Impact of asymptomatic infections on malaria transmission dynamics

Abstract

A considerable proportion of malaria infections are asymptomatic, serving as reservoirs that sustain transmission. This study develops a deterministic mathematical model to examine the spread of malaria with a focus on asymptomatic infections. By analyzing key properties such as positivity, boundedness, and stability of equilibria, the model highlights the epidemiological relevance of these silent carriers. The basic reproduction number ${\mathfrak{R}}_0$ was calculated using the next-generation matrix method, revealing critical dynamics, including backward bifurcation. The quantitative result of the threshold ${\mathfrak{R}}_0$ reveals that asymptomatic cases contribute approximately 30 % to the basic reproduction number, while symptomatic cases account for about 70 %. These findings highlight the need for integrated control strategies that address both symptomatic and asymptomatic malaria cases to effectively manage and reduce malaria transmission.

1. Introduction

Malaria is a disease caused by protozoan parasites known as Plasmodium. It is a significant contributor to affliction and fatalities in many developing countries (Kendie et al., 2021). In the year 2023, the World Health Organization (WHO) reported that the number of global malaria cases increased by five million from 2021 to 2022, reaching 249 million cases. The African region is the most affected, accounting for 95 % of global malaria cases and 96 % of related deaths (W. H. Organization, 2023; Kassie et al., 2024). This situation emphasizes the urgent need for improved prevention, treatment, and research initiatives to tackle this significant public health challenge. To prevent and control malaria, several interventions have been developed, focusing on controlling mosquito populations and treating human infections (Camponovo et al., 2024). In Sub-Saharan Africa, a number of control initiatives have attempted to prevent malaria by managing the vector that acts as the main host for the Plasmodium parasite (Adegbite et al., 2023). In line with this, the World Health Organization (WHO) has approved the implementation of conventional vector control initiatives such as the use of Long Lasting Insecticide-treated Net (LLIN) and Indoor Residual Spray (IRS) to limit underlying challenges of malaria (Sougoufara et al., 2020). Besides, the WHO-approved control initiatives, people in malaria-endemic regions employ traditional malaria control strategies such as air conditioning, protective clothing, mosquito-repellent lotions, and insecticide-treated bed nets (Fischer, 2021). Additionally, artemisinin-based combination therapy (ACT) (Gari & Lindtjørn, 2018; Mattah et al., 2017), and awareness campaign (Febiriana et al., 2024) are crucial in preventing the spread of malaria. However, the persistence of asymptomatic malaria poses a significant challenge to malaria prevention and control efforts (Abebaw et al., 2022; Taffese et al., 2018). Asymptomatic malaria infections, wherein individuals carry the parasite without displaying symptoms, act as hidden reservoirs that perpetuate the transmission of malaria (Prah & Laryea-Akrong, 2024). Recent studies indicate a high prevalence of asymptomatic malaria in high-transmission areas such as Uganda, Nigeria, Ethiopia, and Cameroon. For instance, in Northern Uganda, 34.7 % of children aged 2 to 10 are asymptomatically infected (Agaba et al., 2022). In Ethiopia, asymptomatic cases account for 6.7 % of the general population, with rates among pregnant women ranging from 2.83 % to 11.2 % (Balcha et al., 2023). In Nigeria, asymptomatic Plasmodium falciparum infections represent 19 % of cases (Ibrahim et al., 2023). Also, its significant prevalence poses one of the serious challenges to malaria control efforts in Cameroon (Kaghou et al., 2024).

Various mathematical models have been developed to study malaria transmission dynamics (Diekmann et al., 1990; Feng et al., 2015). In 1911, Ross became the first to develop a malaria model aimed at explaining its transmission dynamics (Ross, 1911). His pioneering work accounted for the interactions between infected humans and mosquito hosts, laying the foundation for research in epidemic modeling. Later, various researchers extended this model by incorporating factors associated to possible control interventions (Al Basir & Abraha, 2023; Adongo et al., 2005; Haringo et al., 2024; Helikumi et al., 2024; ul Rehman et al., 2022; Aldila, 2021; Mandal et al., 2011).

Adegbite eta al (Adegbite et al., 2023). proposed a mathematical and computational model considering mobility and control factors. They suggest that the implementation of WHO-approved initiatives and traditional malaria control tools are effective control strategy against malaria importation. In alignment with this, Korsah et al. (Korsah et al., 2024) devised a malaria transmission model demonstrating that the combined use of insecticidal nets, indoor residual spraying, and personal protective measures can significantly reduce malaria prevalence. Alsulami et al. (Alsulami et al., 2024) developed an SIR-SI model incorporating environmental immunity factors, demonstrating that early detection, treatment, and insecticide use effectively reduce malaria transmission. In (Kushavaha & Sinha, 2024), authors proposed a host-vector dynamical model of vertical and horizontal malaria transmission by introducing preventive treatment in pregnant women. Their findings highlight scheduling regular preventive treatment strategies with quality antimalarial drugs and blocking vertical transmission can significantly reduce malaria infection. In (Misra et al., 2023), authors proposed a malaria model to evaluate the effects of television and social media promotions on malaria dynamics. Their analysis suggests that reducing the mosquito population alone may not successfully reduce the infection. In a related avenue, authors in (Srivastav & Ghosh, 2021) developed a dynamical mathematical model for malaria transmission that integrates saturated treatments with a logistically growing mosquito population. Their findings suggest that providing effective treatment to infected individuals, coupled with an increase in the mosquito death rate, can significantly aid in the eradication of malaria from the population. Banasiak et al. (Banasiak et al., 2023) developed a mathematical model of malaria transmission to analyze the effects of different demographic growths and the use of transmission-blocking drugs in humans. Their key finding reveal that one specific demographic growth pattern resulted in a considerably higher disease burden compared to others.

Xue et al. (Xue et al., 2023) proposed a malaria dynamical model that includes asymptomatic individuals, assuming that exposed humans can progress to either asymptomatic or symptomatic malaria. After some time, malaria-recovered individuals may return to the susceptible class due to waning immunity, which increases their risk of re-exposure to the parasite. However, this aspect has been not addressed in their model. Shi et al. (Shi et al., 2024) developed a compartmental mathematical model for malaria that includes asymptomatic carriers, suggesting they can transmit the infection to mosquitoes without displaying clinical symptoms. However, their model neglects the effects of waning immunity in recovered individuals as well as the potential progression of asymptomatic carriers to symptomatic complications. Cai et al. (Cai et al., 2017) developed a malaria model that incorporates asymptomatic infections and super-infection to explore various prevention interventions. Their model omits crucial assumptions, such as waning immunity in recovered individuals and the dynamics of malaria within the treated compartment. Similarly, the authors in (Chitnis et al., 2008) developed a mathematical model including exposed individuals and mosquitoes, analyzing the effects of disease parameters on transmission and validating their results through numerical simulations. Recently, the authors in (Haringo et al., 2024) proposed a compartmental mathematical model to examine the effects of insecticide usage and drug treatment interventions on malaria. However, this model does not account for asymptomatic cases that can sustain ongoing transmission. Additionally, it lacks real-world malaria data, which is critical for enhancing predictions of future outbreaks. To elucidate the advancements in our study, we also present a comparative schematic that highlights the key assumptions, findings, and limitations in relation to prior works on asymptomatic malaria, as shown in Table 1.

Table 1.

Comparison of researchers’ works on asymptomatic infection, detailing authors, assumptions, findings, and limitations.

Authors	Key assumption	Findings	Limitations
Xue et al. (Xue et al., 2023)	Consider variable standard incidence rates, incorporating exposed and asymptomatic classes	stability transitions occur at ${\mathfrak{R}}_0=1$. Additionally asymptomatic carriers highly influence malaria dynamics	Ignores waning immunity, omits treatment class, neglects potential progression of asymptomatic carriers, and lacks real-world data
Shi et al. (Shi et al., 2024)	Asymptomatic infections often arise from partial immunity to malaria due to repeated exposures	Malaria will become extinct when ${\mathfrak{R}}_0<1$ and persist when ${\mathfrak{R}}_0>1$	In addition to the limitations identified in Xue et al. (Xue et al., 2023), the global stability of the positive periodic solution in this model remains unaddressed
Cai et al. (Cai et al., 2017)	Assumes that asymptomatic individuals can become symptomatic upon reinfection, and symptomatic individuals may revert to asymptomatic status due to incomplete treatment	Malaria prevalence is highest with incomplete treatment of symptomatic individuals and lowest when all individuals transition to symptomatic status and are treated successfully	Treatment is not included as an epidemiological compartment and lacks integration of real-world data
Chitnis et al. (Chitnis et al., 2008)	Assumes a constant immigration into the susceptible compartment and excludes the movement of exposed individuals due to their short time of the exposed stage	Mosquito biting rates can be reduced through interventions like LLIN and IRS, while targeting human recovery rates via prompt diagnosis and treatment of infected individuals can control malaria	Significant epidemiological compartments, primarily asymptomatic cases and treatment, are not considered
Haringo et al. (Haringo et al., 2024)	Assumes that susceptible individuals transition between aware and unaware states based on their level of awareness, reflecting the impact of media on insecticide usage	Integration of awareness-based insecticide usage with malaria treatment holds the potential for malaria elimination	Exclusion of asymptomatic carriers and lack of real-world data integration
Laishram et al. (Laishram et al., 2012)	Insights from various fields contribute to a comprehensive understanding of the asymptomatic infection	The review findings underscore the complexity of malaria control and highlight the necessity of incorporating asymptomatic cases into research and public health strategies	Significant gaps in understanding asymptomatic malaria remain unclear or under-researched

In this study, a mathematical model for malaria infection is being formulated and studied by contemplating the following conditions (see Table 2). Asymptomatic malaria serves as a key reservoir for the parasite without displaying symptoms, yet it has received less attention than symptomatic infections [38, 39, 40]. Therefore, incorporating their transmission dynamics into models is essential. Furthermore, including treatment and recovered classes is critical for understanding malaria dynamics. Treatment significantly reduces infectiousness, thereby lowering transmission rates. Additionally, a distinct recovered class facilitates the development of immunity over time, which is essential for predicting long-term transmission patterns. By addressing these gaps, this model aims to provide critical insights for public health in malaria prevention and control. This study utilizes data from Ethiopia, a country currently facing substantial malaria challenges.

Table 2.

Description of parameters involved in the malaria model (1).

Parameters	Meanings	Units
Λh	Recruitment rate of susceptible humans either by birth or immigration	year−1
Λm	Recruitment of susceptible mosquitoes	year−1
ϵ	Rate of waning of immunity	year−1
σ, q	Constant proportions between 0 and 1	unit-less
β1	The rate of transmission from Im to Sh class	year−1
β2	The rate of transmission from Ah to Sm class	year−1
β3	The rate of transmission from Ih to Sm class	year−1
ϕ	The progression rate asymptomatic humans to either Ih or Rh class	year−1
r	The progression rate expose humans to either Ah or Ih class	year−1
μh	Natural death rate of humans	year−1
η	Drug treatment rate of symptomatic infected humans	year−1
δ	Malaria-induced death rate of symptomatic infected humans	year−1
ρ	Recovery rate of treated humans	year−1
φ	Death rate due to malaria treatment failure	year−1
γ	Rate of insecticides usage	year−1
μm	Natural death rate of mosquitoes	year−1

The remaining parts of this paper are organized as follows: Section 2 provides the formulation of the model. Section 3 focuses on the analysis of the model. Section 4 conducts and discusses the various numerical simulations. The last Section 5 offers the concluding remarks.

