}}\right)+\left({\gamma }_h+{\rho }_h+m_h\right)I_h^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}$

${\mu }_v{N}_v={\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}+{\alpha }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}$

we get:

$\begin{array}{c}\frac{\text{d}{L}_1}{\text{d}t}={\alpha }_h\left(1-\frac{S_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h}\right)\left[S_h^{\mathrm{<mo>\; *\; </mo>}}-S_h\right]+{\tau }_h\left(1-\frac{S_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h}\right)\left[I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right]\\ +\left(1-\frac{S_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h}\right)\left[{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}-{\beta }_h{S}_h{I}_v\right]+\left(1-\frac{I_h^{\mathrm{<mo>\; *\; </mo>}}}{I_h}\right){\beta }_h{S}_h{I}_v-\left(1-\frac{I_h^{\mathrm{<mo>\; *\; </mo>}}}{I_h}\right)\xi I_h\\ +\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}\left(1-\frac{S_v^{\mathrm{<mo>\; *\; </mo>}}}{S_v}\right)\left[{\alpha }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}-{\alpha }_v{S}_v\right]+\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}\left(1-\frac{S_v^{\mathrm{<mo>\; *\; </mo>}}}{S_v}\right)\left[{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}-{\beta }_v{S}_v{I}_v\right]\\ +\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}\left(1-\frac{I_v^{\mathrm{<mo>\; *\; </mo>}}}{I_v}\right){\beta }_v{S}_v{I}_h-\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}\left(1-\frac{I_v^{\mathrm{<mo>\; *\; </mo>}}}{I_v}\right){\alpha }_v{I}_v\\ +\frac{{\tau }_h}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}\left[\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)+\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)\right]\left[\left(S_h^{\mathrm{<mo>\; *\; </mo>}}-S_h\right)+\left(I_h^{\mathrm{<mo>\; *\; </mo>}}-I_h\right)\right]\\ +\frac{{\tau }_h\left({\gamma }_h+{\rho }_h+m_h\right)}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}\left(I_h^{\mathrm{<mo>\; *\; </mo>}}-I_h\right)\left[\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)+\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)\right]\mathrm{,}\end{array}$

since:

${\tau }_h\left(1-\frac{S_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h}\right)\left[I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right]={\tau }_h\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)\frac{{\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)}^2}{S_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}}+\frac{{\tau }_h}{S_h^{\mathrm{<mo>\; *\; </mo>}}}\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)\mathrm{,}$

therefore,

$\begin{array}{c}\frac{\text{d}{L}_1}{\text{d}t}=\frac{-{\alpha }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}}{S_n{S}_h^{\mathrm{<mo>\; *\; </mo>}}}{\left[S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right]}^2-\frac{{\tau }_h\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)}{S_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}}{\left[S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right]}^2+\frac{{\tau }_h}{S_h^{\mathrm{<mo>\; *\; </mo>}}}\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)\\ +{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}\left(1-\frac{S_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h}\right)-{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}\frac{S_h{I}_v{I}_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h}+{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}\\ -\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}\frac{{\alpha }_v}{S_v}{\left[S_v-S_v^{\mathrm{<mo>\; *\; </mo>}}\right]}^2+{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}\left(1-\frac{S_v^{\mathrm{<mo>\; *\; </mo>}}}{S_v}\right)-{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}\frac{S_v{I}_h{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{S_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v}\\ +{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}+\frac{{\tau }_h{\alpha }_h{S}_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_n^{\mathrm{<mo>\; *\; </mo>}}}\frac{{\left(S_n-S_n^{\mathrm{<mo>\; *\; </mo>}}\right)}^2}{S_n{S}_n^{\mathrm{<mo>\; *\; </mo>}}}\\ -\frac{{\tau }_h}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}\left[{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right]{\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)}^2\\ -\frac{{\tau }_h}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}\left[2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right]\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)\mathrm{.}\end{array}$

$\begin{array}{c}\frac{\text{d}{L}_1}{\text{d}t}=-\left[{\alpha }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}+{\tau }_h\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)+\frac{{\tau }_h{\alpha }_h{S}_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}\right]\frac{{\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)}^2}{S_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}}\\ -\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}\frac{{\alpha }_v}{S_v}{\left(S_v-S_v^{\mathrm{<mo>\; *\; </mo>}}\right)}^2\\ -\frac{{\tau }_h}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}\left[{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right]{\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)}^2\\ +{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}\left[4-\frac{S_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h}-\frac{S_h{I}_v{I}_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h}-\frac{S_v^{\mathrm{<mo>\; *\; </mo>}}}{S_v}-\frac{S_v{I}_h{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{S_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v}\right]\\ +\left[\frac{{\tau }_h}{S_h^{\mathrm{<mo>\; *\; </mo>}}}-\frac{{\tau }_h}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}\left[2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right]\right]\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)\end{array}$

