Introduction

Malaria is a severe infectious disease caused by Plasmodium parasites transferred to humans by the bite of infected female Anopheles mosquitos1. Malaria in human beings is caused by five different species of Plasmodium (single-celled protozoa): Plasmodium falciparum, Plasmodium vivax, Plasmodium malariae, Plasmodium ovale, and Plasmodium knowlesi. Plasmodium falciparum has the highest death and morbidity rates. However, Plasmodium vivax is the most widely dispersed species in the world2,3. These two species represent the greatest threat and contribute the most to global malaria prevalence. Plasmodium malariae is the third most prevalent species, with a prevalence range of 15–40%. According to the World Health Organization (WHO) report, globally, 249 million malaria cases were detected in 85 countries in 2022, with an estimated 608,000 malaria-related deaths. In 2022, the African region was home to 94% of malaria cases (233 million) and 95% of malaria deaths (580,000)4. Over the last two decades, global malaria cases have been on a downward trend, dropping from 80 to 57 cases per 1,000 people at risk between 2000 and 2019. Similarly, malaria-related deaths have also declined from 25 to 10 deaths per 100,000 people at risk. Despite malaria elimination initiatives in Sub-Saharan Africa, the disease load remains stable, especially among endemic territories. Despite a decline in death from 444,600 in 2020 to 427,854 in 2021, malaria cases jumped from 165 million to 168 million. Plasmodium falciparum malaria decreased, although Plasmodium malariae and Plasmodium ovale infections increased two to six times5. The COVID-19 pandemic has had a substantial impact on the epidemiology of imported malaria, with travel restrictions reducing the number of travel-related infections. However, statistics indicate that those returning from endemic areas are more likely to contract severe malaria6. Malaria cases documented between 2020 and 2021 were lower than in pre-pandemic years; however, initial data was inconclusive. The number of imported malaria cases increased in 2021 as COVID-related travel restrictions were removed, implying that severe malaria cases may rise.

The European Center for Disease Prevention and Control (ECDC) reported 2432 confirmed malaria cases in 2020 and 4780 in 2021, down from a peak of 8462 in 2019. France, Germany, Spain, and Belgium had the highest instances in 2020 and 20216,7. During the COVID-19 pandemic, the number of imported malaria cases increased as people traveled more. According to research by the Spanish national cooperation network8, the number of severe malaria cases increased significantly between 2020 and 2021. Another study9 found that relaxing COVID-19 limitations led to increased rates of parasitemia, hyperparasitemia, and severe malaria cases. The COVID-19 pandemic presents considerable difficulty in controlling malaria, as both diseases exhibit similar symptoms. Its outgrowths have intensified the incidence of malaria in Africa, threatening millions of lives10. Cyprus is the third-largest island (9.251 km\(^2\)) in the Mediterranean, after Sicily and Sardinia11. Like all countries in the Mediterranean basin, Cyprus has been severely affected by malaria for centuries. In the first half of the 1900s, malaria was endemic in Cyprus. However, with the “Malaria Eradication Project”carried out on the island of Cyprus between April 1946 and July 1949, malaria infection was successfully eradicated from the island12,13. Today, there are no endemic malaria cases in Cyprus, and all cases are imported. All imported malaria cases detected in Cyprus come to the island from endemic countries, mainly to study at universities14. Malaria is among the notifiable diseases in Cyprus. According to Turkish Republic of Northern Cyprus (TRNC) Ministry of Health data, a total of 56 imported malaria cases were detected in Northern Cyprus in the last eight years15. In the southern part of Cyprus, 17 imported malaria patients were reported between 2016 and 202016.

To understand the global dynamics of different diseases, the study of dynamic systems necessitates the development of mathematical models17. Understanding community-based infectious diseases requires an understanding of epidemiological research. Model construction, parameter estimation, parameters’ sensitivity testing, and numerical simulation computation are all aided by mathematical modeling. This aids in comprehending the relationship between infection and disease transmission within the community18. Numerous scholars have effectively showcased mathematical representations of an array of viral diseases19,20,21,22,23. These models help to formulate appropriate policies to control the disease by conducting prospective studies. Although researchers have adopted mathematical models to explain malaria dynamics and management, these efforts have not effectively lowered the disease’s spread24. More model-based research on malaria dynamics and awareness campaigns is required (e.g.,25,26,27).

The order of differential equations limits classical models involving derivatives. To get around these constraints, researchers are turning to a relatively new branch of mathematics called fractional calculus. Mathematical modeling is valuable and widely used across numerous scientific areas because fractional-order models accurately predict real-world phenomena (e.g.,28,29,30). To improve classical calculus for fractional-order modeling, several mathematical strategies have been used (see,31,32,33,34,35). In their study36, Yunus and Olayiwola modeled malaria using fractional-order differential equations and the Caputo fractional derivative, taking into account long-term memory effects and control measures like therapy and enlightenment. Numerical solutions were found using the Laplace-Adomian Decomposition Method, which enabled simulation of epidemic trajectory and assessment of the efficacy of actions in lowering infection levels. According to certain research cited in37,38, vaccination awareness is crucial for preventing infectious diseases like measles and the Ebola virus. The research supports attempts to combat endemic problems by incorporating behavioral elements into a framework of fractional calculus. Two orders, a fractional order and a fractal dimension, have been proposed by Atangana39 for a novel fractional order that combines operators from differential calculus and integral calculus. Researchers have made extensive use of the fractal-fractional model, a unique method for researching infectious diseases40,41,42,43. This will serve as our inspiration as we develop and examine a fractal-fractional model to comprehend the dynamics of malaria transmission.

The goal of this study is to examine the evolution of malaria in Northern Cyprus using a typical SEIR framework and fractal-fractional derivative, emphasizing the region’s distinct demographics and long incubation period. Using a modified model, this study focuses on basic features of the dynamics of malaria. This is how the remainder of the paper is structured. In Section “Model formulation”, the deterministic malaria model is developed. Section “Key features of the proposed model” explains the salient features of the proposed malaria mathematical model, which include positivity, boundedness, and epidemiologically feasible region. Section “Global derivative’s effect” provides information on how the global derivative affects the existence and uniqueness of the malaria model. Section “Qualitative analysis” determines the equilibrium points, the reproductive number \((R_0)\), and its sensitivity analysis. Section “Stability analysis” examines the model’s local and global stability. Section “Numerical solutions” employs the numerical scheme to derive the solution. In Section “Simulation results”, the model’s numerical simulations are run and discussed. The conclusion is found in Section “Conclusion”.

Model formulation

This work is based on the malaria cases real data in North Cyprus between 2014 and 2022. All datasets that were used in this study were taken from some scientific papers and hospitals in North Cyprus, which are publicly available on the website of the Ministry of Health of North Cyprus. Detailed information of cases was taken from the laboratories with ethical approval. The scientific papers that were used are cited in the paper. All of the cases obtained were in the age range of 18–28. 63% of the cases were men, while 37% of them were women. 97% of the cases were imported malaria cases. Moreover, these cases were from countries that have malaria as an endemic disease. The rest of the cases (3%) were local cases. In Fig. 1a,b, the distribution of cases according to years is shown both as numbers and percentages, respectively. The distribution of cases according to months is visualized in Fig. 2a,b as numbers and percentages, respectively. Figure 3a,b are illustrated to indicate the distribution of cases in seasons. These distributions are given both as numbers and percentages, respectively.

Fig. 1
Fig. 1
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The distribution of cases in numbers and percentages according to years.

Fig. 2
Fig. 2
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The distribution of cases in numbers and percentages according to months.

Fig. 3
Fig. 3
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The distribution of cases in numbers and percentages according to seasons.

Atangana39 introduced a new class of fractional notions by combining power-law, exponential law, and modified Mittag-Leffler law with fractal derivatives. This study uses the fractal-fractional operator with the Mittag-Leffler kernel, which is defined below.

Definition 2.1

39,40,43 Let \(u: (a,b) \rightarrow [0, \infty )\) be a continuous, fractal differentiable function of dimension \(\rho\). The fractal-fractional derivative of u using a generalized Mittag-Leffler kernel of order \(\sigma\) is as follows:

$$\begin{aligned} {}_0^{FFM} D^{\sigma ,\rho }_{t}[u(t)]=\frac{\tilde{AB}(\sigma )}{1-\sigma }\frac{d}{dt^{\rho }}\int _0^{t} u(\upsilon )\textbf{E}_{\sigma }[-\frac{\sigma (t-\upsilon )^{\sigma }}{1-\sigma }]d\upsilon , \end{aligned}$$
(1)

where the fractal derivative is given by

$$\begin{aligned} \frac{du(\upsilon )}{d\upsilon ^{\rho }}=\displaystyle \lim _{t\rightarrow \upsilon }\frac{u(t)-u(\upsilon )}{t^{\rho }-\upsilon ^{\rho }}, \end{aligned}$$
(2)

\(\sigma ,\rho \in (0,1]\), and \(\textbf{E}_{\sigma }\) presents Mittag-Leffler function. The corresponding integral is as follows:

$$\begin{aligned} {}_0^{FFM} I^{\sigma ,\rho }_{t}[u(t)]=\frac{(1-\sigma )\rho t^{\rho -1}}{\tilde{AB}(\sigma )}u(t)+ \frac{\sigma \rho }{\tilde{AB}(\sigma )\Gamma (\sigma )}\int _0^{t} \upsilon ^{\rho -1}(t-\upsilon )^{\sigma -1}u(\upsilon )d\upsilon . \end{aligned}$$
(3)

Since Northern Cyprus has distinct social dynamics and longer incubation periods, the evolution of malaria there is represented mathematically using a deterministic compartmental model. This enables us to comprehend the dynamics of malaria transmission more thoroughly. The model considers the populations of human hosts and mosquito vectors at various stages in the following ways:

  • \(S_h\): Susceptible human population;

  • \(E_h\): Exposed human population;

  • \(I_e\): Infected human population (whose countries have an endemic malaria disease);

  • \(I_{ne}\): Infected human population (whose countries have not an endemic malaria disease);

  • R: Recovered human population;

  • \(S_m\): Susceptible mosquito population;

  • \(E_m\): Exposed mosquito population;

  • \(I_m\): Infected mosquito population.

