Modeling the impact of climate change on dengue transmission dynamics in Dire Dawa, Afar, and Somali, Ethiopia: an African regional perspective

Abstract

This study investigates the impacts of climate variables on dengue transmission dynamics using an advanced deterministic model that incorporates climate-related entomological parameters. Key findings indicate that temperature and rainfall significantly influence dengue transmission by affecting mosquito behavior and life cycles. Human behavior plays a key role in influencing the severity and spread of dengue amid climate change. To effectively address climate-related impacts on dengue, strategies must go beyond technological and environmental solutions and also focus on modifying behaviors related to urbanization, vector control, hygiene, and health-seeking practices. Our sensitivity analysis highlights vital parameters such as the biting rate, where a mere 1-unit increase can elevate the basic reproduction number ($R_0$) by one unit. Similarly, transmission probabilities ($\alpha$ and $\beta$) and the mosquito birth rate collectively contribute to approximately 50% of disease amplification. Interestingly, a 1.4-fold increase in mosquito mortality rate ($\mu _m$) significantly reduces $R_0$, underscoring the potential of targeted mortality interventions. Furthermore, we examined all mosquito related climate based parameters that enable us to identify the region that more affected and regions exhibit varying environmental conditions and outcomes, with Somali showing the most favorable conditions for dengue transmission. Among the strategies evaluated, the combined use of pesticide-treated bed nets and prompt treatment of infected individuals proved to be both highly effective and cost-efficient. Our findings emphasize that managing mosquito populations through environmental controls such as enhanced drainage systems and waste management is paramount in mitigating dengue outbreaks. This comprehensive approach offers a promising pathway to safeguard vulnerable communities against the growing threat of climate-influenced dengue transmission.

Introduction

Dengue represents a significant public health challenge, caused by an arthropod vector-borne virus that is transmitted to humans through the bites of infected female mosquitoes of the Aedes species, primarily Aedes aegypti and Aedes albopictus¹. These mosquito species can transmit one of four related dengue virus serotypes, complicating the disease further. Additionally, climate change plays a substantial role in exacerbating this illness^1,2,3.

The transmission of dengue is influenced by climatic conditions and is positively correlated with increased temperature and rainfall^4,5. Such environmental factors determine mosquito breeding sites and the success of their metabolic cycles. Favorable climatic conditions lead to a substantial increase in mosquito populations, thereby heightening the likelihood of human-mosquito interactions. Moreover, temperature significantly impacts the population size, maturation period, blood-feeding activity, and survival rate of Aedes aegypti^4,6. An increase of $0.5^o\hbox {C}$ can potentially boost the mosquito population³. Furthermore, a temperature shift from $12^o\hbox {C}$ to $31^o\hbox {C}$ reduces the time required for mosquito breeding from approximately two months to one week, with pathogen development within mosquitoes taking 55 days at $16^o\hbox {C}$ and decreasing to one week at $28^o\hbox {C}$⁷. Similarly, rainfall influences the abundance of Aedes aegypti and promotes egg hatching, which, in turn, increases the density of this vector species in specific areas through elevated temperatures⁸.

As of early 2024, around 7.6 million cases and 8,021 fatalities were reported in 75 countries^9,10. The disease is most prevalent in tropical and subtropical regions, where the climate supports the development of mosquito populations. This disease also spreads through different African countries that share borders and climate conditions conducive to its proliferation. For instance, even though there is insufficient health surveillance, some regions of Ethiopia, particularly Dire Dawa, Afar, and Somali, are potentially affected by dengue disease since its emergence^11,12,13,14. Since its emergence in 2013, the transmission dynamics of dengue have surged unexpectedly, coinciding with malaria outbreaks, particularly concerning during the rainy season^14,15. In Ethiopia, from January 1, 2023, to January 1, 2025, over 29,494 dengue cases and 21 deaths were reported. Of these cases, 29,028 were reported from the Afar, Dire Dawa, and Somali regions¹⁶. This problem motivates our work in Ethiopia, particularly in regions at risk for dengue. To examine the impacts that enhance the transmission of this disease, mathematical modeling plays a crucial role in generating useful information for disease prevention.

To this end, several mathematical models have been published regarding dengue transmission dynamics, primarily based on the simple SIR model^2,17. Many of these studies have advanced this basic model by employing more sophisticated approaches related to dengue outbreaks, thereby achieving results that assist healthcare in preventing the disease. For example, research has been conducted on the global stability and endemic equilibrium of dengue to explore the stability behavior at equilibrium¹⁷. Another study focused on a single dengue serotype using basic reproduction number analysis to predict disease dynamics². Additionally, a one-strain model was developed and its stability behavior at equilibrium points was analyzed using the basic reproduction number¹⁸. Moreover, the effects of vector control on dengue virus transmission in the context of climate change were also investigated, revealing that effective mosquito management plays a pivotal role in mitigating the disease¹⁹. Another authors²⁰ also explored temperature based dengue modeling to analyze climate factor slightly and achieve a better results.

Despite the numerous studies conducted on modeling the spread of the dengue virus to aid preventive health measures, there remains a lack of comprehensive models that examine the impact of climate change on dengue transmission dynamics. To address this gap, we investigated the effects of seasonal factors based on regional climate change particularly Dire Dawa, Afar, and Somali region in Ethiopia, specifically rainfall and temperature, on the risk of dengue outbreaks. Additionally, we analyzed a model that excludes climate-based parameters and examined its stability characteristics. Ultimately, our model provides a numerical foundation for further exploration and prediction of climate change’s impact on dengue transmission dynamics.

The paper is organized as follows: In Section (3), model description, assumption and formulation are given; In section (4), model analysis without climate change is performed; In section (5), model analysis with climate variation and climate-based parameters are discussed; In section (6) present model validation via existing data; In section (7), Optimal control and cost-effectiveness are analyzed; In section (8), numerical solution and simulation are implemented. Lastly, results, discussion and conclusion are given in section (9), (10), and (11) respectively.

Model description and formulation

The model accounts for both human and mosquito populations, wherein the human population is categorized into susceptible ($S_h$), exposed ($E_h$), infected ($I_h$), and recovered ($R_h$) classes. In contrast, the mosquito population is structured into susceptible ($S_m$), exposed ($E_m$), and infected ($I_m$) classes. Due to the short lifespan of mosquito populations, the recovery state is not included within the mosquito categories.

Moreover, susceptible human ($S_h$) may contract dengue when bitten by the infected class of Aedes mosquitoes (denoted as $I_m$), subsequently transitioning to the exposed class ($E_h$). After an average latency period of 7 days (ranging from 4 to 10 days)¹, individuals in this class progress to the infectious class (referred to as $\delta E_h$). Those who develop immunity and survive the disease move into the recovered class (denoted as $\phi I_h$), while others succumb to the dengue-induced infection (denoted as $dI_h$).

Likewise, the population of susceptible mosquitoes increases at the birth rate denoted as $\theta (T, R_a)$ represents the rate at which immature mosquitoes develop into adults in the $S_m$ class). These mosquitoes can move to the exposed class upon biting an infected person (designated as $I_h$). Conversely, exposed mosquitoes progress to the infectious class (labeled as $\rho (T)E_m$); once infected, it is assumed that they live with the infection for the remainder of their lifetime. Additionally, we assume that the mosquito mortality rate $\mu _m(T)$ and the biting rate b(T) are temperature-dependent.

Furthermore, the assumptions of the model include: all parameters being non-negative, uniform natural death rates across all population subcategories, consideration of the Aedes aegypti species of mosquito, and initial conditions wherein all humans and female Aedes mosquitoes are susceptible to dengue. The transmission of the dengue virus occurs exclusively through direct contact between humans and female Aedes mosquitoes.

Based on the descriptions and assumptions of the model, it is illustrated diagrammatically in Fig. 1. Subsequently, a system of nonlinear differential equations is formulated to represent dengue transmission and analyze its properties. Moreover, the interpretation and description of the parameters are provided in Table 1. Below is the diagrammatic scheme of the mathematical model of dengue transmission dynamics under the influence of climate change:

Fig. 1

Diagrammatical representation of dengue transmission dynamics and impact of climate on aedes mosquito.

Table 1 Description of parameters and their values.

Based on the above assumptions, dengue virus transmission model which represented in Fig. 1 can be written as

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{dS_{h}}{dt}& = \Lambda _{h}-fS_{h}(t)-\mu _h S_{h}(t)\\ \frac{dE_{h}}{dt}& =fS_{h}(t)-(\mu _h +\delta )E_{h}(t)\\ \frac{dI_{h}}{dt}& =\delta E_{h}(t)-(d+\phi +\mu _h) I_{h}(t)\\ \frac{dR_h}{dt}& =\phi I_{h}(t)-\mu _h R_h(t)\\ \frac{dS_{m}}{dt}& = \theta (T, R_a)-gS_{m}(t)-\mu _{m}(T) S_{m}(t)\\ \frac{dE_{m}}{dt}& =gS_{m}(t)-(\mu _{m} +\rho )(T) E_{m}(t)\\ \frac{dI_{m}}{dt}& =\rho (T) E_{m}(t) -\mu _{m}(T) I_{m}(t) \end{array}\right. \end{aligned}$$

(1)

where $f=\frac{\alpha (T) b(T)I_{m}(t)}{N_{h}}$ is dengue infection force from infected Aedes female mosquito to susceptible human $S_{h}$ and $g =\frac{\beta (T) b(T)I_{h}(t)}{N_{h}}$ is transmission force from infected human to susceptible mosquito with initial conditions of the system (1) of state variables, $S_{h}(0)> 0, E_{h}(0)\ge 0, I_{h}(0)\ge 0, R_{h}(0) \ge 0, S_{m}(0)> 0, E_{m}(0) \ge 0, I_{m}(0) \ge 0$. In addition to this, $\theta (T, R)$ stands for mosquito’s birth rate that mainly determined by temperature and rainfall.

Model analysis

In this section, the fundamental model analysis such as invariant region of the solution, positivity and the basic reproduction number of model (1) are discussed.

Invariant region

To determine a region in which the solution of system (1) is bounded, one can see the following theorem.