2. The model formulation

To portray the transmission dynamics of malaria infection, we develop a compartmental mathematical model using a nonlinear system of ordinary differential equations. The model entails human and mosquito populations. The model flow is susceptible-exposed-asymptomatic-infected-treated-recovered-susceptible (S–E–A–I–T–R–S) patterns for the human population and the susceptible-infected (S–I) pattern for the mosquito population. The total human population denoted by N_h is subdivided into six mutually disjoint classes: susceptible S_h, exposed E_h, asymptomatic infected A_h, symptomatic infected I_h, treated T_h, and recovered R_h classes. Therefore, we express the total human population N_h, as:

$$N_h=S_h+E_h+A_h+I_h+T_h+R_h.$$

On the other hand, the mosquito population is sub-divided into two classes: susceptible mosquitoes S_m and infected mosquitoes I_m. The total mosquito population, denoted as N_m, is given by:

$$N_m=S_m+I_m.$$

Individuals who are born or immigrate are assumed to be susceptible, denoted by Λ_h influxed into the susceptible class. As mentioned in (Aguilar & Gutierrez, 2020; Zahid et al., 2023), human-mosquito contact rates greatly vary and depend on factors such as the number of susceptible and infectious species in the area. It is also assumed that total population sizes of humans in the community determine the average number of mosquito bites that humans experience. Thus, humans in a susceptible class become infected if they come into contact with infected mosquitoes at the transmission rate β₁ with the force of infection $\frac{{\beta }_1{S}_h{I}_m}{N_h}$ and then move to the exposed class. We assume that recovery from malaria does not confer lifelong immunity. Therefore, ϵ rate of recovered humans lose their immunity and return to the susceptible class without re-exposure to the parasite. Due to the arrival of secondary infections from the susceptible human class, there is an increase in the number of exposed human class by $\frac{{\beta }_1{S}_h{I}_m}{N_h}$ humans. Exposed humans remain in that class for a mean duration of $\frac{1}{r}$. They progress at a rate r, with a proportion σ entering the asymptomatic infected class A_h, and the remaining proportion (1 − σ) entering the symptomatic infected class I_h. We assume that an asymptomatic case serves as a reservoir for transmission and is a precursor to symptomatic malaria without treatment (Kimenyi et al., 2019; Kun et al., 2002). Thus, we assume a proportion qϕ of asymptomatic infectious individuals are precursors to the symptomatic infected ones, whereas the remaining (1 − q)ϕ are added to the recovery class due to screening followed by on spot treatment tactic (Singh et al., 2022).

It is assumed that symptomatic humans who are treated enter the recovery class at the rate η and die due to treatment failure at a rate φ. The recovery class increases as humans recover from both asymptomatic and symptomatic forms of malaria, with rates ϕ and η respectively. Finally, the human population in each class is decreased by the natural death rate μ_h. To simplify the model, we omit environmental drivers affecting mosquito populations. This approach improves analytical tractability, and facilitates clearer interpretation of results. Thus, we assume a constant recruitment of mosquitoes into the susceptible mosquito class at a rate Λ_m. When a susceptible mosquito S_m bites an asymptomatic infectious human A_h at rate β₂, the corresponding force of infection is given by $\frac{{\beta }_2{S}_m{A}_h}{N_h}$. Similarly, when a susceptible mosquito bites a symptomatic infectious human I_h at rate β₃, the force of infection is $\frac{{\beta }_3{S}_m{I}_h}{N_h}$. Consequently, the human-to-mosquito force of infection is defined as the sum of contributions from the infectious classes A_h and I_h, expressed mathematically as $\frac{{\beta }_2{S}_m{A}_h}{N_h}+\frac{{\beta }_3{S}_m{I}_h}{N_h}$. Finally, mosquitoes population in each class is reduced by either the insecticide usage at the rate γ or by the natural death at the rate μ_m.

Based on the aforementioned assumptions and the interaction between humans and mosquitoes, the governing system of ordinary differential equations model is given as

$$\left\{\begin{array}{l}\frac{d{S}_h}{dt}={\mathrm{\Lambda }}_h-\frac{{\beta }_1{S}_h{I}_m}{N_h}+ϵR_h-{\mu }_h{S}_h,\\ \frac{d{E}_h}{dt}=\frac{{\beta }_1{S}_h{I}_m}{N_h}-(r+{\mu }_h)E_h,\\ \frac{d{A}_h}{dt}=\sigma r{E}_h-(\varphi +{\mu }_h)A_h,\\ \frac{d{I}_h}{dt}=(1-\sigma )r{E}_h+q\varphi A_h-(\eta +\delta +{\mu }_h)I_h,\\ \frac{d{T}_h}{dt}=\eta I_h-(\rho +\phi +{\mu }_h)T_h,\\ \frac{R_h}{dt}=(1-q)\varphi A_h+\rho T_h-(ϵ+{\mu }_h)R_h,\\ \frac{d{S}_m}{dt}={\mathrm{\Lambda }}_m-\frac{{\beta }_2{S}_m{A}_h}{N_h}-\frac{{\beta }_3{S}_m{I}_h}{N_h}-(\gamma +{\mu }_m)S_m,\\ \frac{d{I}_m}{dt}=\frac{{\beta }_2{S}_m{A}_h}{N_h}+\frac{{\beta }_3{S}_m{I}_h}{N_h}-(\gamma +{\mu }_m)I_m\end{array}\right)$$ (1)

subjected to the initial conditions,

$$S_{h0}>0,E_{h0}>0,A_{h0}>0,I_{h0}>0,T_{h0}>0,R_{h0}>0,S_{m0}>0,I_{m0}>0.$$ (2)

We assume all state variables and parameters involved in the model are non-negative.

3. Model analysis

This section provides mathematical and epidemiological well-posedness, including positivity and boundedness, existence of steady states, computing malaria-reproductive number, stability analysis of steady states, and sensitivity analysis of the malaria model (1).

3.1. Positivity and boundedness of the model solutions

Theorem 1

Every solution of the model (1) remains positive for all t > 0 given that the initial conditions provided in (1) are positive.

Proof

Define

$$t_1=sup\{t>0:S_h(t)>0,E_h(t)>0,A_h(t)>0,I_h(t)>0,T_h(t)>0,R_h(t)>0,S_m(t)>0,I_m(t)>0\}.$$

Given that.

S_h(0) > 0, E_h(0) > 0, A_h(0) > 0, I_h(0) > 0, T_h(0) > 0, R_h(0) > 0, S_m(0) > 0, and I_m(0) > 0. Hence, t₁ > 0. If t₁ < ∞, then S_h(t), E_h(t), A_h(t), I_h(t), T_h(t), R_h(t), S_m(t) and I_m(t) all take zero at t₁. Accordingly, from the first equation of model (1), we have

$$\frac{d{S}_h}{dt}+\left({\mu }_h+\frac{{\beta }_1{I}_m}{N_h}\right)S_h={\mathrm{\Lambda }}_h+ϵR_h.$$

Applying the integrating factor, we obtain

$$S_h(t_1)=S_h(0)e^{-{\int }_0^{t_1}\left({\mu }_h+\frac{{\beta }_1{I}_m(\tau )}{N_h(\tau )}\right)d\tau }+e^{-{\int }_0^{t_1}\left({\mu }_h+\frac{{\beta }_1{I}_m(\tau )}{N_h(\tau )}\right)d\tau }\times {\int }_0^{t_1}\left[\left({\mathrm{\Lambda }}_h+ϵR_h(\tau )\right)e^{{\int }_0^t\left({\mu }_h+\frac{{\beta }_1{I}_m(\xi )}{N_h(\xi )}\right)d\xi }\right]d\tau >0.$$

From the second equation of model (1), we have

$$\frac{d{E}_h}{dt}+(r+{\mu }_h)E_h=\frac{{\beta }_1{S}_h{I}_m}{N_h}.$$

Applying integrating in the same way, we have

$$E_h(t_1)=E_h(0)e^{-{\int }_0^{t_1}\left(r+{\mu }_h\right)d\tau }+e^{-{\int }_0^{t_1}\left(r+{\mu }_h\right)d\tau }\times {\int }_0^{t_1}\left[\left(\frac{{\beta }_1{S}_h(\tau )I_m(\tau )}{N_h(\tau )}\right)e^{{\int }_0^t\left(r+{\mu }_h\right)d\xi }\right]d\tau >0.$$

Similarly, we can demonstrate that the state variables A_h(t) > 0, I_h(t) > 0, T_h(t) > 0, R_h(t) > 0, S_m(t) > 0, and I_m(t) > 0.

To ensure the biological relevance of model (1), both human and mosquito populations remain bounded for all t ≥ 0. We now state the following Theorem.

Theorem 2

All solution of the system (1) with initial data (1) are bounded above in a biological viable set Ω.

$$\mathrm{\Omega }=\left\{\left(S_h,E_h,A_h,I_h,T_h,R_h,S_m,I_m\right)\in R_{+}^8:0\le S_h+E_h+A_h+I_h+T_h+R_h\le \frac{{\mathrm{\Lambda }}_h}{{\mu }_h},0\le S_m+I_m\le \frac{{\mathrm{\Lambda }}_m}{{\mu }_m}\right\}.$$

Proof

Adding the first six equations of model (1) gives the total dynamics of the human host, represented by:

$$\frac{d{N}_h}{dt}={\mathrm{\Lambda }}_h-{\mu }_h{N}_h-\delta I_h-\phi T_h\le {\mathrm{\Lambda }}_h-{\mu }_h{N}_h.$$ (3)

Solving differential inequality (3) by using integrating factor method, we get

$$N_h(t)\le \frac{{\mathrm{\Lambda }}_h}{{\mu }_h}+\left(N_h(0)-\frac{{\mathrm{\Lambda }}_h}{{\mu }_h}\right)e^{-{\mu }_{h}t}.$$

Now, at t = 0, we have $N_h(0)\le \frac{{\mathrm{\Lambda }}_h}{{\mu }_h}>0$. Furthermore, for all t > 0, it follows that $N_h(t)\le \frac{{\mathrm{\Lambda }}_h}{{\mu }_h}$. Therefore, if N_h(0) is within the set Ω, it can be concluded that N_h(t) remains in Ω for all t > 0. Then we can have

$$0\le S_h+E_h+A_h+I_h+T_h+R_h\le \frac{{\mathrm{\Lambda }}_h}{{\mu }_h}.$$ (4)

Similarly, by adding the last two equations of model (1), we obtain the total dynamics of the mosquitoes at time t, given by

$$\begin{array}{lll}\frac{d{N}_m}{dt}& =& {\mathrm{\Lambda }}_m-(\gamma +{\mu }_m)N_m\end{array}$$ (5)

$$\begin{array}{lll}\frac{d{N}_m}{dt}& \le & {\mathrm{\Lambda }}_m-{\mu }_m{N}_m.\end{array}$$ (6)

Solving differential inequality (6), we deduce that

$${\mathrm{lim\; sup}}_{t\to \infty }{N}_m(t)\le \frac{{\mathrm{\Lambda }}_m}{{\mu }_m}.$$

Thus, we have

$$0\le S_m+I_m\le \frac{{\mathrm{\Lambda }}_m}{{\mu }_m}.$$ (7)

Regarding equations (4), (7) the set Ω is positively invariant. Thus, analyzing the system dynamics of model (1) within invariant set Ω ensures both biological relevance and mathematical well-posedness of model (1). Therefore, any solution from system (1) with an initial data in (2) will remain within Ω, ∀t > 0.