When developing a mathematical model, we began by making the following fundamental assumptions.

  • For biological considerations, it is assumed that all variables are non-negative.

  • The total human population at time t, denoted by \(N_H(t)\), is given by

    $$\begin{aligned} N_h(t)=S_h(t)+E_h(t)+I_e(t)+I_{ne}(t)+R(t), \end{aligned}$$
    (4)

    and the total mosquito population at time t, denoted by \(N_m(t)\), is given by

    $$\begin{aligned} N_m(t)=S_m(t)+E_m(t)+I_m(t). \end{aligned}$$
    (5)

  • Due to comparable birth and death rates, the overall population of both human beings and mosquitoes remains constant.

  • Those who recover from malaria develop a temporary immunity to reinfection.

  • The susceptible human population is influenced by recruitment rates \(\Lambda\) and \(\Phi\), as well as reinfection or fading immunity of recovered populations at the rate of q.

  • The force of infection, \(\frac{(\beta _{h_1}+\beta _{h_2}) {S}_h {I}_m}{{N}_h}\), shifts the susceptible human population to the exposed class. The exposed class develop infection at rates of \(\alpha _1\) and \(\alpha _2\), and hence relocate to the infectious compartments.

  • Infected people recover at rates of \(\gamma _1\) and \(\gamma _2\).

  • The logistic equation \(\Theta N_m (1-\delta )\left( 1-\frac{N_m}{p}\right)\) describes mosquito recruitment rates in susceptible mosquito population.

  • The force of infection in mosquitoes, \(\frac{(\beta _{m_1}+\beta _{m_2})({I}_{e}+{I}_{ne}) {S}_m}{{N}_m}\), shifts susceptible mosquitoes to the exposed compartment. The exposed mosquitoes develop infection at a rate of \(\alpha _3\) and hence relocate to the infectious mosquito compartment.

  • The natural deaths of humans and mosquitoes are provided by \(\mu _h\) and \(\mu _m\), respectively.

This is consistent with the notion that fractional models have implications and the potential to be useful. The purpose of this work is to create a mathematical model that uses simulation to compare significant changes in the malaria disease analysis. Fractional-order derivatives will demonstrate how well the malaria system captures memory effects and non-local behaviors. The model incorporates a new fractal-fractional operator that improves accuracy, provides deeper insights into disease dynamics, and proposes new intervention techniques. Based on the aforementioned assumptions, below is the newly constructed malaria model with fractal-fractional derivative order:

$$\begin{aligned} \begin{aligned} {}^{FFM}_0{D}_t^{\sigma ,\rho } S_h(t)&= \Lambda +\Phi -\frac{(\beta _{h_1}+\beta _{h_2 } ) S_h I_m}{N_h} +q R-\mu _h S_h,\\ {}^{FFM}_0{D}_t^{\sigma ,\rho } E_h(t)&= \frac{(\beta _{h_1}+\beta _{h_2} ) S_h I_m}{N_h}-(\alpha _1+\alpha _2 ) E_h-\mu _h E_h,\\ {}^{FFM}_0{D}_t^{\sigma ,\rho } I_e(t)&= \alpha _1 E_h-\gamma _1 I_e-\mu _h I_e,\\ {}^{FFM}_0{D}_t^{\sigma ,\rho } I_{ne}(t)&= \alpha _2 E_h-\gamma _2 I_{ne}-\mu _h I_{ne},\\ {}^{FFM}_0{D}_t^{\sigma ,\rho } R(t)&= \gamma _1 I_e+\gamma _2 I_{ne}-q R-\mu _h R,\\ {}^{FFM}_0{D}_t^{\sigma ,\rho } S_m(t)&= \Theta N_m (1-\delta )(1-\frac{N_m}{p})-\frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} -\mu _m S_m,\\ {}^{FFM}_0{D}_t^{\sigma ,\rho } E_m(t)&= \frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} -\alpha _3 E_m-\mu _m E_m,\\ {}^{FFM}_0{D}_t^{\sigma ,\rho } I_m(t)&=\alpha _3 E_m-\mu _m I_m. \end{aligned} \end{aligned}$$
(6)

Initial conditions are as follows:

$$\begin{aligned} S_h(0) = S_{h_0}, \ \ E_h(0)&= E_{h_0}, \ \ I_e(0) = I_{e_0}, \ \ I_{ne}(0) = I_{{ne}_0}, \ \ R(0) = R_{0}, \nonumber \\ S_m(0)&= S_{m_0}, \ \ E_m(0) = E_{m_0}, \ \ I_m(0) = I_{m_0}. \end{aligned}$$
(7)

The dynamical system is illustrated in Fig. 4.

Fig. 4
Fig. 4
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The Schematic diagram illustrating the dynamical system.

The study of a malaria model with distinct infectious classes contributes to a better understanding of the impact of population migrations and local control efforts on global and regional malaria spread, allowing for the assessment of travel’s impact on disease persistence and evaluation of focused treatments. The values of all parameters were derived using data taken from the TRNC Ministry of Health’s website. In addition, some parameter values were determined using the formulations and calculations provided in45,46,47. All parameter values are given in Table 1. The flowchart of the proposed method in the model is given in Fig. 4.

Table 1 Parameters interpretation.
Full size table

Key features of the proposed model

Positivity

The fundamental analysis demonstrates the superiority of the proposed solutions in addressing real-world problems with constructive values, while this subsection explores the conditions that guarantee the model’s advantageousness.

Theorem 3.1

The proposed system’s solutions will remain positive in \(\mathbb {R}_+^8\) for \(t>0\) if the initial conditions satisfy the following.

$$\begin{aligned} (S_{h_0}, E_{h_0}, I_{e_0}, I_{{ne}_0}, R_{0}, S_{m_0}, E_{m_0}, I_{m_0})>0.\end{aligned}$$

Proof

We establish the norm:

$$\begin{aligned} \Vert Y\Vert _{\infty }=\sup _{t \in D_{Y}}|Y(t)|,\end{aligned}$$

then, we have

$$\begin{aligned} \begin{aligned} {}^{FFM}_0{D}_t^{\sigma ,\rho } S_h(t)&= \Lambda +\Phi -\frac{(\beta _{h_1}+\beta _{h_2 } ) S_h I_m}{N_h} +q R-\mu _h S_h \ge -\Big \{\frac{(\beta _{h_1}+\beta _{h_2 } ) |I_m|}{|N_h|}+\mu _h \Big \}S_h\\&\ge -\Big \{\frac{(\beta _{h_1}+\beta _{h_2 } ) \sup _{t\in D_{I_m}}|I_m|}{\sup _{t\in D_{N_h}}|N_h|}+\mu _h \Big \}S_h= -\Big \{\frac{(\beta _{h_1}+\beta _{h_2 } ) {\Vert I_m\Vert }_{\infty }}{{\Vert N_h\Vert }_{\infty }}+\mu _h \Big \}S_h. \end{aligned} \end{aligned}$$
(8)