Theorem 1

The solutions of system of model (1) are contained in the feasible region $\Omega$ where

$\Omega =\Omega _{h}\times \Omega _{m}$ such that

$\Omega _{h}=\{(S_{h}(t), E_{h}(t), I_{h}(t), R_h(t))\in R^{4}_{+}: S_{h}(t)+ E_{h}(t)+ I_{h}(t)+ R_h(t)\le \frac{\Lambda _{h}}{\mu _h}\}$ and

$\Omega _{m}=\{(S_{m}(t), E_{m}(t), I_{m}(t))\in R^{3}_{+}: S_{m}(t)+ E_{m}(t)+ I_{m}(t)\le \frac{\theta (T, R_a)}{\mu _{m}}\}$ is positively bounded.

Proof

From total number of human we have,

$$\begin{aligned} N_{h}(t)= S_{h}(t)+ E_{h}(t)+ I_{h}(t)+ R_h(t). \end{aligned}$$

(2)

Then, differentiating Eq. (2) with respect to time we obtain

$$\begin{aligned} \frac{dN_{h}}{dt}=\frac{dS_{h}}{dt}+\frac{dE_{h}}{dt}+\frac{dI_{h}}{dt}+\frac{dR_{h}}{dt}. \end{aligned}$$

(3)

Substituting (1) into (3) we get,

$$\begin{aligned} \frac{dN_{h}}{dt}= \Lambda _{h} -\mu _h (S_{h}(t) + E_{h}(t) + I_{h}(t) + R_h(t))- d I_{h}(t). \end{aligned}$$

(4)

Substituting Eq. (8) into (4) we get,

$$\begin{aligned} \frac{dN_h}{dt}= \Lambda _{h} -\mu _h N_{h}-d I_{h}. \end{aligned}$$

(5)

In the absence of mortality due to dengue fever disease ($d = 0$), so that Eq. (5) or Without losing the general rule of inequality, Eq. (5) become,

$$\begin{aligned} \frac{dN_{h}}{dt} \le \Lambda _{h} -\mu _h N_{h}. \end{aligned}$$

(6)

Solving the inequality of (6) by applying integral and applying the initial condition $N_{h}(0) = N_{0}$ we get, $N_h(t)\le \frac{\Lambda _{h}}{\mu _h}+\frac{N_{0}}{\mu _h}e^{-t}$, as $t\rightarrow \infty$, $N_{h}(t)\le \frac{\Lambda _{h}}{\mu _h}.$

Thus, all feasible solutions of human population of the system (1) are contained in the domain $\Omega _{h}$.

$$\begin{aligned} \Omega _{h} = \left\{ (S_{h}, E_{h}, I_{h}, R_h)\in R^{4}_{+}:0< N_{h}(t)\le \frac{\Lambda _{h}}{\mu _h}\right\} . \end{aligned}$$

(7)

In the same fashion, let the number of total mosquito is given by:

$$\begin{aligned} N_{m}(t)= S_{m}(t)+ E_{m}(t)+ I_{m}(t). \end{aligned}$$

(8)

Then, differentiating equation of (8) with respect to time we obtain:

$$\begin{aligned} \frac{dN_{m}}{dt}=\frac{dS_{m}}{dt}+\frac{dE_{m}}{dt}+\frac{dI_{m}}{dt}. \end{aligned}$$

(9)

Substituting (1) into (9) we get,

$$\begin{aligned} \frac{dN_{m}}{dt}= \theta (T, R_a) -\mu _{m} (S_{m}(t) + E_{m}(t) + I_{m}(t). \end{aligned}$$

(10)

Replacing equation of (8) into (10) we obtain,

$$\begin{aligned} \frac{dN_{m}}{dt}= \theta (T, R_a) -\mu _{m} N_{m}. \end{aligned}$$

(11)

Without losing the general rule of inequality, Eq. (11) become,

$$\begin{aligned} \frac{dN_{m}}{dt} \le \theta (T, R_a) -\mu _{m} N_{m}. \end{aligned}$$

(12)

Solving the inequality of (12) by applying integral and applying the initial condition $N_{m}(0) = N_{0m}$ we get,

$$\begin{aligned} N_{m}(t)\le \frac{\theta (T, R_a)}{\mu _{m}}+\frac{N_{0m}}{\mu _{m}}e^{-t},\hbox { as } t\rightarrow \infty , N_{m}(t)\le \frac{\theta (T, R_a)}{\mu _{m}}. \end{aligned}$$

Here, since $\theta (T, R_a)$ function built using combination of continuous function, it is continuous. Furthermore, mosquito recruitment increase with temperature up to a point or decrease below threshold. Similarly, rainfall increase mosquito breeding site up to a point. Mathematically, $\frac{\partial {\theta (T, R_a)}}{\partial {T}} \ge 0$ for $T \in (25, 30)^o C$ otherwise it is decreasing. Likewise, $\frac{\partial {\theta (T, R_a)}}{\partial {R_a}} \ge 0$ for $R_a \in (0, 50) mm$ otherwise it is decreasing. This implies $\theta (T, R_a)$ is monotonic function. A function $\theta (T, R_a)$ is also bounded due to density-dependent mortality and recruitment terms is positively bounded.

Thus, the feasible solution set of mosquito dynamics remain in the region $\Omega _{m}$.

$$\begin{aligned} \Omega _{m} = \left\{ (S_{m}, E_{m}, I_{m})\in R^{3}_{+}:0< N_{m}(t)\le \frac{\theta (T, R_a)}{\mu _{m}}\right\} . \end{aligned}$$

(13)

Combining Eq. (7) and (13) we get,

$$\begin{aligned} \Omega&=\Omega _{m}\times \Omega _{h} \nonumber \\&= \left\{ (S_{h}, E_{h}, I_{h}, R_h, S_{m}, E_{m}, I_{m})\in R^{7}_{+}:0< N_{h}(t)\le \frac{\Lambda _{h}}{\mu _h} ;0< N_{m}(t)\le \frac{\theta (T, R_a)}{\mu _m}\right\} . \end{aligned}$$

(14)

Therefore, the system of (1) is mathematically meaningful in the domain (14) and epidemiologically feasible. Hence, it is suffice to investigate the model (1) on the set of (14). $\square$

Positivity of the solutions

We assumed that the initial conditions of the model are nonnegative, and now we want to show, the solutions of the model are also positive for time $t>0$. To elaborate this, we consider the following theorem:

Theorem 2

Let $\Omega = \{(S_{h},E_{h}, I_{h}, R_{h}, S_{m}, E_{m}, I_{m})\:$$\in \mathbb {R}^{7}_{+}: S_{h}(0)> 0, E_{h}(0) \ge 0,I_{h}(0)\:$$\ge 0, R_{h}(0) \ge 0, S_{m}(0)>0,E_{m}(0)\ge 0,I_{m}(0) \ge 0\}$, then the solutions of $\{S_{h}(t), E_{h}(t),I_{h}(t), R_{h}(t), S_{m}(t), E_{m}(t), I_{m}(t) \}\in \mathbb {R}^{7}_{+}$ are positive for $t> 0$.

Proof

From the system of differential equation of (1), taking the first equation we get,

$$\begin{aligned} \frac{dS_{h}}{dt} = \Lambda _{h}-fS_{h}(t)-\mu _{h} S_{h}(t). \end{aligned}$$

(15)

Without loosing the general rule of inequality, Eq.(15) become, $\frac{dS}{dt} \ge -\mu _{h} S_{h}(t)$ (integrating both side we get,) $S_{h}(t)\ge S_{h}(0)e^{-\mu _{h} t}.$ But as $t\rightarrow \infty$, $S_{h}(0)e^{-\mu _{h} t}\rightarrow 0.$ Thus,

$$\begin{aligned} S_{h}(t)\ge 0 \end{aligned}$$

(16)

From second equation of system (1), we do have

$\frac{dE_{h}}{dt}=f S_{h}(t)-\delta E_{h}(t)-\mu _{h}E_{h}(t)$.

$\frac{dE_{h}}{dt} \ge -\mu _{h} E_{h}(t)$

(applying integral with respect to time t we get),

$E_{h}(t) \ge E_{h}(0)e^{-\mu _{h} t}$. But as $t\rightarrow \infty$, $E_{h}(0)e^{-\mu _{h} t}\rightarrow 0$. Thus,

$$\begin{aligned} E_{h}(t)\ge 0. \end{aligned}$$

(17)

Similarly, for third, fourth, fifth, sixth and seventh equations of system of (1), we employed similar procedure.

$$\begin{aligned} I_{h}(t)\ge 0,\; R_{h}(t)\ge 0, \; S_{m}(t)\ge 0,\; E_{m}(t)\ge 0,\; I_{m}(t)\ge 0. \end{aligned}$$

(18)

Therefore, for the given initial conditions holds true, all solutions of equations in model (1) are positive for $t>0$. $\square$

Disease free equilibrium point

The model system in (1) has a critical point in the absence of dengue disease (i.e. $E_{h}=I_{h}=E_{m}=I_{m}=0$). Hence, the disease-free equilibrium point (DFEP) is obtained by setting the right hand side of each equation of (1) equal to zero. Then one can solve for each state in the system. Therefore, the disease free equilibrium point is given by:

$$\begin{aligned} DFEP = \{\frac{\Lambda _{h}}{\mu _{h}}, 0, 0, 0, \frac{\theta }{\mu _{m}}, 0, 0\}. \end{aligned}$$

(19)

This implies that, at DFEP there is no dengue disease.