3.2. Malaria-free equilibrium point and reproduction number

3.2.1. Malaria-free equilibrium point

A malaria-free equilibrium point (MFEP) is defined as a state in which malaria is absent from the population. To determine the MFEP, denoted by E₀, we set the values E_h = A_h = I_h = I_m = 0 in the model described by (1). By solving the resulting system of equations, we determine

$$E_0=\left(\frac{{\mathrm{\Lambda }}_h}{{\mu }_h},0,0,0,0,0,\frac{{\mathrm{\Lambda }}_m}{\gamma +{\mu }_m},0\right).$$ (8)

3.2.2. Basic reproduction number

In the study of infectious disease models, the basic reproduction number, denoted as ${\mathfrak{R}}_0$, refers to the expected number of new cases generated by a single infectious individual in a completely susceptible population (Diekmann et al., 1990). If ${\mathfrak{R}}_0<1$, the disease will decline, as each infected individual generates fewer than one new case. Conversely, if ${\mathfrak{R}}_0>1$, the infection can spread, with each infected individual producing more than one new infection. To compute the reproduction number, ${\mathfrak{R}}_0$ of model (1), we employ the next-generation matrix approach, as outlined in (Van den Driessche & Watmough, 2002). Accordingly, the malarial states are E_h, A_h, I_h, and I_m. Thus, the computation for R₀ starts with rewriting the malarial compartments of the model (1) in the form of:

$$\frac{dX}{dt}=\mathcal{F}-\mathcal{V},$$

where X represents the state variables defined as $X=\left(S_h,E_h,A_h,I_h,T_h,R_h,S_m,I_m\right)$, the secondary infection vectors $\mathcal{F}$ and the transition vectors $\mathcal{V}$ are defined as follows:

$$\mathcal{F}=\left(\begin{array}{l}\frac{{\beta }_1{S}_h{I}_m}{N_h}\\ 0\\ 0\\ \frac{{\beta }_2{S}_m{A}_h}{N_h}+\frac{{\beta }_3{S}_m{I}_h}{N_h}\end{array}\right),\mathcal{V}=\left(\begin{array}{l}(r+{\mu }_h)E_h\\ (\varphi +{\mu }_h)A_h-\sigma r{E}_h\\ (\delta +\eta +{\mu }_h)I_h-(1-\sigma )r{E}_h-q\varphi A_h\\ (\gamma +{\mu }_m)I_m\end{array}\right).$$ (9)

Then the corresponding Jacobin matrices of $\mathcal{F}$ and $\mathcal{V}$ at the MFEP E₀ are, respectively,

$$F=\left(\begin{array}{llll}0& 0& 0& {\beta }_1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& \frac{{\beta }_2{\mu }_h}{\gamma +{\mu }_m}& \frac{{\beta }_3{\mu }_h}{\gamma +{\mu }_m}& 0\end{array}\right),V=\left(\begin{array}{llll}r+{\mu }_h& 0& 0& 0\\ -r\sigma & \varphi +{\mu }_h& 0& 0\\ -r(1-\sigma )& -q\varphi & \delta +\eta +{\mu }_h& 0\\ 0& 0& 0& \gamma +{\mu }_m\end{array}\right).$$ (10)

Now the matrix FV⁻¹ is given by

$$F=\left(\begin{array}{llll}0& 0& 0& \frac{{\beta }_1}{\gamma +{\mu }_m}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{\sigma r{\beta }_2{B}_0}{(r+{\mu }_h)(\varphi +{\mu }_h)}+\frac{B_0{\beta }_{3}r(1-\sigma )(\varphi +{\mu }_h)}{(\delta +\eta +{\mu }_h)(r+{\mu }_h)(\varphi +{\mu }_h)}& \frac{B_0{\beta }_2}{\varphi +{\mu }_h}+\frac{B_{0}q\varphi {\beta }_3(r+{\mu }_h)}{(\delta +\eta +{\mu }_h)(r+{\mu }_h)(\varphi +{\mu }_h)}& \frac{B_0{\beta }_3}{(\delta +\eta +{\mu }_h)}& 0\end{array}\right),$$

where $B_0=\frac{{\mathrm{\Lambda }}_m{\mu }_h}{{\mathrm{\Lambda }}_h(\gamma +{\mu }_m)}$. The reproductive number ${\mathfrak{R}}_0$ for model (1) is defined as the spectral radius of the matrix.

FV⁻¹, given by

$${\mathfrak{R}}_0=\sqrt{\frac{B_0{\beta }_{1}r}{\gamma +{\mu }_m}\left(\frac{\sigma {\beta }_2}{(\varphi +{\mu }_h)(r+{\mu }_h)}+\frac{\left((1-\sigma )\right){\beta }_3}{(\eta +\delta +{\mu }_h)(r+{\mu }_h)}\right)}=\sqrt{{\mathfrak{R}}_{I_m}\left({\mathfrak{R}}_{A_h}+{\mathfrak{R}}_{I_h}\right)}$$ (11)

where ${\mathfrak{R}}_{I_m}=\frac{B_0{\beta }_{1}r}{\gamma +{\mu }_m}$, ${\mathfrak{R}}_{A_h}=\frac{\sigma {\beta }_2}{(\varphi +{\mu }_h)(r+{\mu }_h)}$, and ${\mathfrak{R}}_{I_h}=\frac{(1-\sigma ){\beta }_3}{(\eta +\delta +{\mu }_h)(r+{\mu }_h)}$. In this, each expression ${\mathfrak{R}}_{I_h}$, ${\mathfrak{R}}_{I_A}$, and ${\mathfrak{R}}_{I_m}$ carries its biological interpretations. The term ${\mathfrak{R}}_{I_m}$ implies the expected number of susceptible humans that a single infectious mosquito infects in its infectious period, where $\frac{1}{\gamma +{\mu }_m}$ determines the infectious period of mosquitoes. In addition, B₀β₁r determines the amount of susceptible humans that infected mosquitoes infect per a unit time. On the contrary, the term ${\mathfrak{R}}_{A_h}/{\mathfrak{R}}_{A_h}$ determines an expected number of susceptible mosquitoes that a single asymptomatic/symptomatic infected human infects in its infectious period. In this context, the asymptomatic/symptomatic infected human stays in that class with an average infectious period of $\frac{1}{(\varphi +{\mu }_h)(r+{\mu }_h)}/\frac{1}{(\eta +\delta +{\mu }_h)(r+{\mu }_h)}$. During this period, the asymptomatic/symptomatic person infects σβ₂/(1 − σ)β₃ mosquitoes per a unit time.

3.3. Local stability of the malaria-free equilibrium point

Theorem 3

The MFEP,E₀, of model (1) is locally asymptotically stable if ${\mathfrak{R}}_0<1$.

proof

We open the proof by evaluating the Jacobian matrix$\mathfrak{J}(E_0)$of model (1) at MFEP E₀, that is

$$\mathfrak{J}(E_0)=\left(\begin{array}{llllllll}-{\mu }_h& 0& 0& 0& 0& ϵ& 0& -{\beta }_1\\ 0& -r-{\mu }_h& 0& 0& 0& 0& 0& {\beta }_1\\ 0& r\sigma & -\varphi -{\mu }_h& 0& 0& 0& 0& 0\\ 0& r(1-\sigma )& q\varphi & -\delta -\eta -{\mu }_h& 0& 0& 0& 0\\ 0& 0& 0& \eta & -\rho -\phi -{\mu }_h& 0& 0& 0\\ 0& 0& (1-q)\varphi & 0& \rho & -ϵ-{\mu }_h& 0& 0\\ 0& 0& -B_0{\beta }_2& -B_0{\beta }_3& 0& 0& -\gamma -{\mu }_m& 0\\ 0& 0& B_0{\beta }_2& B_0{\beta }_3& 0& 0& 0& -\gamma -{\mu }_m\end{array}\right).$$ (12)

It is evident that the matrix$\mathfrak{J}(E_0)$is a block diagonal matrix. The first and fourth block sub-matrices constitute its eigenvalues. From the first block, denoted as${\mathfrak{J}}_1(E_0)$, we observe thatλ₁ = −μ_h. The remaining eigenvalues, λ_i for i = 2, 3, 4, 5, 6, 7, 8, are determined from the fourth sub-block matrix, denoted as ${\mathfrak{J}}_4(E_0)$, of the matrix $\mathfrak{J}(E_0)$. Thus, we have

$${\mathfrak{J}}_4(E_0)=\left(\begin{array}{lllllll}-r-{\mu }_h& 0& 0& 0& 0& 0& {\beta }_1\\ r\sigma & -\varphi -{\mu }_h& 0& 0& 0& 0& 0\\ r(1-\sigma )& q\varphi & -\delta -\eta -{\mu }_h& 0& 0& 0& 0\\ 0& 0& \eta & -\rho -\phi -{\mu }_h& 0& 0& 0\\ 0& (1-q)\varphi & 0& \rho & -ϵ-{\mu }_h& 0& 0\\ 0& -B_0{\beta }_2& -B_0{\beta }_3& 0& 0& -\gamma -{\mu }_m& 0\\ 0& B_0{\beta }_2& B_0{\beta }_3& 0& 0& 0& -\gamma -{\mu }_m\end{array}\right).$$ (13)

By expanding the determinant $|{\mathfrak{J}}_4(E_0)-\lambda I|=0$, we obtain the characteristic equation of degree seven

$$\left({\lambda }^4+{\alpha }_1{\lambda }^3+{\alpha }_2{\lambda }^2+{\alpha }_3\lambda +{\alpha }_4\right)(\lambda +\rho +\phi +{\mu }_h)(\lambda +ϵ+{\mu }_h)(\lambda +\gamma +{\mu }_m)=0.$$ (14)

From the characteristic equation (14), we obtain three eigen values: namely λ₅ = −(ρ + φ + μ_h), λ₆ = −(ϵ + μ_h)and λ₇ = −(γ + μ_m). The remaining four eigenvalues: λ₂, λ₃, λ₄ and λ₈ are obtained from the characteristic equation

$${\lambda }^4+{\alpha }_1{\lambda }^3+{\alpha }_2{\lambda }^2+{\alpha }_3\lambda +{\alpha }_4=0,$$ (15)

where

$$\begin{array}{lll}{\alpha }_1& =& (\eta +\delta +{\mu }_h)+(\gamma +{\mu }_m)+(r+{\mu }_h)+(\varphi +{\mu }_h),\\ {\alpha }_2& =& (r+\eta +\varphi +\delta +3{\mu }_h)(\gamma +2{\mu }_h+{\mu }_m)+\varphi (\eta +\delta )+r(\eta +\varphi +\delta ),\\ {\alpha }_3& =& (\gamma +{\mu }_m)(\delta +\eta +{\mu }_h)(\varphi +{\mu }_h)(r+{\mu }_h)(1-{\mathfrak{R}}_0)\\ & +& r{B}_0{\beta }_3\left({(\eta +\delta +{\mu }_h)}^2+{(\gamma +{\mu }_m)}^2+{(r+{\mu }_h)}^2+{(\varphi +{\mu }_h)}^2\right),\\ {\alpha }_4& =& (\eta +\delta +{\mu }_h)(r+{\mu }_h)(\varphi +{\mu }_h)(\gamma +{\mu }_m)\left(1-{\mathfrak{R}}_0\right).\end{array}$$

The coefficients α₁ and α₂ are both positive. Furthermore, when ${\mathfrak{R}}_0<1$, the coefficients α₃ and α₄ will also be positive. Consequently, all roots of the equation (15) will have negative real parts. This implies that the eigenvalues λ₂, λ₃, λ₄, and λ₈ are all negative. Therefore, the MFEP E₀ is locally asymptotically stable within the region Ω when ${\mathfrak{R}}_0<1$.

3.4. Global stability of MFEP

In this subsection, we adopt the approach described by (Castillo-Chavez & Song, 2004) to show MFEP is global asymptotically stable. The model (1) can be rewritten as follows:

$$\left\{\begin{array}{l}\frac{d{Z}_1}{dt}=F(Z_1,Z_2)\\ \frac{d{Z}_2}{dt}=G(Z_1,Z_2),G(Z_1,0)=0,\end{array}\right)$$ (16)

where Z₁ = (S_h, R_h, S_m) and Z₂ = (E_h, A_h, I_h, T_h, I_v) stand for non-infected and infected compartments, respectively. Additionally, the MFEP of the new system is denoted by $E_0=(Z_1^0,0)=\left(\frac{{\mathrm{\Lambda }}_h}{{\mu }_h},0,0,0,0,0,\frac{{\mathrm{\Lambda }}_m}{\gamma +{\mu }_m},0\right)$ representing the malaria-free state of the new system. Assume that.