In case of fractal-fractional derivative44, we have

$$\begin{aligned} & S_h(t)\ge S_{h_0}~\textbf{E}_\sigma \Big [-\frac{\zeta ^{1-\rho }\sigma \Big (\frac{(\beta _{h_1}+\beta _{h_2 } ) {\Vert I_m\Vert }_{\infty }}{{\Vert N_h\Vert }_{\infty }}+\mu _h\Big )t^\sigma }{\tilde{AB}(\sigma ) -(1-\sigma )\Big (\frac{(\beta _{h_1}+\beta _{h_2 } ) {\Vert I_m\Vert }_{\infty }}{{\Vert N_h\Vert }_{\infty }}+\mu _h \Big )}\Big ],~~~~\forall t\ge 0. \end{aligned}$$
(9)
$$\begin{aligned} & \begin{aligned} {}^{FFM}_0{D}_t^{\sigma ,\rho } E_h(t)&= \frac{(\beta _{h_1}+\beta _{h_2} ) S_h I_m}{N_h}-(\alpha _1+\alpha _2 ) E_h-\mu _h E_h \ge -\Big \{\alpha _1+\alpha _2+\mu _h\Big \}E_h\\ \Rightarrow \ \ E_h(t)&\ge E_{h_0}~\textbf{E}_\sigma \Big [-\frac{\zeta ^{1-\rho }\sigma \Big (\alpha _1+\alpha _2+\mu _h\Big )t^\sigma }{\tilde{AB}(\sigma ) -(1-\sigma )\Big (\alpha _1+\alpha _2+\mu _h\Big )}\Big ],~~~~\forall t\ge 0. \end{aligned} \end{aligned}$$
(10)
$$\begin{aligned} & \begin{aligned} {}^{FFM}_0{D}_t^{\sigma ,\rho } I_e(t)&= \alpha _1 E_h-\gamma _1 I_e-\mu _h I_e \ge -\Big \{\gamma _1+\mu _h\Big \}I_e\\ & \Rightarrow \ \ I_e(t) \ge I_{e_0}~\textbf{E}_\sigma \Big [-\frac{\zeta ^{1-\rho }\sigma \Big (\gamma _1+\mu _h\Big )t^\sigma }{\tilde{AB}(\sigma ) -(1-\sigma )\Big (\gamma _1+\mu _h\Big )}\Big ],~~~~\forall t\ge 0. \end{aligned} \end{aligned}$$
(11)
$$\begin{aligned} & \begin{aligned} {}^{FFM}_0{D}_t^{\sigma ,\rho } I_{ne}(t)&= \alpha _2 E_h-\gamma _2 I_{ne}-\mu _h I_{ne}\ge -\Big \{\gamma _2+\mu _h\Big \}I_{ne}\\ &\Rightarrow \ \ I_{ne}(t) \ge I_{{ne}_0}~\textbf{E}_\sigma \Big [-\frac{\zeta ^{1-\rho }\sigma \Big (\gamma _2+\mu _h\Big )t^\sigma }{\tilde{AB}(\sigma ) -(1-\sigma )\Big (\gamma _2+\mu _h\Big )}\Big ],~~~~\forall t\ge 0. \end{aligned} \end{aligned}$$
(12)
$$\begin{aligned} \begin{aligned} {}^{FFM}_0{D}_t^{\sigma ,\rho } R(t)& = \gamma _1 I_e+\gamma _2 I_{ne}-q R-\mu _h R \ge -\Big \{q+\mu _h\Big \}R\\ &\Rightarrow \ \ R(t) \ge R_0 ~\textbf{E}_\sigma \Big [-\frac{\zeta ^{1-\rho }\sigma \Big (q+\mu _h\Big )t^\sigma }{\tilde{AB}(\sigma ) -(1-\sigma )\Big (q+\mu _h\Big )}\Big ],~~~~\forall t\ge 0. \end{aligned} \end{aligned}$$
(13)
$$\begin{aligned} \begin{aligned} {}^{FFM}_0{D}_t^{\sigma ,\rho } S_m(t)&= \Theta N_m (1-\delta )(1-\frac{N_m}{p})-\frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} -\mu _m S_m\\&\ge -\Big \{\frac{(\beta _{m_1}+\beta _{m_2} )(|I_{e}|+|I_{ne}| )}{|N_m|}+\mu _m \Big \}S_m\\&\ge -\Big \{\frac{(\beta _{m_1}+\beta _{m_2} )(\sup _{t\in D_{I_e}}|I_{e}|+\sup _{t\in D_{I_{ne}}}|I_{ne}| )}{\sup _{t\in D_{N_m}}|N_m|}+\mu _m \Big \}S_m\\&= -\Big \{\frac{(\beta _{m_1}+\beta _{m_2} )({\Vert I_{e}\Vert }_{\infty }+{\Vert I_{ne}\Vert }_{\infty } )}{{\Vert N_m\Vert }_{\infty }}+\mu _m \Big \}S_m\\ & \Rightarrow \ \ S_m(t) \ge S_{m_0} ~\textbf{E}_\sigma \Big [-\frac{\zeta ^{1-\rho }\sigma \Big (\frac{(\beta _{m_1}+\beta _{m_2} )({\Vert I_{e}\Vert }_{\infty }+{\Vert I_{ne}\Vert }_{\infty } )}{{\Vert N_m\Vert }_{\infty }}+\mu _m\Big )t^\sigma }{\tilde{AB}(\sigma ) -(1-\sigma )\Big (\frac{(\beta _{m_1}+\beta _{m_2} )({\Vert I_{e}\Vert }_{\infty }+{\Vert I_{ne}\Vert }_{\infty } )}{{\Vert N_m\Vert }_{\infty }}+\mu _m\Big )}\Big ],~~~~\forall t\ge 0. \end{aligned} \end{aligned}$$
(14)
$$\begin{aligned} \begin{aligned} {}^{FFM}_0{D}_t^{\sigma ,\rho } E_m(t)&= \frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} -\alpha _3 E_m-\mu _m E_m \ge -\Big \{\alpha _3+\mu _m\Big \}E_m\\ & \Rightarrow \ \ E_m(t) \ge E_{m_0}~\textbf{E}_\sigma \Big [-\frac{\zeta ^{1-\rho }\sigma \Big (\alpha _3+\mu _m\Big )t^\sigma }{\tilde{AB}(\sigma ) -(1-\sigma )\Big (\alpha _3+\mu _m\Big )}\Big ],~~~~\forall t\ge 0. \end{aligned} \end{aligned}$$
(15)
$$\begin{aligned} \begin{aligned} {}^{FFM}_0{D}_t^{\sigma ,\rho } I_m(t)&=\alpha _3 E_m-\mu _m I_m \ge -\mu _mE_m\\ & \Rightarrow \ \ I_m(t) \ge I_{m_0}~\textbf{E}_\sigma \Big [-\frac{\zeta ^{1-\rho }\sigma \mu _m t^\sigma }{\tilde{AB}(\sigma ) -(1-\sigma )\mu _m}\Big ],~~~~\forall t\ge 0. \end{aligned} \end{aligned}$$
(16)

\(\square\)

Biological feasibility

Theorem 3.2

The model (6) is unique and limited in \(\mathbb {R}^8_{+}\), together with the initial conditions (7).

Proof

We get

$$\begin{aligned} \Big ({}^{FFM}_0{D}_t^{\sigma ,\rho } S_h(t)\Big )\big |_{S_h=0}= & \Lambda +\Phi ~~\ge ~~0,\nonumber \\ \Big ({}^{FFM}_0{D}_t^{\sigma ,\rho } E_h(t)\Big )\big |_{E_h=0}= & \frac{(\beta _{h_1}+\beta _{h_2} ) S_h I_m}{N_h}~~\ge ~~0,\nonumber \\ \Big ({}^{FFM}_0{D}_t^{\sigma ,\rho } I_e(t)\Big )\big |_{I_e=0}= & \alpha _1 E_h ~~\ge ~~0,\nonumber \\ \Big ({}^{FFM}_0{D}_t^{\sigma ,\rho } I_{ne}(t)\Big )\big |_{I_{ne}=0}= & \alpha _2 E_h ~~\ge ~~0,\nonumber \\ \Big ({}^{FFM}_0{D}_t^{\sigma ,\rho } R(t)\Big )\big |_{R=0}= & \gamma _1 I_e+\gamma _2 I_{ne} ~~\ge ~~0,\nonumber \\ \Big ({}^{FFM}_0{D}_t^{\sigma ,\rho } S_m(t)\Big )\big |_{S_m=0}= & \Theta N_m (1-\delta )(1-\frac{N_m}{p}) ~\ge ~0,\nonumber \\ \Big ({}^{FFM}_0{D}_t^{\sigma ,\rho } E_m(t)\Big )\big |_{E_m=0}= & \frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} ~~\ge ~~0,\nonumber \\ \Big ({}^{FFM}_0{D}_t^{\sigma ,\rho } I_m(t)\Big )\big |_{I_m=0}= & \alpha _3 E_m ~~\ge ~~0. \end{aligned}$$
(17)

Since \((S_{h_0},E_{h_0},I_{e_0},I_{{ne}_0},R_0,S_{m_0},E_{m_0},I_{m_0})\in \mathbb {R}_+^8\), (17) states that the solution is not allowed to leave the hyperplane. This proves that the domain of positive invariance is the set \(\mathbb {R}^8_+\). \(\square\)

Theorem 3.3

All solutions of the system (6) are bounded when the initial conditions (7) are positive.

Proof

Since, we have

$$\begin{aligned}N_h(t)=S_h(t)+E_h(t)+I_e(t)+I_{ne}(t)+R(t).\end{aligned}$$

So, we find

$$\begin{aligned} {}^{FFM}_0{D}_t^{\sigma ,\rho } N_h(t)= & {}^{FFM}_0{D}_t^{\sigma ,\rho } S_h(t)+{}^{FFM}_0{D}_t^{\sigma ,\rho } E_h(t)+{}^{FFM}_0{D}_t^{\sigma ,\rho } I_n(t)+{}^{FFM}_0{D}_t^{\sigma ,\rho } I_{ne}(t) +{}^{FFM}_0{D}_t^{\sigma ,\rho } R(t), \nonumber \\= & \Lambda +\Phi -\mu _h (S_h+E_h+I_e+I_{ne}+R),\nonumber \\\le & \Lambda +\Phi -\mu _h N_h. \end{aligned}$$
(18)

The comparison theorem gives us the following:

$$\begin{aligned} 0~\le ~ N_h(t)~\le ~\frac{\Lambda +\Phi }{\mu _h}+\Big (N_h(0)-\frac{\Lambda +\Phi }{\mu _h}\Big )~\textbf{E}_\sigma \Big [-\frac{\zeta ^{1-\rho }\sigma \mu _h t^\sigma }{\tilde{AB}(\sigma ) -(1-\sigma )\mu _h}\Big ],~~~~\forall t\ge 0. \end{aligned}$$
(19)

If \(N_h(0)<\frac{\Lambda +\Phi }{\mu _h}\), then

$$\begin{aligned} \displaystyle \lim _{t\rightarrow \infty } N_h(t)\le & \frac{\Lambda +\Phi }{\mu _h}. \end{aligned}$$
(20)

Consequently,

$$\begin{aligned} 0&\le ~~ N_h(t) ~~ \le&\frac{\Lambda +\Phi }{\mu _h}. \end{aligned}$$
(21)