Basic reproduction number

In the process of computing $R_{0}$ for the system of (1), we start with newly infective classes such as $S_{h}, S_{m}, R_{h}$ and then followed by infected; $E_{h}, E_{m}, I_{h}, I_{m}$. Rewriting these we obtain:

$$\begin{aligned} \mathcal {F} = \left( \begin{array}{c} \frac{b\alpha I_{m}(t)S_{h}(t)}{N_{h}} \\ 0 \\ \frac{b\beta I_{h}(t)S_{m}(t)}{N_{h}}\\ 0 \end{array}\right) . \; \text {and} \; \mathcal {V} = \left( \begin{array}{c} (\mu _{h} +\delta )E_{h} \\ (\phi +\mu _{h} + d)I_{h}-\delta E_{h} \\ (\mu _{m}+\rho )E_{m} \\ rho E_{m}+\mu _{m}I_{m}\end{array}\right) . \end{aligned}$$

(20)

Applying partial derivative with respect to $E_{h}$, $I_{h}$, $E_{m}$ and $I_{m}$ by using Jacobian matrix at the disease-free equilibrium point (19),we obtain,

$$\begin{aligned} \mathcal {F}= \left( \begin{matrix} 0& 0& 0& b\alpha \\ 0 & 0& 0 & 0\\ 0& \frac{b\beta \theta \mu _h}{\mu _{m}\Lambda _{h}}& 0& 0\\ 0& 0& 0& 0 \end{matrix}\right) . \end{aligned}$$

(21)

Likewise, evaluating the Jacobean matrix of equation of the transfer of all individual in stacks by all other means at (19) we get,

$$\begin{aligned} \mathcal {V}=\left( \begin{matrix} (\mu _{h} + \delta )& 0& 0& 0 \\ -\delta & (\phi +\mu _{h}+d)& 0& 0\\ 0& 0& \mu _{m}+\rho & 0\\ 0& 0& -\rho & \mu _{m} \end{matrix}\right) . \end{aligned}$$

(22)

The inverse of matrix V of Eq. (22) become,

$$\begin{aligned} \mathcal {V}^{-1} =\left( \begin{array}{cccc} \frac{1}{\delta + \mu _{h} } & 0& 0 & 0\\ \frac{\delta }{(\phi +\mu _{h}+d)( \mu _{h} + \delta )} & \frac{1}{\phi +\mu _{h}+d}& 0& 0\\ 0& 0& \frac{1}{\rho +\mu _{m}}& 0\\ 0& 0& \frac{\rho }{\mu _{m}(\rho +\mu _{m})}& \frac{1}{\mu _{m}} \end{array}\right) . \end{aligned}$$

(23)

Now, let $Q = \mathcal {F}\mathcal {V}^{-1}$ which we denote as the next generation matrix.

$$\begin{aligned} Q&= \mathcal {F}\mathcal {V}^{-1} \end{aligned}$$

(24)

$$\begin{aligned}&= \left( \begin{array}{cccc} 0& 0& 0& b\alpha \\ 0 & 0& 0 & 0\\ 0& \frac{b\beta \theta \mu _h}{\mu _{m}\Lambda _{h}}& 0& 0\\ 0& 0& 0& 0 \end{array}\right) \left( \begin{array}{cccc} \frac{1}{\delta + \mu _{h} } & 0& 0 & 0\\ \frac{\delta }{(\phi +\mu _{h}+d)( \mu _{h} + \delta )} & \frac{1}{\phi +\mu _{h}+d}& 0& 0\\ 0& 0& \frac{1}{\rho +\mu _{m}}& 0\\ 0& 0& \frac{\rho }{\mu _{m}(\rho +\mu _{m})}& \frac{1}{\mu _{m}} \end{array}\right) . \end{aligned}$$

(25)

This implies that, the next generation matrix of system of (1) is:

$$\begin{aligned} Q = \begin{pmatrix}0& 0& \frac{b\alpha \rho }{\mu _{m}(\rho +\mu _{m})}& \frac{b\alpha }{\mu _{m}}\\ 0& 0& 0& 0\\ \frac{b\beta \mu _{h} \theta \delta }{\mu _{m}\Lambda _{h}(\mu _{h} +\delta )(d+\mu _{h} +\phi )}& \frac{b\beta \mu _{h}\theta }{\mu _{m}\Lambda _{h}(d+\phi +\mu _{h})}& 0& 0\\ 0& 0& 0& 0 \end{pmatrix}. \end{aligned}$$

(26)

Now to get eigenvalue of matrix of (26), we equate determinant of (26) with corresponding eigenvalue to zero. To do so, let $p_{1}=\frac{b\alpha \rho }{\mu _{m}(\rho +\mu _{m})}, p_{2}=\frac{b\alpha }{\mu _{m}}, p_{3}=\frac{b\beta \mu _{h} \theta \delta }{\mu _{m}\Lambda _{h}(\mu _{h} +\delta )(d+\mu _{h} +\phi )}$ and $p_{4}=\frac{b\beta \mu _{h}\theta }{\mu _{m}\Lambda _{h}(d+\phi +\mu _{h})}$, then

$$\begin{aligned} \left| \begin{matrix}-\lambda & 0& p_{1}& p_{2}\\ 0& -\lambda & 0& 0\\ p_{3}& p_{4}& -\lambda & 0\\ 0& 0& 0& -\lambda \end{matrix}\right| =0. \end{aligned}$$

(27)

Solving (27) and substituting the values of $p's$ we get,

$$\begin{aligned} \lambda _{1} =\sqrt{\frac{b^{2}\alpha \beta \delta \rho \mu _{h}\theta }{\mu _{m}^{2}\Lambda _{h}(d+\mu _{h}+\phi )(\delta +\mu _{h})(\rho +\mu _{m})}}, \hspace{0.1in} \lambda _{2}=-\sqrt{\frac{b^{2}\alpha \beta \delta \rho \mu _{h}\theta }{\mu _{m}^{2}\Lambda _{h}(d+\mu _{h}+\phi )(\delta +\mu _{h})(\rho +\mu _{m})}} \hbox { and } \lambda _{3,4}=0 \end{aligned}$$

As the dominant eigenvalue of the next generation matrix Q represents $R_{0}$, which is maximum eigenvalue of $\mathcal {F}\mathcal {V}^{-1}$, thus

$$\begin{aligned} R_{0}=\sqrt{\frac{b^{2}\alpha \beta \delta \rho \mu _{h}\theta }{\mu _{m}^{2}\Lambda _{h}(d+\mu _{h}+\phi )(\delta +\mu _{h})(\rho +\mu _{m})}} \end{aligned}$$

(28)

Simplifying Eq. (28) further and splitting it into $R_{0h}$ and $R_{0m}$ we obtain,

$$\begin{aligned} R_{0}&= \sqrt{\frac{b^{2}\alpha \beta \delta \rho \mu _{h}\theta }{\mu _{m}^{2}\Lambda _{h}(d+\mu _{h}+\phi )(\delta +\mu _{h})(\rho +\mu _{m})}}\nonumber \\&=\sqrt{\frac{b\alpha \delta \mu _{h}}{\Lambda _{h}(d+\mu _{h}+\phi )(\delta +\mu _{h})}}\times \sqrt{\frac{b\beta \rho \theta }{\mu _{m}^{2}(\rho +\mu _{m})}}\nonumber \\&=\sqrt{R_{0h}\times R_{0m}} \end{aligned}$$

(29)

Where $R_{0h}$ is the mean number of human that one Aedes mosquito can produce infection in the whole susceptible human during its infectious lifetime. But, $R_{0m}$ is the average number of Aedes mosquitoes in which one human infects the entire susceptible mosquito dynamics through their infectious period.

Local stability of disease free equilibrium points

Theorem 3

The disease free equilibrium point(DFEP)is locally asymptotically stable in the feasible region $\Omega$, if $R_{0} < 1$ and otherwise unstable.

Proof

To determine local stability at DFEP, we employed Jacobean matrix evaluated at (19). Note that, we ignore equation of recovery since all other states are not depend on it. Taking this under consideration and applying partial derivative with respect to states in the system of (1) become,

$$\begin{aligned} J(19) = \left( \begin{matrix} -\mu _{h} & 0 & 0 & 0& 0& -b\alpha \\ 0 & -(\mu _{h} + \delta )& 0 & 0 & 0& b\alpha \\ 0& \delta & -(d+\phi +\mu _{h})& 0& 0 \\ 0& 0& -\frac{b\beta \mu _{h}\theta }{\mu _{m}\Lambda _{h}}& -\mu _{m}& 0& 0\\ 0& 0& \frac{b\beta \mu _{h}\theta }{\mu _{m}\Lambda _{h}}& 0& -(\rho +\mu _{m})& 0\\ 0& 0& 0& 0& \rho & -\mu _{m} \\ \end{matrix}\right) . \end{aligned}$$

(30)

Now, eliminating the first and fourth columns of (30) we get,

$$\begin{aligned} J_{1}(19) = \left( \begin{matrix} -(\mu _{h} + \delta )& 0 & 0& b\alpha \\ \delta & -(d+\phi +\mu _{h})& 0& 0 \\ 0& \frac{b\beta \mu _{h}\theta }{\mu _{m}\Lambda _{h}}& -(\rho +\mu _{m})& 0\\ 0& 0& \rho & -\mu _{m} \\ \end{matrix}\right) . \end{aligned}$$

(31)

Computing eigenvalues of (31) we obtain:

$$\begin{aligned} (\lambda + \mu _{m} )(\lambda + k_{1} )(\lambda + k_{2} )(\lambda + k_{3} )-\frac{b^{2}\alpha \beta \rho \delta \theta \mu _{h}}{\Lambda _{h}\mu _{m}}=0. \end{aligned}$$

(32)

Expanding Eq. (32) we get,

$$\begin{aligned} \Lambda =\lambda ^{4}+(\mu _{m}+k_{1}+k_{2}+k_{3})\lambda ^{3}+(\mu _{m}k_{1}+\mu _{m}k_{2}+\mu _{2}k_{3}+k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3})\lambda ^{2}\nonumber \\+(\mu _{m}k_{1}k_{2}+\mu _{m}k_{2}k_{3}+k_{1}k_{2}k_{3})\lambda +\mu _{m}k_{1}k_{2}k_{3}-\frac{b^{2}\alpha \beta \rho \delta \theta \mu _{h}}{\Lambda _{h}\mu _{m}}. \end{aligned}$$

(33)

where $k_{1} = \delta +\mu _{h}, k_{2} = \rho +\mu _{m}, k_{3}= d+\phi +\mu _{h}$.

and Rewriting Eq. (33) we get,

$$\begin{aligned} \Lambda =a_{0}\lambda ^{4}+a_{1}\lambda ^{3}+a_{2}\lambda ^{2}+a_{3}\lambda +a_{4}. \end{aligned}$$

(34)

Since the characteristics polynomial which given by (34) is high degree polynomial, we introduce Routh-Hurwitz condition to ensure as all roots of (34) have not positive real parts. Moreover, as no terms are missed and all coefficients of it has the same sign, Routh-Hurwitz conditions are hold. So, we proceed as follow:

$$\begin{aligned} a_{0}=1, \; a_{1}&=\mu _{m}+k_{1}+k_{2}+k_{3},\nonumber \\ a_{2}&=(\mu _{m}k_{1}+\mu _{m}k_{2}+\mu _{m}k_{3}+k_{1}k_{2}+k_{1}k_{3}+k_{2}k_{3}),\nonumber \\ a_{3}&=(\mu _{m}k_{1}k_{2}+\mu _{m}k_{2}k_{3}+k_{1}k_{2}k_{3})\nonumber ,\\ a_{4}&=\mu _{m}k_{1}k_{2}k_{3}-\frac{b^{2}\alpha \beta \rho \delta \theta \mu _{h}}{\Lambda _{h}\mu _{m}}\nonumber \\&=\mu _{m}k_{1}k_{2}k_{3}(1-\frac{b^{2}\alpha \beta \rho \delta \theta \mu _{h}}{\mu _{m}^{2}k_{1}k_{2}k_{3}\Lambda _{h}})\nonumber \\&=\mu _{m}k_{1}k_{2}k_{3}(1-R_{0}) \end{aligned}$$