• $(A_1)\frac{d{Z}_1}{dt}=F(Z_1,0),Z_1^0$ is globally asymptotically stable.

• $(A_2)G(Z_1,Z_2)=A{Z}_2^T-\hat{G}(Z_1,Z_2),\hat{G}(Z_1,Z_2)\ge 0$ for all (Z₁, Z₂) ∈ Ω where matrix A is a Metzle-matrix in which the model makes biologically meaningful in Ω defined earlier. Then E₀ is globally asymptotically stable such that ${\mathfrak{R}}_0<1$.

Having the aforementioned results, we can state the following theorem pertaining to model (1).

Theorem 4

The MFEP of model (1) is a globally asymptotic stable (G.A.S.) when ${\mathfrak{R}}_0<1$ and the assumptions (A₁) and (A₂) hold true.

Proof

Now, we needed to confirm that conditions A₁ and A₂ hold true to prove that the MFEP E₀ is globally asymptotically stable (GAS) when ${\mathfrak{R}}_0<1$. Now the two vector-valued functions F(Z₁, Z₂) and G(Z₁, Z₁) are given as:

$$F(Z_1,Z_2)=\left(\begin{array}{l}{\mathrm{\Lambda }}_h-\frac{{\beta }_1{S}_h{I}_m}{N_h}+ϵR_h-{\mu }_h{S}_h\\ (1-q)\varphi A_h+\rho T_h-(ϵ+{\mu }_h)R_h\\ {\mathrm{\Lambda }}_m-\frac{{\beta }_2{S}_m{A}_h}{N_h}-\frac{{\beta }_3{S}_m{I}_h}{N_h}-(\gamma +{\mu }_m)S_m\end{array}\right),G(Z_1,Z_2)=\left(\begin{array}{l}\frac{{\beta }_1{S}_h{I}_m}{N_h}-(r+{\mu }_h)E_h\\ \sigma r{E}_h-(\varphi +{\mu }_h)A_h\\ (1-\sigma )r{E}_h+q\varphi A_h-(\eta +\delta +{\mu }_h)I_h\\ \eta I_h-(\rho +\phi +{\mu }_h)T_h\\ \frac{{\beta }_2{S}_m{A}_h}{N_h}+\frac{{\beta }_3{S}_m{I}_h}{N_h}-(\gamma +{\mu }_m)I_m\end{array}\right).$$

Now we deal with the simplified systems to condition (A₁)

$$\frac{d{Z}_1}{dt}=F(Z_1,0)⟺\left\{\begin{array}{l}\dot{S_h}={\mathrm{\Lambda }}_h+ϵR_h-{\mu }_h{S}_h\\ \dot{R_h}=-(ϵ+{\mu }_h)R_h\\ \dot{S_M}={\mathrm{\Lambda }}_m-(\gamma +{\mu }_m)S_m.\end{array}\right)$$ (17)

Clearly, system (17) possesses a unique equilibrium point given by $Z_1^0=\left(\frac{{\mathrm{\Lambda }}_h}{{\mu }_h},0,\frac{{\mathrm{\Lambda }}_m}{\gamma +{\mu }_m}\right)$. Since the second and third equations of system (17) are linear and also coupled with the first equation, we obtain solutions.

$$\left\{\begin{array}{l}{S}_h(t)=\frac{{\mathrm{\Lambda }}_h}{{\mu }_h}+e^{-{\mu }_{h}t}\left(S_h(0)+R_h(0)-1\right)-R_h(0)e^{-(ϵ+{\mu }_h)t}\\ R_h(t)=R_h(0)e^{-({\mu }_h+ϵ)t}\\ S_m(t)=S_m(0)e^{-(\gamma +{\mu }_m)t}+\frac{{\mathrm{\Lambda }}_m}{\gamma +{\mu }_m}\left(1-e^{-(\gamma +{\mu }_m)t}\right)\end{array}\right)$$ (18)

It is evident that the solutions $\left(S_h(t),R_h(t),S_m(t)\right)\to \left(\frac{{\mathrm{\Lambda }}_h}{{\mu }_h},0,\frac{{\mathrm{\Lambda }}_m}{\gamma +{\mu }_m}\right)=Z_1^0$ as t → ∞. Thus, $Z_1^0$ is globally asymptotically stable implying condition (A₁) holds true. Now we needed to show $G(Z_1,Z_2)=A{Z}_2^T-\hat{G}(Z_1,Z_2)$, to ensure vector $\hat{G}(Z_1,Z_2)\ge 0$. Thus,

$$A=\frac{\partial G(Z_2^0,0)}{\partial Z_2}=\left(\begin{array}{lllll}-r-{\mu }_h& 0& 0& 0& {\beta }_1\\ r\sigma & -\varphi -{\mu }_h& 0& 0& 0\\ r(1-\sigma )& q\varphi & -\delta -\eta -{\mu }_h& 0& 0\\ 0& 0& \eta & -\rho -\phi -{\mu }_h& 0\\ 0& B_0{\beta }_2& B_0{\beta }_3& 0& -\gamma -{\mu }_m\end{array}\right),$$ (19)

$$\text{and}\hat{G}(H,Z)=\left(\begin{array}{l}{\beta }_1{I}_m\left(1-\frac{S_h}{N_h}\right)\\ 0\\ 0\\ 0\\ {\beta }_2{B}_0\left(1-\frac{S_m}{B_0{N}_h}\right)+{\beta }_3{B}_0\left(1-\frac{S_m}{B_0{N}_h}\right)\end{array}\right).$$

Vector $\hat{G}(Z_1,Z_2)\ge 0$ if $1-\frac{S_m}{B_0{N}_h}\ge 0$. As a result, the MFEP E₀ of model (1) is the GAS equilibrium whenever ${\mathfrak{R}}_0<1$ and hence Theorem 4 holds true.

3.5. Existence of malaria equilibrium point

The Malaria Equilibrium Point (MEP) represents a steady-state solution in which malaria is sustained within the population. The MEP of model (1), denoted as E₁, is expressed as:

$$E_1=\left(S_h^{\ast },E_h^{\ast },A_h^{\ast },I_h^{\ast },T_h^{\ast },R_h^{\ast },S_m^{\ast },I_m^{\ast }\right).$$

It is obtained by equating the system in (1) to zero and solving for the state variables in terms of E_h.

$$\begin{array}{lll}{S}_h^{\ast }& =& {\xi }_1+{\xi }_3{E}_h,A_h^{\ast }=\frac{{\omega }_1}{{\omega }_2}{E}_h,I_h^{\ast }=\frac{{\omega }_2{\omega }_3+{\omega }_1{\omega }_4}{{\omega }_2{\omega }_5}{E}_h,T_h^{\ast }=\frac{\eta ({\omega }_2{\omega }_3+{\omega }_1{\omega }_4)}{{\omega }_2{\omega }_5{\omega }_6}{E}_h,\\ R_h^{\ast }& =& \frac{\eta \rho ({\omega }_2{\omega }_3+{\omega }_1{\omega }_4)+{\omega }_1{\omega }_5{\omega }_6{\omega }_7}{{\omega }_2{\omega }_5{\omega }_6{\omega }_8}{E}_h,S_m^{\ast }={\omega }_9-\frac{{\omega }_0{N}_h^{\ast }}{{\beta }_1\left({\xi }_1+{\xi }_3{E}_h\right)}{E}_h,I_m^{\ast }=\frac{{\omega }_0{N}_h^{\ast }}{{\beta }_1\left({\xi }_1+{\xi }_3{E}_h\right)}{E}_h,\end{array}$$ (20)

where

$$\begin{array}{ll}{\omega }_0& =r+{\mu }_h,{\omega }_1=\sigma r,{\omega }_2=\varphi +{\mu }_h,{\omega }_3=(1-\sigma )r,{\omega }_4=q\varphi ,{\omega }_5=\eta +\delta +{\mu }_h,\\ {\omega }_6& =\rho +\phi +{\mu }_h,{\omega }_7=(1-q)\varphi ,{\omega }_8=ϵ+{\mu }_h,{\omega }_9=\frac{{\mathrm{\Lambda }}_m}{\gamma +{\mu }_m},{\xi }_0=\gamma +{\mu }_m,{\xi }_1=\frac{{\mathrm{\Lambda }}_h}{{\mu }_h},\\ {\xi }_3& =\frac{ϵ\eta \rho {\omega }_2{\omega }_3+ϵ\eta \rho {\omega }_1{\omega }_4+ϵ{\omega }_1{\omega }_5{\omega }_6{\omega }_7+{\omega }_0{\omega }_2{\omega }_5{\omega }_6{\omega }_8}{{\omega }_8{\omega }_5{\omega }_6{\omega }_8{\mu }_h}.\end{array}$$

Plugging equation (20) into the last equation of system (1), it can be revealed that E_h satisfies the following equation:

$$E_h^{\ast }\left(A_0{E}_h^{\ast 2}+A_1{E}_h^{\ast }+A_2\right)=0$$ (21)

where,

$$\begin{array}{lll}{A}_0& =& {\omega }_0\left({\omega }_1(\left(1+{\xi }_2\right){\omega }_5{\omega }_6{\omega }_8+{\omega }_3\left(\eta \rho +\left(\eta +{\omega }_6\right){\omega }_8\right))+{\omega }_1({\omega }_5{\omega }_6({\omega }_7+{\omega }_8)+{\omega }_4(\eta \rho +\left(\eta +{\omega }_6\right){\omega }_8))\right)\\ & & \left({\beta }_3({\omega }_1{\omega }_3+{\omega }_1{\omega }_4)+{\beta }_2{\omega }_1{\omega }_5\right){\omega }_6{\omega }_8+{\xi }_0{\omega }_1\left((1+{\xi }_1){\omega }_5{\omega }_6{\omega }_8+{\omega }_3(\eta \rho +(\eta +{\omega }_6))\right)\\ & & \left(({\omega }_8+{\omega }_1)\left({\omega }_5{\omega }_6\left({\omega }_7+{\omega }_8\right)+{\omega }_4\left(\eta \rho +\left(\eta +{\omega }_6\right){\omega }_8\right)\right)\right),\\ A_1& =& {\omega }_2{\omega }_5{\omega }_6{\omega }_8\left(2{\xi }_0{\xi }_1{\omega }_0\left({\omega }_1((1+{\xi }_1){\omega }_5{\omega }_6{\omega }_8+{\omega }_3(\eta \rho +(\eta +{\omega }_6){\omega }_8))+{\omega }_1({\omega }_5{\omega }_6{\omega }_7+{\omega }_8)\right)\right)\\ & +& \left(\left({\omega }_4(\eta \rho +(\eta +{\omega }_6){\omega }_8)\right)\right)+({\beta }_3({\omega }_2{\omega }_3+{\omega }_1{\omega }_4)+{\beta }_2{\omega }_1{\omega }_5){\omega }_6{\omega }_8({\xi }_1{\omega }_0-{\beta }_1{\xi }_1{\omega }_9),\\ A_2& =& {\xi }_0{\xi }_1^2{\omega }_0{\omega }_2^2{\omega }_5^2{\omega }_6^2{\omega }_8^2(1-{\mathfrak{R}}_0).\end{array}$$

The MFEP, E₀ is obtained when E_h = 0. In contrast, the MEP, E₁ is determined from the equation

$$A_0{E}_h^{\ast 2}+A_1{E}_h^{\ast }+A_2=0.$$ (22)

We observe that the coefficient A₀ is positive. Consequently, it is evident that the existence of all possible positive roots of E_h of the polynomial equation (22) depends on the signs of coefficients A₁ and A₂. To ascertain this, we apply Descartes rule of signs introduced in (Wang, 2004). Now, we present the possibilities for the existence of the positive endemic equilibrium point E_h in Table 3.