Additionally, the total mosquitoes population is

$$\begin{aligned}N_m(t)=S_m(t)+E_m(t)+I_m(t),\end{aligned}$$

then

$$\begin{aligned} {}^{FFM}_0{D}_t^{\sigma ,\rho } N_m(t)= & {}^{FFM}_0{D}_t^{\sigma ,\rho } S_m(t)+{}^{FFM}_0{D}_t^{\sigma ,\rho } E_m(t)+{}^{FFM}_0{D}_t^{\sigma ,\rho }I_m(t),\nonumber \\= & \Theta N_m (1-\delta )(1-\frac{N_m}{p}) -\mu _m (S_m+E_m+I_m),\nonumber \\\le & \Theta N_m (1-\delta )(1-\frac{N_m}{p})-\mu _m N_m,\nonumber \\\le & \big (1-\frac{\mu _m}{\Theta (1-g)}\big )-\frac{1}{p}N_m. \end{aligned}$$
(22)

Therefore,

$$\begin{aligned} 0&\le ~~ N_m(t) ~~ \le&p\big (1-\frac{\mu _m}{\Theta (1-g)}\big ). \end{aligned}$$
(23)

Hence, the system (6)’s solutions are limited to the biological feasible region \(\Upsilon\):

$$\begin{aligned} & \Upsilon =\Big \{(S_h,E_h,I_e,I_{ne},R,S_m,E_m,I_m)\in \mathbb {R}_+^8\mid 0\le N_h(t) \le \frac{\Lambda +\Phi }{\mu _h}, ~0 \le N_m(t) \nonumber \\ & \quad \le p\big (1-\frac{\mu _m}{\Theta (1-g)}\big )\Big \},~~~\forall t\ge 0. \end{aligned}$$
(24)

\(\square\)

Global derivative’s effect

Widely recognized in the literature, among the most frequently recurring integral is the Riemann-Stieltjes integral, of which the standard integral is a specific example. For

$$\begin{aligned} Z(t)= \int Z(t)dt,\end{aligned}$$

the Riemann-Stieltjes integral is

$$\begin{aligned}Z_y (t)= \int Z(t)dy(t).\end{aligned}$$

The global derivative of Z(t) in terms of y(t) is provided by

$$\begin{aligned} \textsf{D}_y Z(t)= & \lim _{\tau \rightarrow 0} \frac{Z(t+\tau )-Z(t)}{y(t+\tau )-y(t)}. \end{aligned}$$
(25)
$$\begin{aligned} \textsf{D}_y Z(t)= & \frac{Z^\prime (t)}{y^\prime (t)}, ~~~~~~~~~y^\prime (t)\ne 0,~~~~\forall t\in \mathbb {D}_{y^\prime }. \end{aligned}$$
(26)

To examine if it has an impact on the malaria model (6), we are going to substitute out the global derivative with the classical derivative.

$$\begin{aligned} {\left\{ \begin{array}{ll} \textsf{D}_y S_h(t) ~=~ \Lambda +\Phi -\frac{(\beta _{h_1}+\beta _{h_2 } ) S_h I_m}{N_h} +q R-\mu _h S_h,\\ \textsf{D}_y E_h(t) ~=~ \frac{(\beta _{h_1}+\beta _{h_2} ) S_h I_m}{N_h}-(\alpha _1+\alpha _2 ) E_h-\mu _h E_h,\\ \textsf{D}_y I_e(t) ~=~ \alpha _1 E_h-\gamma _1 I_e-\mu _h I_e, \\ \textsf{D}_y I_{ne}(t) ~=~ \alpha _2 E_h-\gamma _2 I_{ne}-\mu _h I_{ne},\\ \textsf{D}_y R_h(t) ~=~ \gamma _1 I_e+\gamma _2 I_{ne}-q R-\mu _h R,\\ \textsf{D}_y S_m(t) ~=~ \Theta N_m (1-\delta )(1-\frac{N_m}{p})-\frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} -\mu _m S_m,\\ \textsf{D}_y E_m(t) ~=~ \frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} -\alpha _3 E_m-\mu _m E_m,\\ \textsf{D}_y I_m(t) ~=~ \alpha _3 E_m-\mu _m I_m. \end{array}\right. } \end{aligned}$$
(27)

Here, we assume that y is differentiable. As a result, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} S_h^{\prime }(t) ~=~ y^{\prime }\big \{\Lambda +\Phi -\frac{(\beta _{h_1}+\beta _{h_2 } ) S_h I_m}{N_h} +q R-\mu _h S_h\big \},\\ E_h^{\prime }(t) ~=~ y^{\prime }\big \{\frac{(\beta _{h_1}+\beta _{h_2} ) S_h I_m}{N_h}-(\alpha _1+\alpha _2 ) E_h-\mu _h E_h\big \},\\ I_e^{\prime }(t) ~=~ y^{\prime }\big \{\alpha _1 E_h-\gamma _1 I_e-\mu _h I_e, \\ I_{ne}^{\prime }(t) ~=~ y^{\prime }\big \{\alpha _2 E_h-\gamma _2 I_{ne}-\mu _h I_{ne}\big \},\\ R_h^{\prime }(t) ~=~ y^{\prime }\big \{\gamma _1 I_e+\gamma _2 I_{ne}-q R-\mu _h R\big \},\\ S_m^{\prime }(t) ~=~ y^{\prime }\big \{\Theta N_m (1-\delta )(1-\frac{N_m}{p})-\frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} -\mu _m S_m\big \},\\ E_m^{\prime }(t) ~=~ y^{\prime }\big \{\frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} -\alpha _3 E_m-\mu _m E_m\big \},\\ I_m^{\prime }(t) ~=~ y^{\prime }\big \{\alpha _3 E_m-\mu _m I_m\big \}. \end{array}\right. } \end{aligned}$$
(28)

Let

$$\begin{aligned} \Vert y^{\prime }\Vert _\infty = \sup _{t\in {D_{y^{\prime }}}}|y^{\prime }(t)| < N. \end{aligned}$$
(29)

Examine the following system:

$$\begin{aligned} {\left\{ \begin{array}{ll} \chi _1(t,\omega (t)) ~=~ y^{\prime }\big \{\Lambda +\Phi -\frac{(\beta _{h_1}+\beta _{h_2 } ) S_h I_m}{N_h} +q R-\mu _h S_h\big \},\\ \chi _2(t,\omega (t)) ~=~ y^{\prime }\big \{\frac{(\beta _{h_1}+\beta _{h_2} ) S_h I_m}{N_h}-(\alpha _1+\alpha _2 ) E_h-\mu _h E_h\big \},\\ \chi _3(t,\omega (t)) ~=~ y^{\prime }\big \{\alpha _1 E_h-\gamma _1 I_e-\mu _h I_e\big \}, \\ \chi _4(t,\omega (t)) ~=~ y^{\prime }\big \{\alpha _2 E_h-\gamma _2 I_{ne}-\mu _h I_{ne}\big \},\\ \chi _5(t,\omega (t)) ~=~ y^{\prime }\big \{\gamma _1 I_e+\gamma _2 I_{ne}-q R-\mu _h R\big \},\\ \chi _6(t,\omega (t)) ~=~ y^{\prime }\big \{\Theta N_m (1-\delta )(1-\frac{N_m}{p})-\frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} -\mu _m S_m\big \},\\ \chi _7(t,\omega (t)) ~=~ y^{\prime }\big \{\frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} -\alpha _3 E_m-\mu _m E_m\big \},\\ \chi _8(t,\omega (t)) ~=~ y^{\prime }\big \{\alpha _3 E_m-\mu _m I_m\big \}, \end{array}\right. } \end{aligned}$$
(30)

where \(\omega =(S_h,E_h,I_e,I_{ne},R,S_m,E_m,I_m)\). We will see from this example that the proposed system has a single solution.

To ensure the existence and uniqueness of our system, it is crucial to ensure the following:

  1. 1.

    \(\big | \chi _i(t,\omega (t))\big |^2 ~ \le ~ K_i(1+ |\omega |^2)\), i=1,2,3,...,8.

  2. 2.

    \(\forall\) \({\omega },\tilde{\omega }\), \({\parallel \chi _i(t,\omega (t))- \chi _i(t,\tilde{\omega }(t)) \parallel }_{\infty }^2 \ \le \ \overline{K}_i{\parallel \omega -\tilde{\omega } \parallel }_{\infty }^2\).