(35)

Setting the first two rows of Routh table we get, $\begin{array}{l|cc} \lambda ^{4}& a_{0} & a_{2} \\ \lambda ^{3}& a_{1} & a_{3} \\ \lambda ^{2}& b_{3,1}& a_{4} \\ \lambda ^{1}& b_{4,1} & 0 \\ \lambda ^{0}& b_{5,1} & - \end{array}$

$$\begin{aligned} b_{3,1}=-\frac{1}{a_{1}}\left| \begin{matrix} a_{0}& a_{2}\\ a_{1}& a_{3} \end{matrix}\right| =a_{2}-\frac{a_{0}a_{3}}{a_{1}}=a_{2}-\frac{a_{3}}{a_{1}} \end{aligned}$$

This follows $b_{3,1}>0$ if $a_{2}>\frac{a_{3}}{a_{1}}$, $b_{4,1}=-\frac{1}{b_{3,1}}\left| \begin{matrix} a_{1}& a_{3}\\ b_{3,1}& a_{4} \end{matrix}\right| =a_{3}-\frac{a_{1}a_{4}}{b_{3,1}}$

Similarly, $b_{4,1}>0$, if $a_{3}>\frac{a_{1}a_{4}}{b_{3,1}}$, $b_{5,1}=-\frac{1}{b_{4,1}}\left| \begin{matrix} b_{3,1}& a_{4}\\ b_{4,1}& 0 \end{matrix}\right| =a_{4}$

Now from model assumption, all parameters are positive, and by comparison of terms by their degrees and growth rate, $b_{3, 1}$ and $b_{4,1}$ are positive. However, $b_{5,1}$ depend on the values of $a_{4}$ which is directly associated with $R_{0}$. This means, if $R_{0}<1$, then $a_{4}>0$. This, ensures that, as all of the first column of Routh’s table are positive(i.e all roots of the characteristic Eq. (34) has negative real number). Therefore, DFE is locally asymptotically stable, if $R_{0}<1$, otherwise unstable. $\square$

Existence of endemic equilibrium

The endemic equilibrium is a point whereby dengue disease present in the population which is given by,

$$\begin{aligned} EE = (S_{h}^{\star }, E_{h}^{\star }, I_{h}^{\star }, R_{h}^{\star }, S_{m}^{\star }, E_{m}^{\star }, I_{m}^{\star }). \end{aligned}$$

To obtain it, we equate each of the equations in (1) to zero and solving explicitly for each compartments we get,

$$\begin{aligned} S_{h}^{\star } \left( t \right)&={\frac{\Lambda _{h}}{f^{\star }+\mu _{h}}}.\nonumber \\ E_{h}^{\star } \left( t \right)&={\frac{\Lambda _{h}b}{\delta \,f^{\star }+\delta \,\mu _{h}+f^{\star }\mu _{h}+{\mu _{h}}^{2}}}.\nonumber \\ I_{h}^{\star } \left( t \right)&={\frac{\delta \,b\Lambda _{h}}{d\delta \,f^{\star }+d\delta \,\mu _{h}+df^{\star }\mu _{h}+d{\mu _{h}}^{2}+\delta \,\lambda \mu _{h}+\delta \,\lambda \phi +\delta \,{\mu _{h}}^{2}+\delta \,\mu _{h}\,\phi +f^{\star }{\mu _{h}}^{2}+f^{\star }\mu _{h}\,\phi +{\mu _{h}}^{3}+{\mu _{h}}^{2}\phi }}.\nonumber \\ R_{h}^{\star } \left( t \right)&={\frac{\delta \,\mu _{h}\,b\Lambda _{h}}{\phi \, \left( d\delta \,f^{\star }+d\delta \,\mu _{h}+df^{\star }\mu _{h}+d{\mu _{h}}^{2}+\delta \,f^{\star }\mu _{h}+\delta \,f^{\star }\phi +\delta \,{\mu _{h}}^{2}+\delta \,\mu _{h}\,\phi +f^{\star }{\mu _{h}}^{2}+f^{\star }\mu _{h}\,\phi +{\mu _{h}}^{3}+{\mu _{h}}^{2}\phi \right) }}.\nonumber \\ S^{\star }_{{m}} \left( t \right)&={\frac{\theta }{g^{\star }+\mu _{{m}}}}\nonumber \\ E^{\star }_{{m}} \left( t \right)&={\frac{g^{\star }\theta }{g^{\star }\rho +g^{\star }\mu _{{m}}+\rho \,\mu _{{m}}+{\mu _{{m}}}^{2}}}.\nonumber \\ I^{\star }_{{m}} \left( t \right)&={\frac{g^{\star }\rho \,\theta }{\mu _{{m}} \left( g^{\star }\rho +\theta \mu _{{m}}+\rho \,\mu _{{m}}+{\mu _{{m}}}^{2} \right) }}. \end{aligned}$$

(36)

Where $f^{\star }=\frac{b\alpha I^{\star }_{m}}{N_h^{\star }}$ and $g^{\star }=\frac{b\beta I^{\star }_{h}}{N_h^{\star }}$ are force of dengue infection which depend on total number of human population. From equation on (36) we observe that, the values of $S_{h}^{\star }, E_{h}^{\star }, R_{h}^{\star }$, $S_{m}^{\star }, E^{\star }_{m}, I^{\star }_{m}$ are positive and unique. Moreover, the following theorem elaborate the global stability at these points.

The global stability of endemic equilibrium

Theorem 4

If $R_{0}>1$, then there exist a unique positive endemic equilibrium in invariant set $\mathbb {D}\subset \Omega$ of (1) such that any trajectories starting in $\mathbb {D}$ converges to EE as $t\rightarrow \infty$, resulting EE is globally asymptotically stable, where $\Omega$ is invariant region for system (1).

Proof

To show this theorem, define a logarithmic Lyapunov function measuring deviations from equilibrium:

$$\begin{aligned} V = \sum _i (x_i - x_i^*) \ln x_i, \end{aligned}$$

where $x_i$ represents each compartment $(S_h, E_h, I_h, R_h, S_m, E_m, I_m)$ and $x_i^*$ is its EE value. Explicitly:

$$\begin{aligned} V&= (S_h - S_h^*) \ln S_h + (E_h - E_h^*) \ln E_h + (I_h - I_h^*) \ln I_h + (R_h - R_h^*) \ln R_h \nonumber \\&\quad +(S_m - S_m^*) \ln S_m + (E_m - E_m^*) \ln E_m + (I_m - I_m^*) \ln I_m. \end{aligned}$$

(37)

This function is non-negative and zero only at the (36).

Compute $\frac{dV}{dt}$ along system trajectories by substituting the ODEs:

• 1. Human Susceptible $S_h$:

  $$\begin{aligned} \frac{dV}{dt} \bigg |_{S_h} = \left( \Lambda _h - f S_h - \mu _h S_h \right) \left( 1 - \frac{S_h}{S_h^*} \right) . \end{aligned}$$

  At equilibrium (EE): $\Lambda _h = (f^* + \mu _h) S_h^*.$

  Substituting and factor out terms $\frac{dV}{dt} \bigg |_{S_h}= \mu _h S_h^* \left( 1 - \frac{S_h}{S_h^*} \right) ^2 + f^* S_h^* \left( 1 - \frac{S_h}{S_h^*} \right) - f S_h \left( 1 - \frac{S_h}{S_h^*} \right) .$

  Simplifying using $f = \frac{N_h \cdot \alpha \cdot I_m}{N_h}$ and $f^* = \frac{N_h \cdot \alpha \cdot I_m^*}{N_h}$:

  $$\begin{aligned} \frac{dV}{dt} \bigg |_{S_h} = -\mu _h S_h^* \left( \frac{S_h}{S_h^*} - 1 \right) ^2 + f^* S_h^* \left( 1 - \frac{S_h}{S_h^*} - \frac{I_m}{I_m^*} + \frac{S_h I_m}{S_h^* I_m^*} \right) . \end{aligned}$$

• 2. Human Exposed $E_h$:

  $$\begin{aligned} \frac{dV}{dt} \bigg |_{E_h} = \left( f S_h - (\mu _h + \delta ) E_h \right) \left( 1 - \frac{E_h}{E_h^*} \right) . \end{aligned}$$

  At equilibrium (EE): $f^* S_h^* = (\mu _h + \delta ) E_h^*.$

  Substituting $f^*$ and Rearranging: $\frac{dV}{dt} \bigg |_{E_h} = f^* S_h^* \left( \frac{f S_h}{f^* S_h^*} - \frac{E_h}{E_h^*} \right) \left( 1 - \frac{E_h}{E_h^*} \right) .$

  Let $f \frac{f^*}{f^*} = \frac{I_m}{I_m^*}$, then: $\frac{dV}{dt} \bigg |_{E_h} = f^* S_h^* \left( \frac{S_h I_m}{S_h^* I_m^*} - \frac{E_h}{E_h^*} \right) \left( 1 - \frac{E_h}{E_h^*} \right) .$

• 3. Human Infected $I_h$:

  $$\begin{aligned} \frac{dV}{dt} \bigg |_{I_h} = \left( \delta E_h - (d + \phi + \mu _h) I_h \right) \left( 1 - \frac{I_h}{I_h^*} \right) . \end{aligned}$$

  At EE: $\delta E_h^* = (d + \phi + \mu _h) I_h^*.$

  Substituting we get: $\frac{dV}{dt} \bigg |_{I_h} = \delta E_h^* \left( \frac{E_h}{E_h^*} - \frac{I_h}{I_h^*} \right) \left( 1 - \frac{I_h}{I_h^*} \right) .$

  Use $\frac{E_h}{E_h^*} = \frac{I_h}{I_h^*}$:

  $$\begin{aligned} = -\delta E_h^* \left( \frac{I_h}{I_h^*} - 1 \right) ^2 \le 0. \end{aligned}$$