Table 3.

Existence of possible positive roots of the polynomial equation (22).

Case	A0	A1	A2	${\mathfrak{R}}_0$	Sign change	+Roots
(i)	+	+	+	${\mathfrak{R}}_0<1$	0	0
(ii)	+	–	+	${\mathfrak{R}}_0<1$	2	2
(iii)	+	+	–	${\mathfrak{R}}_0>1$	1	1
(iv)	+	–	–	${\mathfrak{R}}_0>1$	1	1

Thus, we can infer the following points from Table 3.

• (i)There exist two MEP if A₁ < 0, A₂ > 0 and $A_1^2-4A_0{A}_2>0$;
• (ii)There exist a unique MEP if A₂ < 0 (i.e., if ${\mathfrak{R}}_0>1$) regardless of the sign of A₁;
• (iii)There exist a unique MEP if A₁ < 0 and A₂ = 0 or $A_1^2-4A_0{A}_2=0$;
• (iv)There is no MEP if A₁ > 0 and A₂ > 0 or $A_1^2-4A_0{A}_2<0$.

Considering the conditions (i)-(iv), we can formulate the following theorem.

Theorem 5

The malaria model

• (1)
  has 1. A unique MEP if

  • •
    A₂ < 0, regardless of the sign of A₁ if and only if ${\mathfrak{R}}_0>1$;
  • •
    A₁ < 0 and A₂ = 0 or $A_1^2-4A_0{A}_2=0$;
• 2.Two MEP ifA₁ < 0 and A₂ > 0;
• 3.No MEP otherwise.

In case 2, the model system (1) may have endemic equilibrium when ${\mathfrak{R}}_0<1$, which can result in the occurrence of backward bifurcation. Thus, We will perform a bifurcation analysis in the next subsection to ascertain this fact.

3.6. Bifurcation analysis

In this part, we apply the concepts of Centre Manifold Theory introduced in (Castillo-Chavez & Song, 2004), to investigate the stability of malaria equilibrium points. This theory elucidates the nature of both forward and backward bifurcations, providing valuable insights into the dynamic behaviour of these equilibria. A forward bifurcation indicates local asymptotic stability of the malaria-free equilibrium point when ${\mathfrak{R}}_0<1$ and of the malaria equilibrium point when ${\mathfrak{R}}_0>1$, suggesting potential global stability for these states. Conversely, a backward bifurcation arises when both malaria-free and malaria equilibria points coexist, even in the scenario where ${\mathfrak{R}}_0<1$. To explore dynamics of model (1), we transform the state variables as follows:

S_h = x₁, E_h = x₂, A_h = x₃, I_h = x₄, T_h = x₅, R_h = x₆, S_m = x₇, I_m = x₈. So that, X_h = x₁ + x₂ + x₃ + x₄ + x₅ + x₆. In vector representation, let $x={(x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8)}^T$ and $F={(f_1,f_2,f_3,f_4,f_5,f_6,f_7,f_8)}^T$. Then model (1) can be written compactly in the form

$$\frac{dx}{dt}=F(x),$$ (23)

where the function F(x) represents the right hand side of (1). Thus, the model (1) is given by

$$\left\{\begin{array}{l}\frac{d{x}_1}{dt}={\mathrm{\Lambda }}_h-\frac{{\beta }_1{x}_1{x}_8}{X_h}+ϵx_6-{\mu }_h{x}_1\\ \frac{d{x}_2}{dt}=\frac{{\beta }_1{x}_1{x}_8}{X_h}-\left(r+{\mu }_h\right)x_2\\ \frac{d{x}_3}{dt}=\sigma r{x}_2-\left(\varphi +{\mu }_h\right)x_3\\ \frac{d{x}_4}{dt}=(1-\sigma )r{x}_2+q\varphi x_3-\left(\delta +\eta +{\mu }_h\right)x_4\\ \frac{d{x}_5}{dt}=\eta x_4-\left(\rho +\phi +{\mu }_h\right)x_5\\ \frac{d{x}_6}{dt}=(1-q)\varphi x_3+\rho x_5-\left(ϵ+{\mu }_h\right)x_6\\ \frac{d{x}_7}{dt}={\mathrm{\Lambda }}_m-\frac{{\beta }_2{x}_7{x}_3}{N_h}-\frac{{\beta }_3{x}_7{x}_4}{X_h}-\left(\gamma +{\mu }_m\right)x_7\\ \frac{d{x}_8}{dt}=\frac{{\beta }_2{x}_7{x}_3}{X_h}+\frac{{\beta }_3{x}_7{x}_4}{X_h}-\left(\gamma +{\mu }_m\right)x_8\end{array}\right)$$ (24)

Let the transmission rate ${\beta }_1^{\ast }={\beta }_1$ be bifurcation parameter when ${\mathfrak{R}}_0=1$. Then ${\beta }_1^{\ast }$ can be determined as

$${\beta }_1^{\ast }=\frac{{\mathrm{\Lambda }}_h\left(r+{\mu }_h\right)\left(\varphi +{\mu }_h\right)\left(\eta +\delta +{\mu }_h\right){\left(\gamma +{\mu }_m\right)}^2}{r{\mathrm{\Lambda }}_m{\mu }_h\left(\sigma {\beta }_2(\eta +\delta +{\mu }_h)+(1-\sigma )(\varphi +{\mu }_h{\beta }_3)\right)}$$ (25)

The Jacobian matrix $J(E_0,{\beta }_1^{\ast })$ for the transformed system (24), evaluated at E₀ and ${\beta }_1^{\ast }$, is given by:

$$J(E_0,{\beta }_1^{\ast })=\left(\begin{array}{llllllll}-{\mu }_h& 0& 0& 0& 0& ϵ& 0& -{\beta }_1^{\ast }\\ 0& -r-{\mu }_h& 0& 0& 0& 0& 0& {\beta }_1^{\ast }\\ 0& r\sigma & -\varphi -{\mu }_h& 0& 0& 0& 0& 0\\ 0& r(1-\sigma )& q\varphi & -\delta -\eta -{\mu }_h& 0& 0& 0& 0\\ 0& 0& 0& \eta & -\rho -\phi -{\mu }_h& 0& 0& 0\\ 0& 0& (1-q)\varphi & 0& \rho & -ϵ-{\mu }_h& 0& 0\\ 0& 0& -B_0{\beta }_2& -B_0{\beta }_3& 0& 0& -\gamma -{\mu }_m& 0\\ 0& 0& B_0{\beta }_2& B_0{\beta }_3& 0& 0& 0& -\gamma -{\mu }_m\end{array}\right)$$ (26)

Clearly, $J(E_0,{\beta }_1^{\ast })$ has one simple eigenvalue of zero, while the remaining eigenvalues are negative. The right eigenvector corresponding to this simple zero eigenvalue of the matrix $J(E_0,{\beta }_1^{\ast })$ is assumed to be: u = (u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈), where

$$\begin{array}{ll}{u}_1& =\frac{-ϵ\eta \rho {\beta }_1^{\ast }({\omega }_2{\omega }_3+{\omega }_1{\omega }_4)+u_2{\omega }_0(ϵ{\omega }_1-{\omega }_0{\omega }_2){\omega }_5{\omega }_6}{{\mu }_h{\omega }_0{\omega }_2{\omega }_5{\omega }_6},u_2=u_2>0,u_3=\frac{{\omega }_1}{{\omega }_2}{u}_2,\\ u_4& =\frac{({\omega }_2{\omega }_3+{\omega }_1{\omega }_4)}{{\omega }_2{\omega }_5}{u}_2,u_5=-\frac{\eta ({\omega }_2{\omega }_3+{\omega }_1{\omega }_4)}{{\omega }_2{\omega }_5{\omega }_6}{u}_2,u_6=\frac{u_2(\eta \rho {\omega }_2{\omega }_3+{\omega }_1(\eta \rho {\omega }_4-{\omega }_5{\omega }_6{\omega }_7))}{{\omega }_2{\omega }_5{\omega }_6{\omega }_8},\\ u_7& =\frac{{\omega }_0}{{\beta }_1^{\ast }}{u}_2,u_8=-\frac{{\omega }_0}{{\beta }_1^{\ast }}{u}_2.\end{array}$$

In the same way the left eigenvector associated to a simple zero eigenvalue is given by v = (v₁, v₂, v₃, v₄, v₅, v₆, v₇, v₈), where

$$v_1=0,v_2=v_2>0,v_3=\left(\frac{q\varphi {\xi }_0{B}_0{\beta }_3{\beta }_1^{\ast }}{{\omega }_2{\omega }_5{\xi }_0}+\frac{B_0{\beta }_2{\beta }_1^{\ast }}{{\omega }_2{\xi }_0}\right)v_2,v_4=\frac{B_0{\beta }_3{\beta }_1^{\ast }}{{\omega }_5{\xi }_0}{v}_2,v_5=0,u_6=0,u_7=0,v_8=\frac{{\beta }_1^{\ast }}{{\xi }_0}{v}_2.$$

Now we utilize Theorem 4.1 stated by Castillo-Chavez and Song (Castillo-Chavez & Song, 2004) is to compute the bifurcation coefficients, a and b, of the model (1) where,

$$\begin{array}{ll}a& =\sum _{k,i,j}^8{v}_k{u}_i{u}_j\frac{{\partial }^2{f}_k}{\partial x_i\partial x_j}(E_0,{\beta }_1^{\ast }),\\ b& =\sum _{k,i}^8{v}_k{u}_i\frac{{\partial }^2{f}_k}{\partial x_i\partial {\beta }_1^{\ast }}(E_0,{\beta }_1^{\ast }).\end{array}$$ (27)

Then by considering the non-zero second-order partial derivatives of F(x) at E₀ and ${\beta }_1^{\ast }$, we have