We have

$$\begin{aligned} \begin{aligned} |\chi _1(t,\omega (t)|^2&= \Big |y^{\prime }\big \{\Lambda +\Phi -\frac{(\beta _{h_1}+\beta _{h_2 } ) S_h I_m}{N_h} +q R-\mu _h S_h\big \}\Big |^2\\&\le 2|y^\prime |^2\Lambda ^2+2|y^\prime |^2\Phi ^2+2|y^\prime |^2q^2 |R|^2+2|y^\prime |^2\big |-\frac{(\beta _{h_1}+\beta _{h_2 } ) S_h I_m}{N_h}-\mu _h S_h\big |^2\\&\le 2|y^\prime |^2(\Lambda ^2+\Phi ^2+q^2 |R|^2)+4|y^\prime |^2\frac{(\beta _{h_1}+\beta _{h_2 } )^2 |S_h|^2 |I_m|^2}{|N_h|^2}+4|y^\prime |^2\mu _h^2 |S_h|^2\\&\le 2|y^\prime |^2(\Lambda ^2+\Phi ^2+q^2 |R|^2)\Big (1+\frac{2\big (\frac{(\beta _{h_1}+\beta _{h_2 } )^2|I_m|^2}{|N_h|^2}+\mu _h^2\big )}{\Lambda ^2+\Phi ^2+q^2 |R|^2} |S_h|^2\Big ) \le K_1(1+|S_h|^2), \end{aligned} \end{aligned}$$
(31)

under the condition \(\frac{2\big (\frac{(\beta _{h_1}+\beta _{h_2 } )^2|I_m|^2}{|N_h|^2}+\mu _h^2\big )}{\Lambda ^2+\Phi ^2+q^2 |R|^2}<1\), where \(K_1=2|y^\prime |^2(\Lambda ^2+\Phi ^2+q^2 |R|^2)\). Similarly we have a results for others compartments and this proves that function satisfies the linear growth condition. Next, we verify that the Lipschitz requirement is satisfied. Now let \(\xi _1=(t,S_{h_1},E_h,I_e,I_{ne},R,S_m,E_m,I_m)\) and \(\xi _2=(t,S_{h_2},E_h,I_e,I_{ne},R,S_m,E_m,I_m)\). We have

$$\begin{aligned} |\chi _1(t,\xi _1)-\chi _1(t,\xi _2)|^2 & = \Big |y^{\prime }\big \{-\frac{(\beta _{h_1}+\beta _{h_2 }) I_m}{N_h}-\mu _h\big \} (S_{h_1}-S_{h_2})\Big |^2\\& \le |y^{\prime }|^2\big \{\frac{2(\beta _{h_1}+\beta _{h_2 })^2 |I_m|^2}{|N_h|^2}+2\mu _h^2\big \} |S_{h_1}-S_{h_2}|^2, \end{aligned}$$
$$\begin{aligned} \begin{aligned} \sup _{t\in \textsf{D}_{S_h}}|\chi _1(t,\xi _1)-\chi _1(t,\xi _2)|^2&\le \sup _{t\in \textsf{D}_{y^{\prime }}}|y^{\prime }|^2\big \{\frac{2(\beta _{h_1}+\beta _{h_2 })^2 \sup _{t\in \textsf{D}_{I_m}}|I_m|^2}{\sup _{t\in \textsf{D}_{N_h}}|N_h|^2}+2\mu _h^2\big \} \sup _{t\in \textsf{D}_{S_h}}|S_{h_1}-S_{h_2}|^2\\&\le \Vert y^{\prime }\Vert ^2_{\infty }\big \{\frac{2(\beta _{h_1}+\beta _{h_2 })^2 \Vert I_m\Vert ^2_{\infty }}{\Vert N_h\Vert ^2_{\infty }}+2\mu _h^2\big \} \Vert S_{h_1}-S_{h_2}\Vert ^2_{\infty }\le \overline{K}_1\Vert S_{h_1}-S_{h_2}\Vert ^2_{\infty }, \end{aligned} \end{aligned}$$
(32)

where \(\overline{K}_1=\Vert y^{\prime }\Vert ^2_{\infty }\big \{\frac{2(\beta _{h_1}+\beta _{h_2 })^2 \Vert I_m\Vert ^2_{\infty }}{\Vert N_h\Vert ^2_{\infty }}+2\mu _h^2\big \}\).

The system (6) has a unique solution under the following conditions for all compartments in general:

$$\begin{aligned} \max \Bigg \{&\frac{2\big (\frac{(\beta _{h_1}+\beta _{h_2 } )^2|I_m|^2}{|N_h|^2}+\mu _h^2\big )}{\Lambda ^2+\Phi ^2+q^2 |R|^2},~ \frac{2\alpha _1^2+2\alpha _2^2+\mu _h^2}{\frac{(\beta _{h_1}+\beta _{h_2} ) |S_h|^2 |I_m|^2}{|N_h|^2}},~ \frac{2(\gamma _1^2+\mu _h^2)}{\alpha _1^2|E_h|^2},~ \frac{2(\gamma _2^2+\mu _h^2)}{\alpha _2^2|E_h|^2}, \nonumber \\&~~~~~~\frac{\frac{2(\beta _{m_1}+\beta _{m_2} )^2(|I_{e}+I_{ne}|^2 )}{|N_m|^2}+\mu _m^2}{|N_m|^2 (1-\delta )^2(1-\frac{|N_m|}{p})^2}, ~ \frac{2(\alpha _3^2+\mu _m^2)}{\frac{(\beta _{m_1}+\beta _{m_2} )^2\big |I_{e}+I_{ne}\big |^2 |S_m|^2}{|N_m|^2}}, ~\frac{\mu _m^2}{\alpha _3^2 |E_m|^2} \Bigg \} < 1. \end{aligned}$$
(33)

Qualitative analysis

Equilibrium points

In this portion of the article, we give a comprehensive review of equilibrium points. These equilibrium locations can only be obtained by setting the left side of Equation (6) to zero, as the following illustration indicates. According to the suggested model, the disease-free equilibrium point is

$$\begin{aligned} P_{1}(S_h^0,~E_h^0,~I_e^0,~I_{ne}^0,~R^0,~S_m^0,~E_m^0,~I_m^0)=\Big \{\frac{\Lambda +\Phi }{\mu _h},~0,~0,~0,~0,~p\big (1-\frac{\mu _m}{\Theta (1-g)}\big ),~0,~0\Big \}, \end{aligned}$$
(34)

as well as the endemic equilibrium point is \(P_{2}(S_h^{*},E_h^{*},I_e^{*},I_{ne}^*, R^{*}, S_m^{*}, E_m^{*}, I_m^{*})\), where

$$\begin{aligned} S^*_h =\frac{(\Lambda +\Phi +qR^*)N^*_h}{(\beta _{h_1}+\beta _{h_2})I_m^*+\mu _hN^*_h},~~&~~ E^*_h=\frac{(\beta _{h_1}+\beta _{h_2})S_H^*I_m^*}{N_h^*(\alpha _1+\alpha _2+\mu _h)},\nonumber \\ I^*_e =\frac{\alpha _1E^*_h}{\gamma _1+\mu _h}, ~~&~~I^*_{ne}=\frac{\alpha _2E^*_h}{\gamma _2+\mu _h},\nonumber \\ R^*=\frac{\gamma _1I_e^*+\gamma _2I_{ne}^*}{q+\mu _h}, ~~&~~ S^*_m=\frac{\Theta (N_m^*)^2(1-g)(1-\frac{N_m^*}{p})}{(\beta _{m_1}+\beta _{m_2})(I_e^*+I_{ne}^*)S^*_m+\mu _m N_m^*},\nonumber \\ E^*_m =\frac{(\beta _{m_1}+\beta _{m_2})(I_e^*+I_{ne}^*)S^*_m}{(\alpha _3+\mu _m)N^*_m},~~&~~I^*_m=\frac{\alpha _3E_m^*}{\mu _m}. \end{aligned}$$
(35)

Reproductive number and sensitivity of parameters

Understanding stability conditions is facilitated by the reproduction number \((R_0)\), which is an essential tool in epidemiological modeling. It is dictated by the vectors F and V, which stand for the origin of novel diseases and the dissemination of preexisting infections. We analyze these vectors at the disease-free equilibrium point \(D_1\). Consider the system

$$\begin{aligned} \begin{aligned} \frac{d{E_h}}{dt}&= \frac{(\beta _{h_1}+\beta _{h_2} ) S_h I_m}{N_h}-(\alpha _1+\alpha _2 ) E_h-\mu _h E_h,\\ \frac{d{I_e}}{dt}&= \alpha _1 E_h-\gamma _1 I_e-\mu _h I_e,\\ \frac{d{I_{ne}}}{dt}&= \alpha _2 E_h-\gamma _2 I_{ne}-\mu _h I_{ne},\\ \frac{d{I_m}}{dt}&=\alpha _3 E_m-\mu _m I_m. \end{aligned} \end{aligned}$$
(36)

Therefore, we have

$$\begin{aligned} F = \begin{pmatrix} \frac{(\beta _{h_1}+\beta _{h_2} ) S_h I_m}{N_h}\\ 0\\ 0\\ 0\\ \end{pmatrix},~~~~~~~ V = \begin{pmatrix} (\alpha _1+\alpha _2+\mu _h)E_h\\ -\alpha _1 E_h+(\gamma _1+\mu _h) I_e \\ -\alpha _2 E_h+(\gamma _2+\mu _h) I_{ne} \\ -\alpha _3 E_h+\mu _h I_m \\ \end{pmatrix}. \end{aligned}$$

At \(P_{1}(S_h^0,~E_h^0,~I_e^0,~I_{ne}^0,~R^0,~S_m^0,~E_m^0,~I_m^0)\), we find the Jacobian matrices \((\textsf{F},\textsf{V})\) of F and V. And after that, we have

$$\begin{aligned} \textsf{F}\textsf{V}^{-1} = \begin{pmatrix} \frac{\alpha _3(\beta _{h_1}+\beta _{h_2})}{\mu _m(\alpha _1+\alpha _2+\mu _h)} & 0 & 0 & \frac{\beta _{h_1}+\beta _{h_2}}{\mu _m}\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{pmatrix}. \end{aligned}$$

Given that the reproduction number \((R_0)\) of matrix \(\textsf{F} {\textsf{V}}^{-1}\) is the dominating eigenvalue, thus, we have

$$\begin{aligned} R_0=\frac{\alpha _3(\beta _{h_1}+\beta _{h_2})}{\mu _m(\alpha _1+\alpha _2+\mu _h)}. \end{aligned}$$
(37)

Sensitivity analysis is a technique used to assess a model’s resilience by looking at how variations in input parameters impact the model’s results. It works especially well with uncertain models and ambiguous data. It measures how inputs and outputs relate to one another, highlighting important factors that have a big influence on the model’s output. This aids researchers in identifying model limitations, honing important elements, and reaching well-informed conclusions. By computing the partial derivatives with respect to the pertinent parameters, we can examine the sensitivity of \(R_{0}\), it is evident that the value of \(R_{0}\) is quite sensitive when we change the settings. The parameters \(\alpha _3, \beta _{h_1}, \beta _{h_2}\) in our analysis show growth, whereas the parameters \(\alpha _1,\alpha _2,\mu _h,\mu _m\) show contraction. As a result, for efficient infection management, prevention should come before treatment. Figure 5 represents a graphic representation of the link between \({R}_0\) and the given parameters.