• 4. Mosquito Susceptible $S_m$:

  $$\begin{aligned} \frac{dV}{dt} \bigg |_{S_m} = \left( \theta - g S_m - \mu _m S_m \right) \left( 1 - \frac{S_m}{S_m^*} \right) . \end{aligned}$$

  At EE: $\theta = (g^* + \mu _m) S_m^*.$

  Substituting: $\frac{dV}{dt} \bigg |_{S_m} = -\mu _m S_m^* \left( \frac{S_m}{S_m^*} - 1 \right) ^2 + g^* S_m^* \left( 1 - \frac{S_m}{S_m^*} - \frac{I_h}{I_h^*} + \frac{S_m I_h}{S_m^* I_h^*} \right) .$

  Similarly,

• 5. Mosquito Exposed $E_m$:

  $$\begin{aligned} \frac{dV}{dt} \bigg |_{E_m} = g^* S_m^* \left( \frac{S_m I_h}{S_m^* I_h^*} - \frac{E_m}{E_m^*} \right) \left( 1 - \frac{E_m}{E_m^*} \right) . \end{aligned}$$

• 6. Mosquito Infected $I_m$:

  $$\begin{aligned} \frac{dV}{dt} \bigg |_{I_m} = -\rho E_m^* \left( \frac{I_m}{I_m^*} - 1 \right) ^2 \le 0. \end{aligned}$$

• 7. Recovered Humans $R_h$:

  $$\begin{aligned} \frac{dV}{dt} \bigg |_{R_h} = -\phi I_h^* \left( \frac{I_h}{I_h^*} - 1 \right) ^2 \le 0. \end{aligned}$$

Cross terms like $f^* S_h^* \left( \frac{S_h I_m}{S_h^* I_m^*} - \frac{E_h}{E_h^*} \right)$ and canceling due to equilibrium relationships like:

$$\begin{aligned} \frac{S_h I_m}{S_h^* I_m^*} = \frac{E_h}{E_h^*} \text { and } \frac{S_m I_h}{S_m^* I_h^*} = \frac{E_m}{E_m^*}. \end{aligned}$$

Finally, as $\frac{dV}{dt}$ is a sum of non-positive quadratic terms like:

$$\begin{aligned} -\mu _h S_h^* \left( \frac{S_h}{S_h^*} - 1 \right) ^2 - \mu _m S_m^* \left( \frac{S_m}{S_m^*} - 1 \right) ^2. \end{aligned}$$

we can apply cancellation of cross terms due to the equilibrium structure.

Thus, $\frac{dV}{dt} \le 0,$

Since $\frac{dV}{dt}\le 0$, all trajectories starting in $\mathbb {D}$ converges to EE for $t> 0$ when $R_0> 1$. Thus, EE is globally asymptotically stable. Equivalently, $\frac{dV}{dt}= 0$ follows by LaSalle’s Invariance Principle, the endemic equilibrium is globally asymptotically stable. $\square$

Bifurcation analysis

While stability is well-established within the theoretical framework, bifurcation analysis often encounters gaps related to the precise characterization of transition points and the nature of emergent solutions. Specifically, traditional approaches may lack clarity regarding the local versus global bifurcations and the conditions under which qualitative changes in system behavior occur. The Fig. in 2 helps fill these gaps by providing a detailed visualization, illustrating how parameter variations lead to the formation, disappearance, or transformation of equilibrium states. This visual representation clarifies the nonlinear interactions near critical thresholds, thereby bridging the gap between abstract theoretical predictions and their practical manifestations. Consequently, it enhances our understanding of the stability landscape and the mechanisms underlying bifurcation phenomena, offering a more comprehensive picture that integrates local dynamical behaviors with overarching system stability considerations.

Fig. 2

Bifurcation diagram.

Moreover, Fig. 2 illustrates a critical threshold at $R_0 = 1$, where the system transitions from a disease-free equilibrium (DFE) to an endemic state. For $R_0 < 1$, the DFE remains stable, indicating that the infection cannot sustain itself within the population, as reflected by the stable green line. However, as $R_0$ surpasses 1, the stability of the DFE is lost, and an endemic equilibrium emerges, shown by the stable red line, with the infected population increasing steadily. This bifurcation point signifies a forward (trans-critical) bifurcation, emphasizing the importance of controlling $R_0$ below 1 to prevent disease persistence. The diagram underscores how small changes in $R_0$ around the critical value can lead to significant shifts in disease dynamics, providing valuable insights for public health interventions aimed at disease eradication.

Sensitivity analysis of model parameters

As we are not often know the exact values of parameters of epidemic models, it is an appropriate to determine the parameters that most affect the dynamics of the model. To carry out sensitivity analysis, we employed the following formula.

$$\begin{aligned} S^{R_{0}}_{x_{i}} =\frac{\partial R_{0}}{\partial x_{i}}(\frac{x_{i}}{R_{0}}). \end{aligned}$$

(38)

, where $x_{i}$ represents $i^{th}$ basic parameters, with $i=1,2...n$. But from Eqs. (28) and (29), and computing for each parameter by using the formula (38) and simplifying further we get,

$$\begin{aligned} S^{R_{0}}_{\alpha }&=\frac{\partial R_{0}}{\partial \alpha }\left( \frac{\alpha }{R_{0}}\right) \nonumber \\ S^{R_{0}}_{\alpha }&=\frac{1}{2}.\; \text {Similarly},\nonumber \\ S^{R_{0}}_{\beta }&=S^{R_{0}}_{\theta }=\frac{1}{2},\; S^{R_{0}}_{b} =1,\; S^{R_{0}}_{\Lambda _{h}} = -1,\nonumber \\ S^{R_{0}}_{\rho }&=\frac{\mu _{m}-\rho }{2(\mu _{m}+\rho )},\; S^{R_{0}}_{\delta } =\frac{\mu _{m}-\delta }{2(\mu _{m}+\delta )},\; S^{R_{0}}_{\mu _{m}} = -\frac{2\rho +3\mu _{m}}{\rho +\mu _{m}},\; S^{R_{0}}_{\phi } = -\frac{\phi }{d+\phi +\mu _h}\nonumber \\ S^{R_{0}}_{\mu _{h}}&= \frac{(\mu _{h}+\delta )(d+\mu _{h}+\phi )-2\mu _{h}(2\mu _{h}+\delta +d+\phi )}{2(\delta +\mu _{h})(d+\mu _{h}+\phi )}, \; S^{R_{0}}_{d}= -\frac{d}{(d+\phi +\mu _{h})}. \end{aligned}$$

(39)

This can be summarized graphically as follows:

From the sensitivity index analysis, we observed that the parameters $\alpha$, $\beta$, and $\theta$ account for 50% of the maximization of the dengue outbreak. The parameter b is highly sensitive and significantly influences disease spread, while $\rho$, $\delta$, and d show low sensitivity. A sensitivity value of −1 indicates that an increase in the human recruitment rate ($\Lambda _h$) and recovery rate ($\phi$) leads to a decrease in disease prevalence by one unit. When the sensitivity analysis indicates that increasing $\Lambda _h$ reduces disease prevalence (with a sensitivity value of −1), it suggests that adding more people to the population can dilute the proportion of infected individuals, especially if the additional individuals are initially uninfected. Alternatively, it might also reflect that a higher influx of susceptible humans influences the dynamics of disease transmission, possibly by providing a larger pool of individuals who are susceptible but not yet infected, which can affect overall outbreak patterns. Likewise, increasing $\phi$ decreases disease prevalence (sensitivity value of −1), it implies that faster recovery reduces the number of infectious individuals in the population at any given time, thereby limiting the spread of the disease. This highlights the importance of medical interventions and treatment strategies that accelerate recovery, ultimately helping to control and reduce the outbreak. Similarly, an increase in the mosquito’s mortality rate ($\mu _m$) has a comparable effect. This implies that increasing the mosquito mortality rate would reduce dengue transmission by 1.4 units. This happens because the mosquito mortality rate and the dengue reproduction number are inversely proportional. Conversely, a sensitivity value of $+1$ means that an increase in the mosquito biting rate (b) results in a corresponding increase in disease cases, as illustrated in Fig. 3, meaning increases in this parameter potentially fold dengue transmission.

Fig. 3

Sensitivity of the model parameters.

Model analysis with climate-based parameters

To analyze climate-based parameters, we adopted on system of Eq. (1), so that climate-based entomological parameters are defined. To do so, the monthly mean temperature and rainfall of three regions in Ethiopia were mentioned. The monthly mean temperature and rainfall averages for Dire Dawa, Afar, and the Somali Region in Ethiopia that defined in table available as Supplementary materials, which we consider for our investigation based on climate data obtained from²¹.

Mosquito birth rate

Since the birth rate of mosquitoes is influenced by rainfall and temperature, we employed a sophisticated analytical framework of^22,23 to examine the birth rate of Aedes mosquitoes along with other climate-related entomological parameters. These parameters include the biting rate, mosquito infectious rate, transmission probabilities, and mosquito mortality rate in the Dire Dawa, Afar, and Somali regions. We utilized the parameter values outlined in Table 1, along with regional temperature and rainfall data from the Table available in Supplementary materials. The results of our analyses are summarized for each formula used in our study.

The life cycle of mosquitoes is highly influenced by climate change, particularly temperature and rainfall^5,24. Therefore, the following formulas address how climate change influence mosquito population dynamics and analysis demonstrate the potential impact of temperature and rainfall on the mosquito’s birth rate and how these factors affect their life cycle.

$$\begin{aligned} \theta (T,R_{a})=\frac{\pi _{e}H_{s}(T, R_{a})S_{l}(T,R_{a})S_{p}(R)_{a}}{\tau _{e}+\tau _{l}m(T) +\tau _{p}}. \end{aligned}$$

(40)

Where $\pi _{e}$ represents the number of eggs laid per female Aedes mosquito. The functions $H_{s}(T, R_{a})$, $S_{l}(T, R_{a})$, and $S_{p}(R_{a})$ denote the survival probabilities of eggs, larvae, and pupae, respectively. Additionally, $\tau _{e}$, $\tau _{l}m(T)$, and $\tau _{p}$ represent the average durations of the egg, larval, and pupal stages. Here, T and $R_{a}$ stand for the mean temperature and rainfall, respectively.