$$\begin{array}{ll}& \frac{{\partial }^2{f}_2}{\partial x_1\partial x_8}=\frac{{\partial }^2{f}_2}{\partial x_8\partial x_1}=\frac{{\beta }_1({\xi }_1-1)}{{\xi }_1^2},\frac{{\partial }^2{f}_2}{\partial x_2\partial x_8}=\frac{{\partial }^2{f}_2}{\partial x_8\partial x_2}=\frac{{\partial }^2{f}_2}{\partial x_3\partial x_8}=\frac{{\partial }^2{f}_2}{\partial x_8\partial x_3}=\frac{{\partial }^2{f}_2}{\partial x_4\partial x_8}=\frac{{\partial }^2{f}_2}{\partial x_8\partial x_4}=\frac{{\partial }^2{f}_2}{\partial x_5\partial x_8}\\ & =\frac{{\partial }^2{f}_2}{\partial x_8\partial x_5}=\frac{{\partial }^2{f}_2}{\partial x_6\partial x_8}=\frac{{\partial }^2{f}_2}{\partial x_8\partial x_6}=\frac{{\beta }_1}{{\xi }_1},\frac{{\partial }^2{f}_8}{\partial x_1\partial x_3}=\frac{{\partial }^2{f}_8}{\partial x_3\partial x_1}=\frac{{\partial }^2{f}_8}{\partial x_2\partial x_3}=\frac{{\partial }^2{f}_8}{\partial x_3\partial x_2}=\frac{{\partial }^2{f}_8}{\partial x_3\partial x_5}=\frac{{\partial }^2{f}_8}{\partial x_5\partial x_3}=\frac{{\partial }^2{f}_8}{\partial x_3\partial x_6}\\ & =\frac{{\partial }^2{f}_8}{\partial x_6\partial x_3}=-\frac{{\beta }_2}{B_0{\xi }_1},\frac{{\partial }^2{f}_8}{\partial x_1\partial x_4}=\frac{{\partial }^2{f}_8}{\partial x_4\partial x_1}=\frac{{\partial }^2{f}_8}{\partial x_2\partial x_4}=\frac{{\partial }^2{f}_8}{\partial x_4\partial x_2}=\frac{{\partial }^2{f}_8}{\partial x_4\partial x_5}=\frac{{\partial }^2{f}_8}{\partial x_5\partial x_4}=\frac{{\partial }^2{f}_8}{\partial x_4\partial x_6}=\frac{{\partial }^2{f}_8}{\partial x_6\partial x_4}\\ & =-\frac{{\beta }_3}{B_0{\xi }_1},\frac{{\partial }^2{f}_8}{\partial x_3\partial x_3}=-\frac{2{\beta }_2}{B_0{\xi }_1},\frac{{\partial }^2{f}_8}{\partial x_3\partial x_4}=\frac{{\partial }^2{f}_8}{\partial x_4\partial x_3}=-\frac{{\beta }_2+{\beta }_3}{B_0{\xi }_1},\frac{{\partial }^2{f}_8}{\partial x_3\partial x_7}=\frac{{\partial }^2{f}_8}{\partial x_7\partial x_3}=\frac{{\beta }_2}{{\xi }_1},\frac{{\partial }^2{f}_8}{\partial x_4\partial x_7}\\ & =\frac{{\partial }^2{f}_8}{\partial x_7\partial x_4}=\frac{{\beta }_3}{{\xi }_1},\frac{{\partial }^2{f}_8}{\partial x_4\partial x_4}=-\frac{{\beta }_3}{B_0{\xi }_1},\frac{{\partial }^2{f}_2}{\partial x_8\partial {\beta }_1^{\ast }}=1,\end{array}$$

we obtain the bifurcation parameters a and b respectively

$$a=-\left(\frac{d_0(\eta -(\rho +\phi +{\mu }_h))}{{\mu }_h{\xi }_0^2}-d_1+(\eta -(\rho +\phi +{\mu }_h)){\beta }_1{\omega }_9({\omega }_2+\eta \rho +{\omega }_8{\omega }_1)\right)\frac{2u_2^2{v}_2}{{\omega }_2^2{\omega }_5^2{\omega }_6{\omega }_8{\xi }_1^2},\text{where}$$ (28)

$$\begin{array}{ll}{d}_0& ={\xi }_1{\xi }_0{\omega }_0{\omega }_2{\omega }_5+{\beta }_1{\omega }_9({\beta }_3({\omega }_2{\omega }_3+{\omega }_1{\omega }_4)+{\beta }_2{\omega }_1{\omega }_5),\\ d_1& =ϵ{\xi }_0(\eta \rho {\omega }_2{\omega }_3+{\omega }_1(\eta \rho {\omega }_4+{\omega }_5{\omega }_6{\omega }_7))+\frac{{\beta }_1{\omega }_6{\omega }_8}{{\omega }_9}({\beta }_3{\xi }_1({\omega }_2{\omega }_3+{\omega }_1{\omega }_4+{\beta }_3{\xi }_1{\omega }_1{\omega }_5),\text{and}\\ b& =\frac{{\omega }_0}{{\beta }_1^{\ast }}{u}_2{v}_2\end{array}$$ (29)

The bifurcation coefficient b is always positive (see Fig. 1). Accordingly, whenever a > 0 for η < ρ + φ + μ_h, backward bifurcation occurs. For this condition, the existence of backward bifurcation may depend on the parameters: drug treatment rate η, recovery rate of treated humans ρ, the rate of death due to treatment failure φ, rate of insecticides usage γ, and natural death rate of humans μ_h. In particular, we choosing γ = 0.64, φ = 0.096, and ρ = 0.075. Utilizing the parameter values from Table 4, we illustrate backward bifurcation as depicted in Fig. 2a, which demonstrates the existence of two malaria equilibrium points for ${\mathfrak{R}}_0=0.84$. Notably from Fig. 2a, varying these parameters under the condition η < ρ + φ + μ_h, results in the emergence of two endemic equilibrium points for ${\mathfrak{R}}_0^{\ast }<{\mathfrak{R}}_0<1$. Among these equilibria, one point corresponds to a lower infection level and is unstable, while the other, associated with a higher infection level, is stable in conjunction with the stable MFEP. Thus, reducing ${\mathfrak{R}}_0$ just below unity during an epidemic period will not be sufficient to eradicate the infection. Instead, malaria control strategies should be implemented to ensure that ${\mathfrak{R}}_0$ is significantly below unity.

Fig. 1.

Schematic diagram showing dynamic transmission ofmalaria

Table 4.

Parameters and their values.

Parameters	value	Reference
Λh	$\frac{1000}{67.8}$	Estimated
μh	$\frac{1}{67.8}$	WHO and Life expectancy and healthy life expectancy (2020)
Λm	$\frac{1000\times 365}{21}$	Estimated
μm	$\frac{365}{21}$	Estimated
β1	0.0047	Fitted
β2	0.0344	Fitted
β3	0.101	Fitted
ϵ	0.0055	Fitted
r	0.35	Assumed
ϕ	0.04	Cai et al. (2017)
ρ	0.00475	Fitted
δ	0.08	Fitted
η	0.17	Fitted
φ	0.006	Haringo et al. (2024)
γ	0.062	Fitted
σ	0.2	Assumed from 0 to 1
q	0.1	Assumed from 0 to 1

Fig. 2.

Plot depicting bifurcation of the model(1).

Epidemiologically, the phenomena of backward bifurcation underscores the importance of maintaining balance between treatment capacity, recovery and over all mortality in humans. Thus, malaria intervention mechanisms may be needed to enhance drug treatment, refine the recovery outcomes and decrease malaria induced death. Conversely, for the parameter values presented in Table 4, the bifurcation coefficient a < 0 satisfying the condition η > ω₆ indicates the occurrence of a forward bifurcation, as illustrated in Fig. 2b. This forward bifurcation confirms the existence of a unique endemic equilibrium at ${\mathfrak{R}}_0=1.97$. In this scenario, malaria can only become endemic if ${\mathfrak{R}}_0>1$, resulting in a globally asymptotically stable endemic equilibrium point. Therefore, reducing ${\mathfrak{R}}_0$ just below unity can effectively suppress the spread of the disease within the framework of forward bifurcation.

Considering the birth of backward-bifurcation, we prove the global stability of the unique MEP when ${\mathfrak{R}}_0>1$.

3.7. Global stability of malaria equilibrium point

To prove the stability of the malaria equilibrium point (MEP) denoted as E₁, we state and prove the following theorem.

Theorem 6

The malaria equilibrium,E₁, of the malaria model (1) is globally asymptotically stable whenever ${\mathfrak{R}}_0>1$.

Proof

To prove the global stability of the MEP of model (1), we construct the following positive-definite function

$$V(X)=\frac{1}{2}{\left((S_h-S_h^{\ast })+(E_h-E_h^{\ast })+(A_h-A_h^{\ast })+(I_h-I_h^{\ast })+(T_h-T_h^{\ast })+(R_h-R_h^{\ast })\right)}^2+\frac{1}{2}{\left((S_m-S_m^{\ast })+(I_m-I_m^{\ast })\right)}^2,$$ (30)

where $X=\left(S_h,E_h,A_h,I_h,T_h,R_h,S_m,I_m\right)$.

Clearly, $V:R_{+}^8\to R$ is continuously differentiable function. It is obvious that V(X) = 0 if X = X∗ and V(X) > 0 if X ≠ X∗.

By differentiating function (30) along with model equation (1), we get

$$\begin{array}{ll}\frac{dV(X)}{dt}& =\left(S_h-S_h^{\ast }\right)\frac{d{S}_h}{dt}+\left(E_h-E_h^{\ast }\right)\frac{d{E}_h}{dt}+\left(A_h-A_h^{\ast }\right)\frac{d{A}_h}{dt}+\left(I_h-I_h^{\ast }\right)\frac{d{I}_h}{dt}+\left(T_h-T_h^{\ast }\right)\frac{d{T}_h}{dt}\\ & +\left(R_h-R_h^{\ast }\right)\frac{d{R}_h}{dt}+\left(S_m-S_m^{\ast }\right)\frac{d{S}_m}{dt}+\left(I_m-I_m^{\ast }\right)\frac{d{I}_m}{dt}\end{array}$$

After some simplifications, we have

$$\frac{dV(X)}{dt}=(N_h-N_h^{\ast })\frac{d{N}_h}{dt}+(N_m-N_m^{\ast })\frac{d{N}_m}{dt}.$$ (31)

Using equations (3), (5), equation (31) becomes

$$\begin{array}{lll}\frac{dV(X)}{dt}& =& (N_h-N_h^{\ast })\left({\mathrm{\Lambda }}_h-{\mu }_h{N}_h-\delta I_h-\phi T_h\right)+(N_m-N_m^{\ast })\left({\mathrm{\Lambda }}_m-(\gamma +{\mu }_m)N_m\right)\\ & \le & {\mu }_h\left(N_h-N_h^{\ast }\right)\left(-N_h+\frac{{\mathrm{\Lambda }}_h-\delta I_h-\phi T_h}{{\mu }_h}\right)+\left(N_m-N_m^{\ast }\right)\left({\mathrm{\Lambda }}_m-\left(\gamma +{\mu }_m\right)N_m\right)\\ & \le & {\mu }_h\left(\frac{{\mathrm{\Lambda }}_h}{{\mu }_h}-N_h^{\ast }\right)\left(N_h^{\ast }-N_h\right)+\left(N_m-N_m^{\ast }\right)\left(\frac{{\mathrm{\Lambda }}_m}{\gamma +{\mu }_m}-N_m^{\ast }\right)\left(\gamma +{\mu }_m\right)\\ & =& -{\mu }_h{\left(\frac{{\mathrm{\Lambda }}_h}{{\mu }_h}-N_h^{\ast }\right)}^2-{\left(N_m-N_m^{\ast }\right)}^2\left(\gamma +{\mu }_m\right)\\ & \le & 0.\end{array}$$ (32)

This indicates that $\frac{dV(X)}{dt}\le 0$ within the region Ω. By applying LaSalle's invariance principle as outlined in (La Salle, 1966), we find that the malaria equilibrium state of the model (1) is globally asymptotically stable in Ω when ${\mathfrak{R}}_0>1$. This implies that, for any initial conditions, the model (1) will converge to the malaria-free equilibrium state when ${\mathfrak{R}}_0<1$ and to the malaria equilibrium state when ${\mathfrak{R}}_0>1$.

4. Numerical simulations

In this section we focus on parameter estimation, sensitivity analysis, model simulation, and exploring the effects of some model parameters. To investigate the parameter estimation and model simulation of the malaria model (1), we used t = 50 years based on the assertion of global malaria eradication by the year 2050 using values of parameters in Table 3a. Moreover, we used the following initial conditions for the populations of humans and mosquitoes: S_h(0) = 564.23, E_h(0) = 100, A_h(0) = 0, I_h(0) = 322.7, T_h(0) = 10, R_h(0) = 3, S_m(0) = 800, I_m(0) = 200.