Fig. 5
Fig. 5
Full size image

\({R}_{0}\)’s sensitivity to various parameters.

Stability analysis

Local stability

Theorem 6.1

When \(R_0\) is less than 1, the disease-free equilibrium point of the proposed malaria model is found to be locally asymptotically stable; however, when \(R_0\) is greater than 1, it becomes unstable.

Proof

To assess the stability of the model (6) at the disease-free equilibrium, assume the Jacobian matrix at \(P_{1}(S_h^0,~E_h^0,~I_e^0,~I_{ne}^0,~R^0,~S_m^0,~E_m^0,~I_m^0)\) is as follows:

$$\begin{aligned} \textrm{J}(P_1)=\left( \begin{array}{cccccccc} -\mu _h & 0 & 0 & 0 & q & 0 & 0 & -k_1 \\ 0 & -l_1 & 0 & 0 & 0 & 0 & 0 & k_1 \\ 0 & \alpha _1 & -l_2 & 0 & 0 & 0 & 0 & 0 \\ 0 & \alpha _2 & 0 & -l_3 & 0 & 0 & 0 & 0 \\ 0 & 0 & \gamma _1 & \gamma _2 & -l_4 & 0 & 0 & 0 \\ 0 & 0 & -k_2 & -k_2 & 0 & -\mu _m & 0 & 0 \\ 0 & 0 & k_2 & k_2 & 0 & 0 & -l_5 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \alpha _3 & -\mu _m \\ \end{array} \right) , \end{aligned}$$
(38)

where

$$\begin{aligned} & l_1= \alpha _1+\alpha _2+\mu _h, ~~~~~ l_2= \gamma _1+\mu _h, ~~~~~ l_3= \gamma _2+\mu _h,~~~~~l_4=q+\mu _h,\\ & l_5=\alpha _3+\mu _m,~~~~~k_1=\beta _{h_1}+\beta _{h_2},~~~~~k_2=\beta _{m_1}+\beta _{m_2}.\end{aligned}$$

Thus, the characteristics equation is

$$\begin{aligned} \big |\textrm{J}(P_1)-\lambda I\big |=0. \end{aligned}$$
(39)

We can write

$$\begin{aligned} \begin{vmatrix} -\mu _h-\lambda&0&0&0&q&0&0&-k_1 \\ 0&-l_1-\lambda&0&0&0&0&0&k_1 \\ 0&\alpha _1&-l_2-\lambda&0&0&0&0&0 \\ 0&\alpha _2&0&-l_3-\lambda&0&0&0&0 \\ 0&0&\gamma _1&\gamma _2&-l_4-\lambda&0&0&0 \\ 0&0&-k_2&-k_2&0&-\mu _m-\lambda&0&0 \\ 0&0&k_2&k_2&0&0&-l_5-\lambda&0 \\ 0&0&0&0&0&0&\alpha _3&-\mu _m-\lambda \\ \end{vmatrix}=0. \end{aligned}$$
(40)

Upon solving the determinant matrix mentioned above, we obtain the following eigenvalues \((\lambda )\):

$$\begin{aligned} \lambda _1&=-\mu _h,~~~~~~~~\lambda _2= -l_1,~~~~~~~~\lambda _3=-l_2, ~~~~~~~~~\lambda _4=-l_3, \nonumber \\ \lambda _5&=-l_4,~~~~~~~~\lambda _6= -\mu _m,~~~~~~~~\lambda _7=-l_5, ~~~~~~~~~\lambda _8=-\mu _m. \end{aligned}$$
(41)

Since all of the matrix’s eigenvalues have negative real components, the system is locally asymptotically stable, which means that its solutions will eventually converge to malaria-free equilibrium point. \(\square\)

Global Stability Using Lyapunov for Endemic Equilibrium

Lemma 6.1

Let \(H \in \mathbb {R}^{+}\) be a continuous function such that for any \(t \ge t_0\)

$$\begin{aligned} ^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }\left( H-H^*-H^* \ln \frac{H}{H^*}\right) \le \left( 1-\frac{H^*}{H}\right) {}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho } H(t),~~~~~H^{*} \in \mathbb {R}^{+}, \forall \sigma , \rho \in (0,1). \end{aligned}$$
(42)

Theorem 6.2

If the \(R_0>1\), the endemic equilibrium \(P_2\) of system (6) is globally asymptotically stable; otherwise, they are unstable.

Proof

Initially, we establish the Lyapunov function:

$$\begin{aligned} W(S_h^{*}, E_h^{*},I_e^{*},I_{ne}^*, R^{*}, S_m^{*}, E_m^{*}, I_m^{*})& =(S_h-S_h^*-S_h^* \log \frac{S_h}{S_h^*})+(E_h-E_h^*-E_h^* \log \frac{E_h}{E^*_h})+(I_e-I_e^*-I_e^* \log \frac{I_e}{I^*_e})\nonumber \\& \quad {}+(I_{ne}-I_{ne}^*-I_{ne}^* \log \frac{I_{ne}}{I^*_{ne}}) +(R-R^*-R^* \log \frac{R}{R^*})+(S_m-S_m^*-S_m^* \log \frac{S_m}{S_m^*})\nonumber \\& \quad{}+(E_m-E_m^*-E_m^* \log \frac{E_m}{E^*_m})+(I_m-I_m^*-I_m^* \log \frac{I_m}{I^*_m}). \end{aligned}$$
(43)

We have

$$\begin{aligned} {}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }W&\le (\frac{S_h-S_h^*}{S_h}){}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }S_h(t)+(\frac{E_h-E_h^*}{E_h}){}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }E_h(t)+(\frac{I_e-I^*_e}{I_e}){}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }I_e(t)\nonumber \\&{} \quad +(\frac{I_{ne}-I^*_{ne}}{I_{ne}}){}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }I_{ne}(t)+(\frac{R-R^*}{R}){}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }R(t)+(\frac{S_m-S_m^*}{S_m}){}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }S_m(t)\nonumber \\& \quad{}+(\frac{E_m-E_m^*}{E_m}){}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }E_m(t)+(\frac{I_m-I^*_m}{I_m}){}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }I_m(t). \end{aligned}$$
(44)

We get

$$\begin{aligned} {}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }W&\le (\frac{S_h-S_h^*}{S_h})\Big \{\Lambda +\Phi -\frac{(\beta _{h_1}+\beta _{h_2 } ) S_h I_m}{N_h} +q R-\mu _hS_h\Big \}\nonumber \\& \quad+(\frac{E_h-E_h^*}{E_h}) \Big \{\frac{(\beta _{h_1}+\beta _{h_2} ) S_h I_m}{N_h}-(\alpha _1+\alpha _2 ) E_h-\mu _h E_h\Big \}\nonumber \\& \quad+(\frac{I_e-I^*_e}{I_e})\Big \{\alpha _1 E_h-\gamma _1 I_e-\mu _h I_e\Big \}\nonumber \\& \quad+(\frac{I_{ne}-I^*_{ne}}{I_{ne}})\Big \{\alpha _2 E_h-\gamma _2 I_{ne}-\mu _h I_{ne}\Big \}\nonumber \\& \quad+(\frac{R-R^*}{R})\Big \{\gamma _1 I_e+\gamma _2 I_{ne}-q R-\mu _h R\Big \}\nonumber \\& \quad+(\frac{S_m-S_m^*}{S_m})\Big \{\Theta N_m (1-\delta )(1-\frac{N_m}{p})-\frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m}-\mu _m S_m\Big \}\nonumber \\& \quad+(\frac{E_m-E_m^*}{E_m})\Big \{\frac{(\beta _{m_1}+\beta _{m_2} )(I_{e}+I_{ne} ) S_m}{N_m} -\alpha _3 E_m-\mu _m E_m\Big \}\nonumber \\& \quad+(\frac{I_m-I^*_m}{I_m})\Big \{\alpha _3 E_m-\mu _m I_m\Big \}. \end{aligned}$$
(45)