Now, let’s highlight the existing literature on the average developmental durations for the larval stage, as used by²² and⁵:

$$\begin{aligned} \tau _{l}(T)&=\frac{1}{0.0554T-0.06737}\nonumber \\&\text {which is true if}\; 0.0554T- 0.06737> 0 \nonumber \\ \implies \;&T> \frac{0.06737}{0.0554}\approx 1.216 ^oC \end{aligned}$$

(41)

In reality, Aedes mosquito’s larva did not develop at this temperature²⁴. Additionally, evaluating the regional optimal mean temperature we obtained: Dire Dawa: $\tau _{l}(28.2^o C)=0.067$, Afar: $\tau _{l}(28.6^o C)= 0.066$ Somali: $\tau _{l}(27.5^o C)= 0.069.$ In reality, Aedes aegypti larvae generally take about 7 to 10 days to develop from hatch to pupation at optimal temperatures (26$\phantom{0}^o$C to 30$\phantom{0}^o$C)^25,26. Thus, we leave the formula (41) and employed the other formula that align with the reality, so that the average developmental duration $\tau _lm(T)$ in days can be estimated using the formula which is adopted to²⁷ and match with the fitted one:

$$\begin{aligned} \tau _lm(T) = \frac{K}{T - T_{min}} \end{aligned}$$

(42)

where $\tau _{l}m(T)$ be development time of larva in days as a function of T, T = ambient temperature in $^o\hbox {C}$, $T_{min}$ = lower developmental threshold temperature in $^o\hbox {C}$, which is often estimated around $10^o\hbox {C}$, and K = degree-days required for development (species-specific constant) which is for Aedes aegypti, values around 100 −150 degree-days. Hence, we used the average of degree-days which is 125.

Determining th average development duration of larva for each region we get: Dire Dawa: $\tau _{l}(28.2^o C)=6.87$ days, Afar: $\tau _{l}(28.6^o C)= 6.72$ days, Somali: $\tau _{l}(27.5^o C)= 7.14$ days. These values indicate that larvae in Dire Dawa and Afar develop slightly faster than those in Somali. The differences may suggest variations in environmental conditions or resource availability that could affect growth rates.

$$\begin{aligned} S_{l}(T, R_{a})=(e^{0.06737-0.0554T})\frac{4S_{lmax}}{R^{2}_{fl}}R_{a} (R_{fl}-R_{a}). \end{aligned}$$

(43)

Where $R_{fl}$ is the rainfall which can flush out the larva, $S_{lmax}$ maximum survival probability of larva, $R_a$ stands for mean rainfall in the region. Summary of Survival Probabilities of larva in each region:

Dire Dawa: $S_l \approx 0.0334$, Afar:$S_l \approx 0.04702$, Somali: $S_l \approx 0.05767$. From this analysis, we can conclude that the survival probability of larvae is highest in the Somali region, followed by Afar, and is lowest in Dire Dawa.

The probability of survival rate of pupa is given as

$$\begin{aligned} S_{p}(R_{a})=\frac{4S_{pmax}}{R^{2}_{fp}}R_{a} (R_{fp}-R_{a}) . \end{aligned}$$

(44)

Where $R_{fp}$ is the rainfall which can flush out the pupa, $S_{pmax}$ maximum survival probability of pupa, $R_{a}$ stands for mean rainfall in the region. This formula indicates that the survival probability depends on both $R_a$ and $R_{f_p}$ through a quadratic relationship in terms of $R_a$.

Dire Dawa: $S_p \approx 0.329$, Afar: $S_p \approx 0.473$, and Somali:$S_p \approx 0.549$.

From this analysis, we can conclude that the survival probability of pupae is highest in the Somali region, followed by Afar, and is lowest in Dire Dawa.

Mosquito hatching success

Aedes mosquito hatching succuss depend on both temperature and rainfall. Hence, the formula is adopted to^22,28 and given as

$$\begin{aligned} H_{s}(T, R_{a})=\pi _{5}(1-e^{\pi _{6}(T-40)})\frac{4H_{emax}}{R^{2}_{fe}}R_{a} (R_{fe}-R_a ). \end{aligned}$$

(45)

Where $R_{fe}$ is the rainfall which can flush out the pupa, $H_{emax}$ maximum survival probability of pupa, $R_a$ stands for mean rainfall in the region whose values are given in table 1. To implement this formula with climate data from three regions we obtain the following results.

The hatching success of each region is summarized as follows:

Dire Dawa: $H_s \approx 0.607$, Afar: $H_s \approx 0.0877$, Somali:$H_s \approx 0.1297$

From this analysis, we conclude the success of the hatching rate is highest in Dire Dawa, followed by Somali, and is lowest in Afar.

Generally, the birth rate of Aedes mosquito generalized below:

Table 2 shows that,the highest value of Aedes mosquito birth rate $\theta$ is observed in Dire Dawa, followed by Somali, and the lowest is in Afar.

Table 2 Summary of Mosquito’s birth rate via climate change for different regions.

Adult female mosquito death rate

As life of adult female mosquito determined by temperature, its death rate which was adopted to the quadratic fitting by²⁴ is given as

$$\begin{aligned} \mu _{m}(T)&= a_{1}T^2 + a_{2}T + a_{3}, \; \nonumber \\&\text {where T is mean temperature},\; a_1=0.000193, a_2= -0.00901, a_3 = 0.123. \end{aligned}$$

(46)

But to analyze the Aedes mosquito’s local death rate, we used the values of temperature described in table available as Supplementary materials so that the death rate results of each region is:

$$\begin{aligned} \text {Dire Dawa:} \;\mu _m(28.2)\approx 0.022, \;\text {Afar:} \;\mu _m(28.6)\approx 0.023,\; \text {Somali:} \;\mu _m(27.5)\approx 0.021. \end{aligned}$$

From this computations, the death rates of mosquitoes vary across the three regions, with Afar exhibiting the highest death rate at approximately 0.023 per a day, followed closely by Dire Dawa at 0.022 per a day. Somali records the lowest death rate at 0.021 per a day. This indicates that, on average, the lifespan of Aedes mosquito in Dire Dawa, Afar, and Somali accounts 45.45, 43.48, and 47.62 days respectively. These results suggest that the environmental conditions affecting mortality rates differ, potentially indicative of other ecological factors influencing mosquito populations in each region. Understanding these dynamics is crucial for informing public health strategies related to mosquito control and dengue transmission risk.

Mosquito biting rate

Mosquito feed human blood for the purpose of meal. Due to this, it search for human to bite them. However, the rate of biting depend on temperature^24,29. So that, the biting rate of mosquito is give as

$$\begin{aligned} b(T)=k(T^{2}-T_{min}T)(\sqrt{T_{max}-T}). \end{aligned}$$

(47)

Where $k=2.03\times 10^{-4}$, $T_{min}=10.25^{o}C, T_{max}=38.32^{o}C$ and T is mean temperature.

Implementing the biting rate given in (47) with the area under the study we get:

First, we find the mean temperature of the high transmission months in each region.

$$\begin{aligned} \text {Mean}T_{Dire\;Dawa}= & \frac{28 + 29 + 29 + 28 + 27}{5} = 28.2 \, ^\circ C\\ \text {Mean}T_{Afar}= & \frac{26 + 27 + 29 + 31 + 30}{5} = 28.6 \, ^\circ C\\ \text {Mean}T_{Somali}= & \frac{29 + 28 + 27 + 26}{4} = 27.5 \, ^\circ C \end{aligned}$$

Now, let’s calculate the biting rate for each region using the mean temperatures.

Computing biting rate for Dire Dawa we get:

$$\begin{aligned} b(28.2)&= 2.03 \times 10^{-4} \left( (28.2^2 - 10.25 \cdot 28.2)\sqrt{(38.32 - 28.2)}\right) \\&\approx 3.269 \times 10^{-1} \text { (or approximately 0.3269 per day)}. \end{aligned}$$

Following the same processes for Afar:

$$\begin{aligned} b(28.6)&= 2.03 \times 10^{-4} \left( (28.6^2 - 10.25 \cdot 28.6)\sqrt{(38.32 - 28.6)} \right) \\ b(28.6)&= 3.389 \times 10^{-1} \approx 0.3389 \,\text {per day}. \end{aligned}$$

Similarly, for Somali:

$$\begin{aligned} b(27.5)&= 2.03 \times 10^{-4} \left( (27.5^2 - 10.25 \cdot 27.5)\sqrt{(38.32 - 27.5)} \right) \\ b(27.5)&= 3.168 \times 10^{-1} \approx 0.3168 \text {per day}. \end{aligned}$$

Hence, based on the mean temperatures of high transmission risk months we observe that:

Afar has the highest biting rate at approximately 0.3389. Dire Dawa follows with 0.3269. Somali has the lowest biting rate at approximately 0.3168. This analysis suggests that even though all regions have significant vulnerability, Afar appears to have the highest potential for dengue transmission based on biting rates during peak months.

Bi-transmission probabilities

Based on²⁸ resource, the transmission probability of infection from mosquito to human and vice-verse are given respectively as,

$$\begin{aligned} \alpha (T)&= 0.001044T(T-12.286) \sqrt{32.461-T},\;(12.286< T< 32.461)^o C. \nonumber \\&\text {Plugging the mean temperature of the areas under consideration we get,} \nonumber \\ \alpha (T)_{Dire\;Dawa}&= 0.967,\;\alpha (T)_{Afar}= 0.957,\;\alpha (T)_{Somali}= 0.972.\nonumber \\ \beta (T)&=\left\{ \begin{array}{ll} 0.729T-0.9037, \;(12.4\le T\le 26.1)^o C\nonumber \\ 1, \; (26.1<T<32.5)^o C \end{array}\right. \\&\text {Here, since temperature range in}\;(26.1<T<32.5)^o C,\nonumber \\ \beta (T)_{Dire\;Dawa}&= \beta (T)_{Afar}=\beta (T)_{Somali}=1. \end{aligned}$$

(48)

Here this results are only for selective months particularly, June–August as yearly optimal temperature in these regions.

Progression rate

The rate at which vulnerable adult female mosquito move into infected class due to a pathogen transmitted from infected human which consists Extrinsic Incubation Period(EIP). EIP is the time taken to the dengue virus inside mosquito before it can be transmitted, that can determined by external factors such as climate change, particularly temperature.

$$\begin{aligned} EIP(T)&= e^{5.2 - 0.123T} \; \text {Where T is optimal temperature of a given region}\nonumber \\ \rho (T)&= \frac{1}{EIP}= e^{-(5.2 - 0.123T)} . \end{aligned}$$

(49)

We need to evaluate $\rho (T)$ at the mean temperatures previously calculated for the high-risk months of each region under Eq. (47).