4.1. Parameter estimation

In this section, we employ Ethiopian malaria incidence data to fit model (1) and estimate key parameters of malaria model (1). Since malaria has been endemic in Ethiopia for many years, the accurate estimation of model parameters is essential for predicting the course of an outbreak and evaluating the impact of interventions. Here, malaria incidence is defined as the number of new malaria cases per 1, 000 individuals at risk in Ethiopia. The incidence data, sourced from the World Bank, ranges from the years 2000–2022 (W. Bank) and is illustrated in Fig. 3a. For model fitting and parameter estimation, we utilize the least-squares fitting method as described in (Johnson & Faunt, 1992). Consequently, the model (1) can be written as follows

$$\frac{dx}{dt}=F(t,x,\mathrm{\Theta }),x(t_0)=x_0,$$ (33)

where x and Θ are the state variables and unknown parameters, respectively. Therefore, the sum of least-squares error is represented by

$$\mathrm{\Psi }(\mathrm{\Theta })=\sum _{i=1}^n{(x_i-{\overline{x}}_i)}^2,$$

where ${\overline{x}}_i$ is the real data and x_i = x(t_i, Θ) is the solution of the model (1) and n denotes the number of the data. In order to obtain the parameter, the goal is to minimize the objective function.

$${\min }_{\mathrm{\Theta }}\mathrm{\Psi }(\mathrm{\Theta }),$$

subjected to equation (33).

Fig. 3.

Plot showing malaria prevalence data, curve fit and residual plot.

The constant influx of malaria-prone humans Λ_h and the natural mortality rate of humans μ_h are estimated from Ethiopian demographic data. The value for μ_h is calculated by taking the reciprocal of the average life expectancy in Ethiopia, which is approximately 67.8 years (WHO & Life expectancy and healthy life expectancy, 2020). Thus, we have ${\mu }_h=\frac{1}{67.8}$ deaths year^-1. To estimate annually constant influx of malaria-prone humans Λ_h, we explore a scenario introduced in (Madueme & Chirove, 2023) where the human population has reached a steady state Λ_h = 1000 × μ_h. For the constant influx of malaria-prone mosquitoes, we assume a hypothetical average population of 1,000, which satisfies the relationship $\frac{{\mathrm{\Lambda }}_m}{{\mu }_m}=1000$. This assumption is made due to the lack of quantified estimates for actual mosquito populations. The average lifespan of mosquitoes is calculated as ${\mu }_m=\frac{365}{21}{\text{year}}^{-1}$. In Ethiopia, the prevalence of asymptomatic malaria infections is lower compared to that of symptomatic infections (Cherkos et al., 2024). This discrepancy leads to a higher chance of exposed individuals progressing to symptomatic malaria. Consequently, we assume a proportion σ = 0.2, indicating that the number of symptomatic infections exceeds that of asymptomatic carriers. Asymptomatic carriers who receive on-the-spot treatment after screening recover directly, while those without treatment may experience higher parasite loads and progress to symptomatic malaria (Singh et al., 2022). The parameter φ is sourced from published literature, while parameters r and ϕ are reasonably assumed. The remaining parameters β₁, β₂, β₃, γ, η, ρ, δ and ϵ are obtained through a fitting process. The model-fit results, illustrated in Fig. 3b, demonstrate that the model closely aligns with malaria incidence data in Ethiopia from 2000 to 2022.

4.2. Sensitivity analysis

To identify the most dominant parameters of the model described in (1), we conduct a sensitivity analysis in terms of reproduction number ${\mathfrak{R}}_0$. This is achieved using the elasticity index method. Following Chitnis et al. (Chitnis et al., 2008), the normalized elasticity index of reproduction number ${\mathfrak{R}}_0$ in relation to parameter ϱ is defined by:

$${\mathrm{\Upsilon }}_{\varrho }^{{\mathfrak{R}}_0}≔\frac{\partial {\mathfrak{R}}_0}{\partial \varrho }\times \frac{\varrho }{{\mathcal{R}}_0},$$ (34)

where ${\mathrm{\Upsilon }}_{\varrho }^{{\mathfrak{R}}_0}$ is the elasticity index that assess the relative change in ${\mathfrak{R}}_0$ when parameter ϱ changes. The elasticity index of the reproduction number ${\mathfrak{R}}_0$ in relation to model parameters β₁, Λ_m, Λ_h, γ, and r is computed accordingly

$${\mathrm{\Upsilon }}_{{\beta }_1}^{R_0}={\mathrm{\Upsilon }}_{{\mathrm{\Lambda }}_m}^{R_0}=0.5,{\mathrm{\Upsilon }}_{{\mathrm{\Lambda }}_h}^{R_0}=-0.5,{\mathrm{\Upsilon }}_{\gamma }^{R_0}=-\frac{\gamma }{\gamma +{\mu }_m},{\mathrm{\Upsilon }}_r^{R_0}=\frac{{\mu }_h}{2(r+{\mu }_h)}.$$

For the remaining model parameters, analytical elasticity indices can be calculated using the same methodology. Moreover, Table 5 presents the numerical elasticity indices of ${\mathfrak{R}}_0$, evaluated at the parameter values provided in Table 4.

Table 5.

Elasticity index of ${\mathfrak{R}}_0$ calculated with parameters value in Table 4.

Elasticity indexes	Values	Elasticity indexes	Values
${\mathrm{\Upsilon }}_{\gamma }^{R_0}$	−0.566	${\mathrm{\Upsilon }}_{{\beta }_1}^{R_0}$	+0.50
${\mathrm{\Upsilon }}_{{\mathrm{\Lambda }}_h}^{R_0}$	−0.500	${\mathrm{\Upsilon }}_{{\mathrm{\Lambda }}_m}^{R_0}$	+0.50
${\mathrm{\Upsilon }}_{{\mu }_m}^{R_0}$	−0.434	${\mathrm{\Upsilon }}_{{\mu }_h}^{R_0}$	+0.25
${\mathrm{\Upsilon }}_{\eta }^{R_0}$	−0.145	${\mathrm{\Upsilon }}_{{\beta }_3}^{R_0}$	+0.23
${\mathrm{\Upsilon }}_{\delta }^{R_0}$	−0.066	${\mathrm{\Upsilon }}_{{\beta }_2}^{R_0}$	+0.27
${\mathrm{\Upsilon }}_{\varphi }^{R_0}$	−0.055	${\mathrm{\Upsilon }}_r^{R_0}$	+0.02
		${\mathrm{\Upsilon }}_{\sigma }^{R_0}$	+0.22

From Table 5, the positive elasticity index indicates that when a parameter value increases, so will the value of ${\mathfrak{R}}_0$. On the other hand, the negative elasticity index indicates that the value of ${\mathfrak{R}}_0$ will fall as the parameter value increases. Specifically, regarding ${\mathrm{\Upsilon }}_{{\mathrm{\Lambda }}_h}^{{\mathfrak{R}}_0}=-0.5$, increasing constant influx of malaria-prone humans Λ_h by 10 %, will raise ${\mathfrak{R}}_0$ by 10 %. Contrarily, ${\mathrm{\Upsilon }}_{{\mathrm{\Lambda }}_m}^{{\mathfrak{R}}_0}=0.5$, a 10 % rise in the constant influx of malaria-prone mosquitoes Λ_m directs to a 10 % lower in ${\mathfrak{R}}_0$. The same applies to the elasticity indices of the remaining parameters. Parameters with high elasticity indices should be carefully monitored, as small variations can lead to significant changes in ${\mathfrak{R}}_0$. For further illustration, these indices can be shown, simultaneously, in the bar graph given in Fig. 4.

Fig. 4.

Bar graph illustrating the elasticity indices for each parameter of ${\mathfrak{R}}_0$.

In light of the results presented in Table 5 and the elasticity bar graph drawing presented by Fig. 4, parameters γ, Λ_h, η, β₁, Λ_m, β₃, and β₂ have the most significant impact on the basic reproduction number, ${\mathfrak{R}}_0$. This suggests that increasing γ, Λ_h, and η reduces the spread of malaria within the population, while an increase in β₁, Λ_m, β₃, and β₂ enhances malaria transmission in the community.

4.3. Model simulation

In this, we present numerical solutions of model (1) to support the theoretical findings obtained from previous sections. We start with illustrations that reinforce the epidemiological concept of backward bifurcation, specifically when the basic reproduction number ${\mathfrak{R}}_0$ is below unity. To do this, we choose γ = 0.64, ρ = 0.075, and φ = 0.096, with parameter values presented in Table 4. Upon substituting these values into equation (11), we found ${\mathfrak{R}}_0=0.84$. Thus, Fig. 5a–c reveal that when the basic reproduction number, ${\mathfrak{R}}_0$, is below unity, the number of infectious individuals in each species becomes few, which does not guarantee the eradication of malaria as the result of the existence of backward bifurcation. In this scenario the transmission of the malaria becomes minimal, resulting in a small number of susceptible humans becoming infected.

Fig. 5.

The profiles of human and mosquito populations when ${\mathfrak{R}}_0<1$.

Alternatively, by using the assertion in Table 4, we calculate ${\mathfrak{R}}_0=1.97$. This value brings the presence a distinct positive malaria equilibrium point, which is globally asymptotically stable as depicted in Figure (6). Thus, this globally asymptotically stable malaria equilibrium point signifies a persistent state of infection in the population, where the dynamics will eventually settle regardless of initial conditions. This concept is crucial for understanding how malaria dynamics behave in populations and for planning effective public health interventions.

Fig. 6.

The profiles of humans and mosquitoes population when ${\mathfrak{R}}_0>1$.

4.4. The effect of model parameters

Now, we explore the effect of model parameters in the transmission dynamics of malaria. Thus, we vary parameters: the rate of insecticide usage γ, the rate of malaria treatment η, the rates of malaria transmission (β₁, β₂ and β₃), and the rate of recovery of treated humans ρ, while the remaining parameters are presented in Table 4. The primary aim of this analysis is to observe how these parameters influence the dynamics of malaria transmission in both human and mosquito populations.

4.4.1. The effect of varying insecticide usage γ

Figure (7) illustrates how the insecticide usage rate, γ, affects malaria dynamics while keeping the model parameters in Table 4 unvaried. As shown in Fig. 7b and c, high insecticide usage significantly reduces mosquito populations, subsequently decreasing the number of asymptomatic humans, as depicted in Fig. 7a. Conversely, Fig. 7b and c shows that low insecticidal usage allows mosquito populations to grow, leading to an increase in the number of asymptomatic humans, as illustrated in Fig. 7a. Thus, high insecticidal usage can reduce both mosquito populations and asymptomatic malaria cases, while inadequate usage may worsen the situation. These results underscore the critical role of sustained and effective insecticide usage in controlling both mosquito populations and asymptomatic malaria cases.

Fig. 7.

The effect of varying insecticide usage γ for γ = [0.062, 0.12, 0.24, 0.64].

4.4.2. Effect of varying transmission rate β₁, β₂ and β₃

In this section, we analyze the variation in human and mosquito populations in response to variations in transmission rates β₁, β₂, and β₃. Figures (8)-(10) illustrate the effects of altering these transmission rates on the dynamics of the malaria model (1). In each figure, variations in transmission rates significantly impact both human and mosquito populations. High transmission rates result in an increase in both infected species, while the number of susceptible humans decreases significantly. Conversely, low transmission rates lead to a decline in infected species and an increase in susceptible humans.

Fig. 8.

The effect of varying transmission rate β₁ for β₁ = [0.0047, 0.023, 0.087, 0.25].

In particular, Fig. 8a reveals that the number of malaria-susceptible humans increases as the transmission rate β₁ decreases and vice versa. However, from Fig. 8b and c, it can be seen that when the transmission rate β₁ increases, the number of asymptomatic humans and infected mosquitoes also increase. These values decrease as the transmission rate β₁ decreases, but this trend does not continue throughout the simulation period for the infected mosquito class. Reducing human–mosquito contact effectively limits malaria transmission by lowering infection levels in both hosts and vectors. In contrast, increased contact facilitates transmission, amplifying asymptomatic malaria prevalence across human and mosquito populations.