Now, using \(S_h=S_h-S_h^*\), \(E_h=E_h-E_h^*\), \(I_e=I_e-I_e^*\), \(I_{ne}=I_{ne}-I_{ne}^*\), \(R=R-R^*\), \(S_m=S_m-S_m^*\), \(E_m=E_m-E_m^*\), and \(I_m=I_m-I_m^*\). After rearranging and simplification the equality above can be written as

$$\begin{aligned} {}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }W\le \eta _1-\eta _2, \end{aligned}$$
(46)

where

$$\begin{aligned} \eta _1&= \Lambda +\Phi +q (R-R^*)+\frac{(\beta _{h_1}+\beta _{h_2} ) (S_h-S_h^*) (I_m-I_m^*)}{(N_h-N_h^*)}+\alpha _1 (E_h-E_h^*)+\alpha _2 (E_h-E_h^*)\nonumber \\&~~~~~+\gamma _1 (I_e-I_e^*)+\gamma _2 (I_{ne}-I_{ne}^*) +\Theta (N_m-N_m^*) (1-\delta )(1-\frac{(N_m-N_m^*)}{p})\nonumber \\&~~~~~+\frac{(\beta _{m_1}+\beta _{m_2} )\big ((I_e-I_e^*)+(I_{ne}-I_{ne}^*) \big ) (S_m-S_m^*)}{(N_m-N_m^*)} +\alpha _3(E_m-E_m^*), \end{aligned}$$
(47)

and

$$\begin{aligned} \eta _2&=(\Lambda +\Phi )\frac{S_h^*}{S_h}+\frac{(\beta _{h_1}+\beta _{h_2 } )(S_h-S_h^*)^2 (I_m-I_m^*)}{S_h(N_h-N_h^*)}+q (R-R^*)\frac{S_h^*}{S_h}+\mu _h \frac{(S_h-S_h^*)^2}{S_h}\nonumber \\&~~~ +(\alpha _1+\alpha _2+\mu _h )\frac{(E_h-E_h^*)^2}{E_h}+\frac{(\beta _{h_1}+\beta _{h_2} ) (S_h-S_h^*) (I_m-I_m^*)E_h^*}{E_h(N_h-N_h^*)} +\alpha _1 (E_h-E_h^*)\frac{I_e^*}{I_e}\nonumber \\&~~~ +(\gamma _1+\mu _h)\frac{(I_e-I_e^*)^2}{I_e}+\alpha _2 (E_h-E_h^*)\frac{I_{ne}^*}{I_{ne}}+(\gamma _2+\mu _h)\frac{(I_{ne}-I_{ne}^*)^2}{I_{ne}} +\gamma _1 (I_e-I_e^*)\frac{R^*}{R}+\gamma _2 (I_{ne}-I_{ne}^*)\frac{R^*}{R}\nonumber \\&~~~+\Theta (N_m-N_m^*) (1-\delta )\frac{S_m^*}{S_m}(1-\frac{(N_m-N_m^*)}{p})+\frac{(\beta _{m_1}+\beta _{m_2} )\big ((I_e-I_e^*)+(I_{ne}-I_{ne}^*) \big ) (S_m-S_m^*)^2}{S_m(N_m-N_m^*)}\nonumber \\&~~~+\mu _m \frac{(S_m-S_m^*)^2}{S_m}+(\alpha _3+\mu _m)\frac{(E_m-E_m^*)^2}{E_m}+\frac{(\beta _{m_1}+\beta _{m_2} )\big ((I_e-I_e^*)+(I_{ne}-I_{ne}^*) \big )E^*_m (S_m-S_m^*)}{E_m(N_m-N_m^*)} \nonumber \\&\quad +\alpha _3\frac{I-m^*}{I_m}(E_m-E_m^*)+\mu _m \frac{(I_m-I_m^*)^2}{I_m}. \end{aligned}$$
(48)

It is evident that in the case where \({\eta _1<\eta _2}\),

$$\begin{aligned}{}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }W<0.\end{aligned}$$

But, if \(S_h=S_h^*\), \(E_h=E_h^*\), \(I_e=I_e^*\), \(I_{ne}=I_{ne}^*\), \(R=R^*\), \(S_m=S_m^*\), \(E_m=E_m^*\), and \(I_m=I_m^*\), then

$$\begin{aligned}{}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }W=0.\end{aligned}$$

It is our understanding that the largest compact invariant set for the proposed model (6) in

$$\begin{aligned} \{(S_h^{*},E_h^{*},I_e^{*},I_{ne}^*, R^{*}, S_m^{*}, E_m^{*}, I_m^{*})\in \Upsilon , {}^{F F M}_0 \textrm{D}_{t}^{\sigma , \rho }W=0\},\end{aligned}$$

is the endemic equilibrium \(D_2(S^*,E^*,I^*,Q^*,R^*)\) of the system (6). In the event that \({\eta _1<\eta _2}\), we can conclude that \(P_2(S_h^{*},E_h^{*},I_e^{*},I_{ne}^*, R^{*}, S_m^{*}, E_m^{*}, I_m^{*})\) is globally asymptotically stable in \(\Upsilon\). \(\square\)

Numerical solutions

The work proposes a numerical scheme for the suggested model (6) that used the fractal-fractional operator with the Mittag-Leffler kernel and procedure for evolution is shown in Fig. 6.

$$\begin{aligned} {\left\{ \begin{array}{ll} {}^{FFM}_{0}D^{\sigma ,\rho }_{t}(S_h(t))~=~\chi _1\big (t,\omega (t)\big ),\\ {}^{FFM}_{0}D^{\sigma ,\rho }_{t}(E_h(t))~=~\chi _2\big (t,\omega (t)\big ),\\ {}^{FFM}_{0}D^{\sigma ,\rho }_{t}(I_e(t))~=~\chi _3\big (t,\omega (t)\big ),\\ {}^{FFM}_{0}D^{\sigma ,\rho }_{t}(I_{ne}(t))~=~\chi _4\big (t,\omega (t)\big ),\\ {}^{FFM}_{0}D^{\sigma ,\rho }_{t}(R(t))~~=~\chi _5\big (t,\omega (t)\big ),\\ {}^{FFM}_{0}D^{\sigma ,\rho }_{t}(S_m(t))~=~\chi _6\big (t,\omega (t)\big ),\\ {}^{FFM}_{0}D^{\sigma ,\rho }_{t}(E_m(t))~=~\chi _7\big (t,\omega (t)\big ),\\ {}^{FFM}_{0}D^{\sigma ,\rho }_{t}(I_m(t))~=~\chi _8\big (t,\omega (t)\big ), \end{array}\right. } \end{aligned}$$
(49)

where \(\omega =(S_h,E_h,I_e,I_{ne},R,S_m,E_m,I_m)\). Let the abstract description of (49) be

$$\begin{aligned} {\left\{ \begin{array}{ll} {}^{FFM}_{0}D^{\sigma ,\rho }_{t} \omega (t)=\chi (t,\omega (t)),~~~~~~~~~\sigma ,\rho \in (0,1],~~~t\in [0,T],\\ ~~~~~~~~~~~~~~~~~~~\omega (0)=\omega _{0}, \end{array}\right. } \end{aligned}$$
(50)

where \(\chi =(\chi _1,\chi _2,\chi _3,\chi _4,\chi _5,\chi _6,\chi _7,\chi _8)\).

Following from integral (3), we have the Voltera formula:

$$\begin{aligned} \omega \left( t_{\kappa +1}\right) = \omega (0)+\frac{1-\sigma }{\tilde{AB}(\sigma )} t_\kappa ^{1-\rho } \chi \left( t_\kappa , \omega \left( t_\kappa \right) \right) +\tilde{AB} \sum _{r=2}^\kappa \int _{t_0}^{t_0+1} \chi \left( \upsilon , \omega (\upsilon )\right) \upsilon ^{1-\rho }\left( t_{\kappa +1}-\upsilon \right) ^{\sigma -1} d \upsilon . \end{aligned}$$
(51)

Since, Newton polynomial can be written as:

$$\begin{aligned} \begin{aligned} \omega \left( t_{\kappa +1}\right)&=\omega (0) +\frac{1-\sigma }{\tilde{AB}(\sigma )} t_\kappa ^{1-\rho } \chi \left( \textrm{t}_\kappa , \omega (t_\kappa )\right) +\tilde{AB} \sum _{r=2}^\kappa \chi \left( \textrm{t}_\kappa , \omega (t_\kappa )\right) t_{\kappa -2}^{1-\rho }\int _{t_0}^{t_0+1}\left( t_{\kappa +1}-\upsilon \right) ^{\sigma -1} d \upsilon \\&\quad +\tilde{AB}\sum _{r=2}^\kappa \frac{1}{\Delta t}\left[ t_{\kappa -1}^{1-\rho } \chi \left( \textrm{t}_{\kappa -1}, \omega ^{\kappa -1}(t)\right) \right. \left. -t_{\kappa -1}^{1-\rho }\chi \left( \textrm{t}_{\kappa -2}, \omega ^{\kappa -2}(t)\right) \right] \\ &\quad \times \int _{t_0}^{t_0+1}\left( \upsilon -t_{\kappa -2}\right) \left( t_{\kappa +1}-\upsilon \right) ^{\alpha -1} d \upsilon \\&\quad +\frac{\sigma }{\tilde{AB}(\sigma ) \Gamma (\sigma )} \sum _{r=2}^\kappa \frac{1}{\Delta t^2}\left[ t_0^{1-\rho } \chi \left( \textrm{t}_{r}, {\omega }^{r}(t)\right) \right. \left. -2 t_{\kappa -1}^{1-\rho } \chi \left( \textrm{t}_{\kappa -1}, \omega ^{\kappa -1}(t)\right) +t_{\kappa -2}^{1-\rho } \chi \left( \textrm{t}_{\kappa -2}, \omega ^{\kappa -2}(t)\right) \right] \\&\quad \times \int _{t_0}^{t_0+1}\left( \upsilon -t_{\kappa -2}\right) \left( \upsilon -t_{\kappa -1}\right) \left( t_{\kappa +1}-\upsilon \right) ^{\sigma -1} d \upsilon , \end{aligned} \end{aligned}$$
(52)