Mean Temperatures from Previous Analysis

$$\begin{aligned} \text {Dire Dawa: } T = 28.2^\circ C, \; \text {Afar: } T = 28.6^\circ C, \; \text {Somali: } T = 27.5^\circ C \end{aligned}$$

Using the mean temperatures, we will compute the extrinsic incubation period EIP(T) for each region:

$$\begin{aligned} \text {Dire Dawa:}\;EIP(28.2)&= e^{5.2 - 0.123 \times 28.2} \approx 5.65\;\text {days}.\\ \text {Afar:}\; EIP(28.6)&= e^{5.2 - 0.123 \times 28.6} \approx 5.378\;\text {days}.\\ \text {Somali:}\; EIP(27.5)&= e^{5.2 - 0.123 \times 27.5} \approx 6.16 \;\text {days}. \end{aligned}$$

Its physical meaning is for example in Dire Dawa, EIP(28.2) = 5.65 days implies that, after a mosquito bites an infected human and becomes exposed, it takes 5.65 days for dengue virus to develop inside it before it can infect another human.

$$\begin{aligned} \rho (28.2)_{Dire \; Dawa}&= \frac{1}{EIP_{Dire\; Dawa}} \approx 0.177\;\text {per a day}. \\ \rho (28.6)_{Afar}&= \frac{1}{EIP_{Afar}} \approx 0.186\;\text {per a day}. \\ \rho (27.5)_{Somali}&= \frac{1}{EIP_{Somali}} \approx 0.162\;\text {per a day}. \end{aligned}$$

Based on the calculations of the progression rates, the region with the highest incubation completion is Afar with approximately 0.186 while Dire Dawa follows with 0.177, and Somali has the lowest such transition rate at 0.162. A value of 0.186 would mean that, on average, an exposed Aedes mosquito will become infectious after approximately 5.378 days, which has similar meaning for the rest one.

Thus, Afar is the region with the shortest EIP which occur at optimal temperature according to the given formula and mean temperature assessments.

Human death rate

To compute human death rate, we used the average life span (LE) of Dire Dawa, Afar, and Somali regions which is 67 years.

$$\begin{aligned} \mu _h=\frac{1}{LE}=\frac{1}{67\times 365.25}=0.00004\;\text {per a day}. \end{aligned}$$

Human birth rate

$$\begin{aligned} \Lambda _h= \mu _h\times N_h= 0.00004\times 8,445,050=345\;\text {person per day}. \end{aligned}$$

, $N_h$ be total populations of Dire Dawa, Afar, and Somali region.

Seasonal transmission rate

Dengue transmission exhibits seasonal variation driven by the cyclical interplay between mosquito and human populations. In this study, transmission rate is split into biting rate and transmission probabilities. Collectively, the transmission rate from mosquito to human, $\alpha ^{\prime }$, is modulated by a cosine function with a period of one year, peaking during warmer months. This is represented as $\alpha ^{\prime } = \alpha _0(1 + \omega _1 \cos (\frac{2\pi (t-\Phi )}{365}))$, where $\alpha _0 = 0.25$ is the baseline transmission rate, $\omega _1 = 0.3$ is the amplitude of seasonal variation (in tropical and subtropical region), and t is time in days. Similarly, human-to-mosquito transmission, $\beta ^{\prime }$, also varies seasonally, with $\beta ^{\prime } = \beta _0(1 + \omega _2 \cos (\frac{2\pi (t-\Phi )}{365}))$, indicating that transmission from infected humans to mosquitoes is also higher during warmer months, with $\beta _0 = 0.29$ and $\omega _2 = 0.3$ representing the baseline and amplitude of seasonal variation. We employed the value of $\alpha _0$ and $\beta _0$ from fitting whereas $\omega _1$ and $\omega _2$ from³⁰. But the phase shift ($\Phi$) indicates the season where the disease become peak. For instance in Ethiopia, the peak season of dengue is August or September (day 220 to 260) in average 240 and the observed data also shows August. Hence, $\Phi = 240$. The detailed impact of this seasonal change is elaborated graphically under section (8).

Where $\pi _1$ stands for Entropy contribution to hatch whereas $\pi _{2}$ represent temperature influence factor to hatch, and $\tau$ stands for mosquito death due to insecticide treated-spray.

Model validation

To validate the model described in Eq. (1), we conducted a comprehensive analysis utilizing various tools, including model fitting techniques and additional statistical metrics. This approach allows us to assess the model’s performance and reliability, ensuring that it accurately represents the underlying data and fulfills the research objectives. To analyze these tools, monthly collected two years of current dengue case data from the Ethiopian Public Health Institute (EPHI)¹⁶.

Based on the reported cases available in Supplementary materials, the model was fitted to the data as follows. Before using the raw data, the data were scaled to reflect cases per 1,000 people for the simulation.

Figure 4 shows that, the model that fitted to the prevalence of dengue cases and the y-axis represented number of infected human scaled down by 1,000 people. For instance, $I_h =10$ on vertical axis stands 10,000 infected people. The graph demonstrates a strong alignment between the model’s predictions and actual dengue cases over two years, highlighting a notable peak around the 10th month. The close correspondence during this surge indicates the model effectively captures seasonal or outbreak-driven fluctuations in infection rates. This pattern underscores the importance of targeted interventions during peak periods to mitigate transmission and highlights the model’s utility in forecasting and managing dengue outbreaks.

Fig. 4

The model fitted to the existing dengue data.

Statistical analysis

Based on the fitted model applied to observed cases using MATLAB software with RK45 method, we discuss the implications of the fitted parameters, RSSE (Residual Sum of Squares Error), R-squared, AIC (Akaike Information Criterion), confidence intervals, and p-values.

Fitted parameters: The fitted parameters represent the estimated values of the model parameters after optimization by making climate variable constant. However, climate based parameters were computed lonely for each region under consideration. The results are approximately, similar. These include: $\alpha : 0.7809,\; \beta : 0.9103,\; b: 0.3206,\; \delta : 0.1450,\;$$\rho : 0.1750,\; \theta : 0.5760,\; \phi : 0.0149,\; d: 0.000027 \;$$(\text {dengue induced death rate}),\; \mu _m: 0.0220 \; (\text {mosquito death rate}),$ and $\text {scaling factor: } 0.2104.$

This implies that the parameters influencing the disease reproduction rate ($\alpha$ and $\beta$) are relatively high, suggesting that transmission may be slightly spread within the population. However, the high value of biting rate ensures as many works are expected from public health sector. The parameters $\delta$ and $\rho$ provide insights into the progression rates from exposed to infectious stages in humans and mosquitoes, respectively. Additionally, the $\phi$ parameter indicates the recovery rate for humans. The low death rates for both humans (d) and mosquitoes ($\mu _m$) suggest that the mosquito population and human infections are relatively stable and not particularly lethal.

The residual sum of squares error (RSSE) value is 0.000135, which indicates that the model predictions are closely aligned with the observed data, suggesting a good model fit. On the other hand, the obtained $R^2$ value is 0.9998938, which is close to 1. This implies that the model explains almost all the variability in the observed data. This suggests an excellent fit and indicates that changes in the predictor variables (the model parameters) are well accounted for in predicting the response variable (the observed data).

AIC (Akaike information criterion): The AIC is calculated as:

$$\begin{aligned} AIC = 2n(E_p) + m \ln {(\frac{RSSE}{m})} = \text {Value: } -248.7, \end{aligned}$$

where $E_p$ is the number of parameters, and m stands for observed datasets. A lower AIC value generally indicates a better model fit when comparing neglected models. The negative value here demonstrates that the model performs exceptionally well in terms of goodness-of-fit.

Most parameter confidence intervals are not wide. But $\phi$: [−1.9094, 1.9879]) which suggests the value of $\phi$ lies exactly between −1.9094 and 1.9879.

The p-values of most parameters are less than 0.05, indicating statistical significance at conventional levels. This suggests that these parameters are critical predictors in the model and should not be retained in any further analyses or refinements. However, the p-value for d : 0.6856 which is greater than 0.05, indicating that its contribution is relatively less significant when considering the model collectively.

Overall, the model appears to fit the data very well, satisfying almost all statistical criteria, thereby ensuring that the model is biologically interpretable and useful in real-world applications.

A model with optimal control

In this section, we extended the basic model given in system (1) by incorporating three optimal controls. These control variables and corresponding costs are defined as:

• a) Insecticide treated bed-nets (labeled as $u_{1}(t)\in [0, 1]$) is a preventive measure at time t and protect infected Aedes mosquito ($I_{m}$) from susceptible person($S_{h}$) and to eradicate mosquito populations. The term $u_{1}=0$ stands for no use of pesticide treated bed-nets while $u_{1}=1$ stands for complete use of pesticide treated bed-nets. The cost under consideration includes pesticide treated bed-net and it’s distribution which denoted as $C_{1}$.

• b) Providing insecticides-spray ($u_{2}(t)\in [0, 1]$) to kill adult mosquito and it’s immature part. This control serves in destroying larva site and adult mosquito. The cost of implementation includes purchase of spraying tools and cost of chemicals, which is labeled as $C_{2}$.

• c) Treatment ($u_{3}(t)\in [0, 1]$) (i.e treating individuals who developed symptoms of the disease). Treatment of infected person can eliminate the prevalence of disease and also diminish the progress of vector. The cost of giving treatment includes medicine and clinical care which termed as $C_{3}$.