$\begin{array}{c}\frac{\text{d}{L}_1}{\text{d}t}=-\left[{\alpha }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}-{\tau }_h{I}_h^{\mathrm{<mo>\; *\; </mo>}}+{\tau }_h{I}_h+\frac{{\tau }_h{\alpha }_h{S}_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}\right]\frac{{\left(S_h-S_h^{\mathrm{<mo>\; *\; </mo>}}\right)}^2}{S_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}}\\ -\frac{{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{{\beta }_v{S}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}}\frac{{\alpha }_v}{S_v}{\left(S_v-S_v^{\mathrm{<mo>\; *\; </mo>}}\right)}^2\\ -\frac{{\tau }_h}{\left(2{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right)S_h^{\mathrm{<mo>\; *\; </mo>}}}\left[{\alpha }_h+{\gamma }_h+{\rho }_h+m_h\right]{\left(I_h-I_h^{\mathrm{<mo>\; *\; </mo>}}\right)}^2\\ +{\beta }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}\left[4-\frac{S_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h}-\frac{S_h{I}_v{I}_h^{\mathrm{<mo>\; *\; </mo>}}}{S_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h}-\frac{S_v^{\mathrm{<mo>\; *\; </mo>}}}{S_v}-\frac{S_v{I}_h{I}_v^{\mathrm{<mo>\; *\; </mo>}}}{S_v^{\mathrm{<mo>\; *\; </mo>}}{I}_h^{\mathrm{<mo>\; *\; </mo>}}{I}_v}\right]\mathrm{.}\end{array}$

If $I_h^{\mathrm{<mo>\; *\; </mo>}}/S_h^{\mathrm{<mo>\; *\; </mo>}}\le {\alpha }_h/{\tau }_h$, then $\frac{\text{d}{L}_1}{\text{d}t}\le 0$ for all $S_h\mathrm{,}{I}_h\mathrm{,}{R}_h\mathrm{,}{S}_v\mathrm{,}{I}_v$. The solutions of system (1-5) limit to $ℳ$, the largest invariant subset of $\left\{\left(S_h^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{I}_h^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{R}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{S}_v^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}{I}_v^{\mathrm{<mo>\; *\; </mo>}}\right)\in R\mathrm{<mo>\; |\; </mo>}\frac{\text{d}{L}_1}{\text{d}t}=0\right\}$. Therefore by LaSalle invariant principle, if $I_h^{\mathrm{<mo>\; *\; </mo>}}/S_h^{\mathrm{<mo>\; *\; </mo>}}\le {\alpha }_h/{\tau }_h$, then Endemic equilibrium ${\mathcal{D}}_1$ is globally asymptotically stable.

Finally, we show that $I_h^{\mathrm{<mo>\; *\; </mo>}}/S_h^{\mathrm{<mo>\; *\; </mo>}}\le {\alpha }_h/{\tau }_h$. Using $S_h^{\mathrm{<mo>\; *\; </mo>}}$ and $I_h^{\mathrm{<mo>\; *\; </mo>}}$ of Equations ((3d), (3e)), we have

${\alpha }_h{S}_h^{\mathrm{<mo>\; *\; </mo>}}-{\tau }_h{I}_h^{\mathrm{<mo>\; *\; </mo>}}\equiv \frac{{\tau }_h\xi {\alpha }_h{\alpha }_v^2}{E_1}\left[ℰ+1-{ℛ}_0^2\right]$

Therefore, $I_h^{\mathrm{<mo>\; *\; </mo>}}/S_h^{\mathrm{<mo>\; *\; </mo>}}\le {\alpha }_h/{\tau }_h$, if $1<{ℛ}_0^2\le ℰ+1$.

In this section, the global stability of the system’s equilibria will be illustrated numerically. The simulations were done using the appropriate commands/package in Matlab. We use the values of the parameters presented in Table 1 . In addition, we make a comparison between single-infection and co-infection.

The interaction rates of human/mosquitos had an impact on increasing the infectious disease as time increased. The environmental effect on Malaria spread is a very complex factor, climate change, heavy rainfall and humidity provide good breeding areas for mosquitos to breed. The early diagnosis of these areas and the treatments or the controlling activities such as insecticidal spraying will help in decreasing the transmission rate and hence controlling the epidemic.