After some calculations, we obtain

$$\begin{aligned} \begin{aligned} \omega (\textrm{t}_{\kappa +1})&=\omega (0)+\frac{1-\sigma }{\tilde{AB}(\sigma )} \textrm{t}_\kappa ^{1-\rho } \chi (\textrm{t}_\kappa , \omega (t_\kappa ))\\&\quad {}+\frac{\sigma (\Delta \textrm{t})^\sigma }{\tilde{AB}(\sigma ) \Gamma (\sigma +1)} \sum _{r=2}^\kappa \chi (\textrm{t}_\kappa , \omega (t_\kappa )) \textrm{t}_{\mathrm {r-2}}^{1-\rho }[(\kappa -r+1)^\sigma -(\kappa -r)^\sigma ] \\&\quad +\frac{\sigma (\Delta \textrm{t})^\sigma }{\tilde{AB}(\sigma ) \Gamma (\sigma +2)} \sum _{r=2}^\kappa \frac{1}{\Delta \textrm{t}} \Big \{\textrm{t}_{r-1}^{1-\rho } \chi (\textrm{t}_{\kappa -1}, \omega ^{\kappa -1}(t)) -\textrm{t}_{r-2}^{1-\rho } \chi (\textrm{t}_{\kappa -2}, \omega ^{\kappa -2}(t))\Big \}\\&\quad \times [(\kappa -r+1)^\sigma (\kappa -r+3+2 \sigma )-(\kappa -r)^\sigma (\kappa -r+3+3 \sigma )]\\&\quad +\frac{\sigma (\Delta \textrm{t})^\sigma }{\tilde{AB}(\sigma ) \Gamma (\sigma +3)} \sum _{r=2}^\kappa \frac{1}{2 \Delta \textrm{t}^2}\Big \{\textrm{t}_{r}^{1-\rho } \chi (\textrm{t}_{r}, {\omega }^{r}(t))-2 \textrm{t}_{r-1}^{1-\beta } \chi (\textrm{t}_{\kappa -1}, \omega ^{\kappa -1}(t)) +\textrm{t}_{r-2}^{1-\rho } \chi (\textrm{t}_{\kappa -2}, \omega ^{\kappa -2}(t))\Big \}\\&\quad \times [(\kappa -r+1)^\sigma \{2(\kappa -r)^2+(3 \sigma +10)(\kappa -r)+2 \sigma ^2+9\sigma +12\}\\&\quad -(\kappa -r)^\sigma \{2(\kappa -r)^2+(5 \sigma +10)(\kappa -0)+6 \sigma ^2+18 \sigma +12\}]. \end{aligned} \end{aligned}$$
(53)
Fig. 6
Fig. 6
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The flowchart of proposed method.

Simulation results

From the Figs. 1 and 2, it can be concluded that the number of malaria cases are in rise since 2020. Moreover, the vast majority of the obtained cases are imported malaria cases which came from the countries that has malaria as an endemic disease. A statistical study was conducted in China, in 2020, to indicate the importance of imported malaria cases and to emphasize the recent increase of cases in China. From the Figs. 2 and 3, it is obvious that most of the cases obtained in autumn months, especially in October. Human and mosquito population density, resource availability, infrastructure, and growth potential all have a major impact on the dynamics of malaria transmission in human society. The work uses numerical simulations under various fractional orders and fractal dimensions to show the malaria model’s accuracy and viability. The two-order fractal-fractional form operator is more helpful for data and practical difficulties, but the fractal-fractional malaria model is essential for understanding epidemiological model. Given the following initial conditions and parameter values listed in Table 1, the fractal-fractional malaria model is computed by numerical simulations.

$$\begin{aligned} & N_h(0)=50461,~~~~~~~N_m(0)=10478,~~~~~~~S_h(0)=50000,~~~~~E_h(0)=305,~~~~~I_e(0)=73,\\ & R(0)=78, ~~~~~S_m(0)=10000,~~~~~E_m(0)=400,~~~~~I_m(0)=78,~~~~~I_{ne}(0)=5.\end{aligned}$$

To understand how parameters affect the results of state variables associated with malaria, we ran several simulations.

  • Figures 7, 8, 9, 10, 11, 12, 13 and 14 display our model’s graphical solution for all compartments at different fractional orders \((\sigma )\) and fractal dimensions \((\rho )\). The dynamics of human populations and mosquito vectors against different fractional orders are comprehensively illustrated in the paper.

  • The fractal dimension of 1.0 is used to simulate Figs. 7a, 8, 9, 10, 11, 12, 13 and 14a, whereas Figs. 7b, 8, 9, 10, 11, 12, 13 and 14b are simulated against fractal dimension of 0.9.

  • Figure 7 shows the mathematical simulation for susceptible human populations versus different fractional orders. while the rate of infection declines, the susceptible human population first rises and then falls at higher values of \(\sigma\), suggesting that while the rate of infection declines, the number of susceptible individuals naturally rises.

  • Figures 8, 9 and 10 illustrate how the infection against greater fractional orders is spreading concurrently because susceptible persons are spreading the infection to exposed and affected classes.

  • Figure 11 illustrates the fast expanding recovered class in comparison to larger fractional orders that indicate the infected individuals’ recovery rate.

  • The graphs in Figs. 12, 13 and 14 show how mosquitoes grow into three different classes, each of which shows a positive relationship with fractional order. At higher values of \(\sigma\), the population of mosquito vectors increases quickly in all classes.

The fractal-fractional order model offers a wide range of calibrations for varying levels of infectivity in populations of mosquitoes and humans. The study discovered that an increase in oscillations causes the memory effects of infected vectors and humans to rise as the derivative order \((\sigma )\) approaches 1. Furthermore, an apparent change in the dynamics of different compartments occurs with a decrease in the fractal dimension \((\rho )\). The figures demonstrate how, within the constrained domain, the solution approaches a steady state as the value of \(\sigma\) declines. To get to this point, each graph travels a non-linear path, emphasizing the significance of fractional order. Greater \(\sigma\) values correspond to a faster rate of convergence. The fractal dimension \(\rho\)’s self-similar pattern and declining amplitudes suggest that the malaria system’s paths are moving in the direction of stability. Researchers can find trends and patterns that can direct population management and sustainable development strategies by analyzing the impact of different parametric values on population density.

Fig. 7
Fig. 7
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Dynamics and forecasting \(S_h(t)\) at higher quartile coefficient values index.

Fig. 8
Fig. 8
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Dynamics and forecasting \(E_h(t)\) at higher quartile coefficient values index.

Fig. 9
Fig. 9
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Dynamics and forecasting \(I_e(t)\) at higher quartile coefficient values index.

Fig. 10
Fig. 10
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Dynamics and forecasting \(I_{ne}(t)\) at higher quartile coefficient values index.

Fig. 11
Fig. 11
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Dynamics and forecasting R(t) at higher quartile coefficient values index.

Fig. 12
Fig. 12
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Dynamics and forecasting \(S_m(t)\) at higher quartile coefficient values index.

Fig. 13
Fig. 13
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Dynamics and forecasting \(E_m(t)\) at higher quartile coefficient values index.

Fig. 14
Fig. 14
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Dynamics and forecasting \(I_m(t)\) at higher quartile coefficient values index.

Conclusion

To better understand malaria transmission dynamics in Northern Cyprus, researchers used a deterministic eight compartmental mathematical model. The model reveals distinct social dynamics and an extended incubation period, with reproductive number serving as the primary metric. To better understand the model’s behavior under constrained conditions, sensitivity and stability analyses were performed. Fractal geometry and fractional calculus were employed to improve forecasting accuracy and comprehension of population dynamics. The study employed a Newton polynomial numerical approach to obtain solutions and runs simulations to examine the global impact of parameters on malaria virus symptomatic and asymptomatic rates. Fractal geometry and fractional calculus were employed to improve forecasting accuracy and comprehension of population dynamics. Human and mosquito growth rates are proportional to the fractional order \((\sigma )\), with the memory effect increasing as the derivative order decreases from 1. This research is critical for comprehending viral transmission dynamics and developing effective control measures. The study underscores the significance of control strategies for imported malaria cases during entry into a country, suggesting that policymakers should implement awareness-raising measures, test applications during entry, and vaccination programs to combat the endemic of malaria. The uncertainty of Cyprus’ epidemiological status, particularly in the north, makes it difficult to establish a baseline and reliable historical data for a malaria model. Most cases are imported malaria cases from endemic countries, with diagnosis dates typically falling in October, coinciding with university openings. The model’s accuracy is based on precise, region-specific data on human, vector, and disease prevalence, which may be scarce in Northern Cyprus. Complex models are required to account for unique factors such as tourism-induced vector introduction and diverse immune status, but overcomplicated models can be overly parameterized and difficult to analyze. The model must account for local conditions such as climate, water bodies, and human behavior, which can be difficult to generalize from models developed for other regions. Limited or noisy data can make it difficult to identify specific parameter values in models, affecting both analysis and prediction. To overcome over-parameterization and improve parameter estimation, model reduction techniques may be required, which simplify complex models while preserving essential transmission dynamics. To improve this model, localized data such as climate, mosquito density, and treatment-seeking behaviors should be included. Complex epidemiological factors such as spatial heterogeneity, treatment efficacy, and traditional medicine should be addressed. Future research should employ numerical optimization approaches and algorithms, such as non-linear least squares curve fitting or machine learning methods, to reduce the discrepancy between anticipated epidemic curves and real-world data while also estimating unknown parameters and fractional orders. Experimenting with other fractional derivatives and numerical methods can help identify transmission drivers and assess intervention strategies.