Similarly, Figure (9) illustrates the effect of malaria transmission rate from asymptomatic infected humans to susceptible mosquitoes for different β₂ values. Notably, from Fig. 9b and c, it is evident that the endemicity of malaria increases in both asymptomatic infected human and infected mosquito populations as the β₂ value rises. However, Fig. 9a highlights that the number of susceptible humans decreases as the transmission rate β₂ increases, and vice versa. In general, a high transmission rate from asymptomatic infected humans to susceptible mosquitoes increases the reservoir of infectious vectors, sustaining silent malaria transmission. Conversely, a low transmission rate from asymptomatic individuals reduces mosquito infections, limiting covert transmission and aiding control efforts.

Fig. 9.

The effect of varying transmission rate β₂ for β₂ = [0.0344, 0.0688, 0.125, 0.25].

Finally, Figure (10) illustrates the effect of transmission rate from infected humans on susceptible mosquitoes for different β₃ values. From Fig. 10c, it is evident that the number of infectious mosquitoes increase as the transmission rate β₃ increase and vice-versa. From the data in Fig. 10b, it can be seen that, decrease in the malaria transmission rate leads to a increase in the number of infected human in class after 20 years of the outbreak. This epidemiological phenomenon occurs when the inflow of asymptomatic cases to symptomatic infection decreases or when the treatment rate for symptomatic infections increases. In general, a high transmission rate from infected humans to susceptible mosquitoes increases the infectious mosquito population, thereby intensifying malaria transmission. Conversely, a low transmission rate limits mosquito infections and helps suppress disease spread.

Fig. 10.

The effect of varying transmission rate β₃ for β₃ = [0.01, 0.050.101, 0.087, 0.15].

4.4.3. Effect of varying treatment rate η

Figure (11) illustrates the varying effect of malaria treatment rate on the human population while keeping all other parameters in Table 4 unvaried. Specifically, in Fig. 11a and c, it is evident that both susceptible and treated humans increase as the treatment rate increases. Here, with increased malaria drug treatment, a greater number of individuals will recover from the infection. However, these individuals may subsequently lose their natural immunity, leading to an increased susceptibility among those who have recovered. This phenomenon can be attributed to a loss of immunity rate of ϵ as discussed in Section (2). This dynamic behavior indicates that effective drug treatment alone does not guarantee the elimination of malaria. Therefore, it is essential to combine effective treatment with other recommended preventive measures to achieve comprehensive malaria elimination in the population. In contrast, Fig. 11b demonstrates a decrease in the number of infectious humans with the increasing malaria treatment rate.

Fig. 11.

The effect of varying treatment rate η for η = [0.02, 0.17, 0.45, 0.77].

4.4.4. Effect of varying recovery rate ρ

Figure (12) illustrates the varying effect of recovery rate in relation to model (1). Based on the graphical result in Fig. 12c, it is observed that the number of recovered humans increases as the recovery rate ρ increases. However, from Fig. 12a and b, it can be seen that a decrease in the recovery rate leads to an increase in the number of susceptible and treated humans. This reduction in both susceptible and treated individuals diminishes the potential for malaria transmission. Consequently, rapid recovery decreases susceptibility and treatment demand, enhancing malaria control within the population. Understanding this epidemiological context is crucial for effective malaria transmission management.

Fig. 12.

The effect of varying recovery rate ρ for ρ = [0.00475, 0.0084, 0.0124, 0.225].

4.4.5. Effect of varying progression rate ϕ

Figure (13) illustrates the impact of the asymptomatic malaria progression rate ϕ on the classes of asymptomatic humans A_h, symptomatic infectious humans I_h, and recovered humans R_h. Specifically, Fig. 13a highlights that a higher progression rate of asymptomatic infection correlates with an increased number of recoveries. In contrast, Fig. 13c indicates that there is no significant variation in the symptomatic infectious class I_h despite considerable changes in the progression rate ϕ. This suggests that progression from asymptomatic to recovery does not significantly influence symptomatic infection levels.

Fig. 13.

The effect of varying progression rate ϕ for ϕ = [0.004, 0.04, 0.08, 0.4].

4.5. The role of asymptomatic infection

4.5.1. Dependency of asymptomatic infection on parameters

In this, we perform simulations to assess the dependency of asymptomatic prevalence of the malaria dynamics on insecticide usage rate γ, drug treatment rate η, progression rate of asymptomatic humans, and transmission rate β₁ on the equilibrium prevalence $A_h^{\ast }=\frac{\sigma r}{\varphi +{\mu }_h}{E}_h$, where E_h is equilibrium prevalence of exposed class that can be computed from characteristic equation (22). Fig. 14a highlights the equilibrium prevalence of asymptomatic malaria as the function of insecticide usage rate γ and the rate of progression to class ϕ. The Figure reveals that the prevalence of asymptomatic infections decreases when both the insecticide usage rate and the progression rate from asymptomatic to either the recovery or infectious class increase. In this context, the population may face the highest disease burden due to an increase in reservoirs within the infectious class. This has importance from an epidemiological perspective, informing public health strategies to control malaria in the community. Similarly, Fig. 14b illustrates the equilibrium prevalence of asymptomatic malaria A_h as a function of the drug treatment rate η and the progression rate ϕ. The data indicate that the equilibrium prevalence of asymptomatic humans A_h increases as η and ϕ values decrease, while it decreases when both parameters increase. Specifically, when the treatment of infected individuals rises and the recovery rates of asymptomatic individuals increase, the number of recoveries rises. This results in a considerable decrease in the prevalence of malaria by slowing the flow of asymptomatic patients to the symptomatic class. In contrast, Fig. 14c shows the equilibrium prevalence of asymptomatic infectious humans A_h as a function of β₁ and r. The prevalence is lowest when β₁ = r = 0, indicating that the asymptomatic class is nonexistent due to a lack of infection contributions, leading to its disappearance. Conversely, the prevalence of A_h is highest when β₁ = r = 1, suggesting that the asymptomatic class increases over time when both the transmission rate of mosquitoes β₁ and the progression rate of the exposed class r are elevated, allowing it to persist and potentially grow.

Fig. 14.

3D plots depicting dependence of equilibrium prevalence A_h on parameters γ, η, ϕ, β₁ and r.

4.5.2. Asymptomatic infection contribution on ${\mathfrak{R}}_0$

The expression for ${\mathfrak{R}}_0$ accounts for contributions from both asymptomatic and symptomatic infections, highlighting the significant role of asymptomatic cases in malaria dynamics. This approach enhances our understanding of how asymptomatic individuals affect overall transmission patterns and the effectiveness of control strategies. The average infection contributions from asymptomatic and symptomatic individuals can be expressed as ${\mathfrak{R}}_{I_m}{\mathfrak{R}}_{A_h}$ and ${\mathfrak{R}}_{I_m}{\mathfrak{R}}_{I_h}$, respectively. Having the estimated parameters provided in Table 4, we find that the asymptomatic class contributes approximately 29.8 %, while the symptomatic class accounts for 70.2 % of the basic reproduction number. This indicates that asymptomatic individuals represent about one-third of total infections in the community. Thus, to effectively control the malaria, it is essential to minimize the number of asymptomatic individuals through extensive vector control strategies and increasing transition them into the either infectious or recovery compartment for appropriate management.

5. Conclusion

Despite malaria control efforts, asymptomatic malaria remains a key reservoir for the parasite. It often goes undetected due to a lack of symptoms and has received less attention than symptomatic infections (Prah & Laryea-Akrong, 2024). This study introduces a deterministic mathematical model of malaria transmission, specifically designed to analyze the impact of asymptomatic malaria on disease spread. We analyzed fundamental mathematical properties such as positivity, boundedness, feasibility, and stability of equilibria. Our analysis demonstrates that all solutions remain positive and are constrained by the initial conditions, thereby affirming the model's mathematical well-posedness and its relevance to epidemiological contexts.

The basic reproduction number, ${\mathfrak{R}}_0$, was determined using the next-generation matrix method, as described in (Van den Driessche & Watmough, 2002). We established both local and conditional global stability of the malaria-free equilibrium point (MFEP). The model displays complex dynamics influenced by human treatment, recovery, and mortality rates. Under a certain conditions with in these parameters, the model (1) reveals the existence of two malaria equilibria even when ${\mathfrak{R}}_0<1$, leading to backward bifurcation at ${\mathfrak{R}}_0=1$. This implies that merely reducing the basic reproduction number to below unity is insufficient for eradicating malaria infections. The basic reproduction number, denoted as ${\mathfrak{R}}_0$, was determined using the next-generation matrix method. We showed that when ${\mathfrak{R}}_0<1$, the malaria-free equilibrium point (MFEP) is locally asymptotically stable; conversely, when ${\mathfrak{R}}_0>1$, the MFEP becomes unstable. The system exhibits a backward bifurcation at ${\mathfrak{R}}_0=1$, indicating that there exists a stable malaria equilibrium point (MEP) along with a stable MFEP for ${\mathfrak{R}}_0\in ({\mathfrak{R}}_0^{\ast },1)$ where ${\mathfrak{R}}_0^{\ast }<1$. This suggests that simply reducing ${\mathfrak{R}}_0$ to below unity is insufficient for malaria eradication.

The model has been validated against malaria incidence data per thousand people from 2000 to 2022 in Ethiopia, enabling the estimation of critical parameters. To identify key parameters, we conducted a elasticity index analysis. The elasticity index analysis revealed that increasing the insecticidal usage rate γ, the influx rate of susceptible humans Λ_h, and the treatment rate η effectively reduces malaria transmission within the population. Conversely, an increase in transmission rates (β₁, β₂, and β₃) and influx rate of susceptible mosquitoes Λ_m enhances malaria transmission within the community.

Model simulations indicate that all transmission rates significantly influence malaria progression. Control measures should prioritize reducing contact rates between susceptible humans and infected mosquitoes, as well as between susceptible mosquitoes and malaria-infected individuals (both asymptomatic and symptomatic). To reduce contact rates, vector control strategies primarily focus on maximizing insecticide use through insecticide-treated nets (ITNs) and indoor residual spraying (IRS) to eliminate adult mosquitoes. Additional measures include traditional malaria control methods such as air conditioning, protective clothing, and mosquito repellent lotions in endemic regions (Fischer, 2021).

The quantitative result of threshold ${\mathfrak{R}}_0$ value shows that asymptomatic cases contribute approximately 30 % to the basic reproduction number, while symptomatic cases account for about 70 %. These findings underscore the significant role of asymptomatic individuals in the transmission of malaria in Ethiopia. Addressing both symptomatic and asymptomatic cases is essential for effective malaria control and reducing community transmission. However, asymptomatic infections pose a major challenge due to the absence of clinical symptoms, making detection difficult. This necessitates targeted interventions such as mass screening and prompt treatment in endemic areas to identify and treat hidden parasite reservoirs.

During the current work, we have identified some shortcomings that require immediate improvement. In the course of theoretical analysis, asymptomatic malaria cases are often overlooked in research and reporting, leading to ineffective interventions and a failure to reduce malaria transmission effectively. Moreover, many malaria transmission models fail to account for the role of asymptomatic infections, limiting their accuracy. To improve model accuracy and control strategies, it is essential to incorporate these infections alongside key epidemiological factors. Additionally, the common assumptions of time-invariant recruitment rates and exclusion of environmental drivers constrain model realism and predictive power. Integrating time-varying recruitment rates and environmental variables enhances model complexity by capturing the dynamic interplay among human populations, mosquito vectors, and environmental conditions, thereby improving the accuracy of transmission dynamics and informing more effective malaria control.

CRediT authorship contribution statement

Andualem Tekle Haringo: Writing – review & editing, Writing – original draft, Software, Methodology, Investigation, Formal analysis. Legesse Lemecha Obsu: Writing – review & editing, Validation, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. Feyissa Kebede Bushu: Writing – review & editing, Supervision, Investigation, Formal analysis.

Data availability

Data is available upon request.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Handling Editor: Dr Yiming Shao