After incorporating, $u_{1}, u_{2}$ and $u_{3}$ in dengue infection model of (1), we obtain the following optimal control model:

$$\begin{aligned} \left\{ \begin{array}{ll} \frac{dS_{h}}{dt}& =\Lambda _{h}-\frac{\alpha b(1-u_{1})I_{m}(t) S_{h}(t)}{N_{h}}-\mu _{h} S_{h}(t)\\ \frac{dE_{h}}{dt}& =\frac{\alpha b(1-u_{1})I_{m}(t) S_{h}(t)}{N_{h}}-(\mu _{h}+ \delta )E_{h}(t)\\ \frac{dI_{h}}{dt}& =\delta E_{h}(t)-(d(1-u_{3})+u_{3}\phi +\mu _{h}) I_{h}(t)\\ \frac{dR_{h}}{dt}& =\phi I_{h}(t)-\mu _{h} R_{h}(t)\\ \frac{dS_{m}}{dt}& = \theta (1-u_{2})-\frac{\beta b(1-u_{1})I_{h}(t) S_{m}(t)}{N_{h}}-(\mu _{m}+u_{2}\tau )S_{m}(t)\\ \frac{dE_{m}}{dt}& =\frac{\beta b(1-u_{1})I_{h}(t) S_{m}(t)}{N_{h}}-(\mu _{m} +\rho (1-u_{2})) E_{m}(t)\\ \frac{dI_{m}}{dt}& =\rho (1-u_{2})E_{m}(t) -\mu _{m} I_{m}(t) \end{array}\right. \end{aligned}$$

(50)

with initial conditions of system 1 are:

$$\begin{aligned} S_{h}(0)> 0, E_{h}(0)\ge 0, I_{h}(0)\ge 0, R_{h}(0) \ge 0, S_{m}(0)> 0, E_{m}(0) \ge 0, I_{m}(0) \ge 0. \end{aligned}$$

The control set is defined as:

$U = {(u_{1}(t), u_{2}(t), u_{3}(t)): 0 \le u_{1}(t) \le 1, 0 \le u_{2}(t) \le 1, 0 \le u_{3}(t) \le 1, 0 \le t \le t_{f} }$, which allows us to explore the optimal levels.

Now, our main purpose is to obtain the optimal levels of the controls and related state variables that optimize the objective function. In this study, we employed Pontryagin’s minimum principle; so that the objective function is given as:

$$\begin{aligned} O_{f}=min_{(u_{1},u_{2},u_{3})}\int _{0}^{t_{f}}(w_{1}E_{h}+w_{2}I_{h}+w_{3}E_{m}+w_{4}I_{m}+\frac{1}{2}\sum _{i=1}^{3}u_{i}^{2}c_{i})dt. \end{aligned}$$

(51)

The coefficients associated with state variables ($w_{k}, k=1, 2,..4$) stands for societal cost(i.e., economic loss due to disease burden) and ($c_{i}, i=1, 2, 3$) are costs of interventions. Now, we aimed to minimize the number of infective and costs of corresponding control in objective function of Eq. (51).

The Hamiltonian and optimality system

To obtain the Hamiltonian function (H), we integrate the objective function (performance index) with respect to t and add the sum of product of each state with corresponding co-state as follow.

$$\begin{aligned} H =\frac{dO_{f}}{dt}+\lambda _{1}\frac{dS_{h}}{dt}+\lambda _{2}\frac{dE_{h}}{dt}+\lambda _{3}\frac{dI_{h}}{dt}+\lambda _{4}\frac{dR_{h}}{dt}+\lambda _{5}\frac{dS_{m}}{dt}+\lambda _{6}\frac{dE_{m}}{dt}+\lambda _{7}\frac{dI_{m}}{dt}. \end{aligned}$$

This implies that

$$\begin{aligned} H(S_{h}, E_{h}, I_{h}, R_{h}, S_{m}, E_{m}, I_{m}, u_{i}, \lambda _{k} ,t)&= L(E_{h}, I_{h}, E_{m}, I_{m}, u_{1}, u_{2}, u_{3}, t)\nonumber \\&\quad +\lambda _{1}\frac{dS_{h}}{dt}+\lambda _{2}\frac{dE_{h}}{dt}+\lambda _{3}\frac{dI_{h}}{dt}+\lambda _{4}\frac{dR_{h}}{dt}\nonumber \\ &+\lambda _{5}\frac{dS_{m}}{dt}+\lambda _{6}\frac{dE_{m}}{dt}+\lambda _{7}\frac{dI_{m}}{dt}. \end{aligned}$$

(52)

where $L(E_{h}, I_{h}, E_{m}, I_{m}, u_{1}, u_{2}, u_{3}, t) = (w_{1}E_{h}+w_{2}I_{h}+w_{3}E_{m}+w_{4}I_{m}+\frac{1}{2}\sum _{i=1}^{3}u_{i}^{2}c_{i})$ and $\lambda _{1}, \lambda _{2}, \lambda _{3}, \lambda _{4}, \lambda _{5}, \lambda _{6}$ and $\lambda _{7}$ are adjoint variables or co-state functions. Moreover, the existence of control set with corresponding state solutions, co-state functions and characterization of controls are elaborated in the following theorem.

Theorem 5

There exists an optimal control set $u_{1}, u_{2}, u_{3}$ and corresponding solutions, $S_{h}(t), E_{h}(t), I_{h}(t), R_{h}(t), S_{m}(t), E_{m}(t)$ & $I_{m}(t)$, that minimizes $O_{f}(u_{1}, u_{2}, u_{3} )$ over the set of control U. Furthermore, there exists adjoint functions, $\lambda _{1} , ..., \lambda _{7}$ such that,

$$\begin{aligned} \left\{ \begin{array}{cl} \frac{d\lambda _{{1}} (t) }{{ dt}}=& \frac{( -I_{{m}} ( E_{h}(t ) +I_{h}( t) +R_{h}( t) ) \left( -1+u_{1} \right) b +\mu _{h}\,{N_{h}}^{2}) \lambda _{{1}}(t)}{N_{h}^2}+B_1 \\ \frac{d\lambda _{{2}}( t) }{{ dt}}=& \frac{ \delta \,{N_h}^{2}+\mu _h\,( S_h( t)) ^{2}+ ( 2\,\mu _h\,I_h( t) +2\,\mu _h\,E_h ( t) +2\,\mu _{h}\,R_{h}( t) -bE_{{m}}( t) ( -1+u_{1})) S_{h}( t)}{N_{h}^2}+B_2\\ \frac{d\lambda _{{3}}(t) }{{ dt}}=& -{ w_{1}}+{\frac{\lambda _{{1}}(t) b( -1+u_{1}) S_{h}(t) I_{{m}}(t)}{{N_{h}}^{2}}}-\frac{\lambda _{{5}}(t) b( -1+u_{1}) S_{{m}}(t)(S_{h}(t) +E_{h}(t) +R_{h}(t)) }{{N_{h}}^{2}}+B_3 \\ \frac{d\lambda _{{4}}(t) }{{ dt}}=& {\frac{\lambda _{{4}}(t) \mu \,{N_{h}}^{2}+( -1+u_{1}) b(E_{{m}}(t)(\lambda _{{1}}(t) -\lambda _{{2}}( t)) S_{h}(t) +I_{h}( t) S_{{m}}(t)( \lambda _{{5}}(t) -\lambda _{{6}}(t)))}{{N_{h}}^{2}}}\\ \frac{d\lambda _{5}(t)}{{ dt}}=& -\lambda _{5}(t)(-{\frac{b(1-u_{1}) I_{h}(t)}{N_{h}}}-\tau \,u_{2}-\mu _{{m}})+{\frac{\lambda _{6}(t) b(-1+u_{1}) I_{h}(t) }{N_{h}}}\\ \frac{d\lambda _{6}(t)}{{ dt}}=& -{ w_{2}}-{\frac{\lambda _{{1}}(t) b( -1+u_{1}) S_{h}(t) }{N_{h}}}+{\frac{\lambda _{{2}}(t) b( -1+u_{1}) S_{h}(t) }{N_{h}}}+B_4\\ \frac{d\lambda _{{7}}(t)}{{ dt}}=& \mu _{{m}}\lambda _{{7}}(t) \end{array}\right. \end{aligned}$$

(53)

Where $B_1=\frac{( -1+u_{1}) b(( E_{{m}}( t) \lambda _{{2}} ( t) +( \lambda _{{5}}(t) -\lambda _{{6}}( t)) S_{{m}}( t)) I_{h}( t) +E_{{m}}( t) \lambda _{{2}}(t)( E_{h}( t) +R_{h}( t)) ) }{{N_{h}}^{2}}$,

$B_2=\frac{\mu _{h}\,(E_{h}( t) +I_{h}( t) +R_{h}( t)) ^{2} \lambda _{{2}} (t)-\lambda _{{3}}( t) \delta \,{N_{h}}^{2}+c( -1+u_{1}) b( E_{{m}}( t) \lambda _{{1}} ( t) S_{h}( t)}{N_{h}^2}+ \frac{I_{h}(t) S_{{m}}(t) ( \lambda _{{5}}(t) -\lambda _{{6}}(t))}{{N_{h}}^{2}}$,

$B_3={-\frac{\lambda _{{2}}(t) b( -1+u_{1}) S_{h}(t) I_{{m}}(t)- (( d-\phi ) u_{3}-d-\mu _{h}) \lambda _{{3}}(t) -\lambda _{{4}}(t) \phi }{{N_{h}}^{2}}} +\frac{\lambda _{6}(t) b(-1+u_{1}) S_{{m}}(t)(S_{h}(t) +E_{h}(t) +R_{h}(t))}{{N_{h}}^{2}}$,

$B_4=\lambda _{{6}}(t)( \mu _{h}+ ( 1-u_{2}) \rho )+\lambda _{{7}}(t)( -1+u_{2}) \rho$

With transversality conditions, $\lambda _{i}(t_{f}) = 0, i = 1,..., 7$. and the characterized control set of $(u_{1}^{\star }, u_{2}^{\star }, u_{3}^{\star })$ is:

$$\begin{aligned} u_{1}^{\star }= & max\left\{ 0, min\left( 1,-{\frac{( E_{{m}}(t)( \lambda _{{1}}(t) -\lambda _{{2}}(t)) S_{h}(t) +I_{h}( t) S_{{m}}(t)( \lambda _{{5}}(t) -\lambda _{{6}}(t))) b}{N_{h}c_{{1}}}}\right) \right\} \\ u_{2}^{\star }= & max\left\{ 0,min\left( 1,-{\frac{I_{h}(t)( d-\phi ) \lambda _{{3}}(t) }{c_{{2}}}}\right) \right\} \\ u_{3}^{\star }= & max\left\{ 0,min\left( 1,{\frac{\lambda _{{5}}(t) \tau \,S_{{m}}(t) -\lambda _{{6}}(t) \rho \,I_{{m}}(t)+\lambda _{{7}}(t) \rho \,E_{{m}}(t) +\theta \,\lambda _{{5}}(t) }{c_{{3}}}}\right) \right\} . \end{aligned}$$

Proof

To prove this theorem, we employed the Pontryagin’s minimum principle. Accordingly, to obtain the system of adjoint variables, we differentiate the Hamiltonian function given on Eq. (52) with respect to each states of compartment and to obtain critical point of control we solve differential of Hamiltonian function with respect to each control equal to zero. $\square$

System of optimal control

Optimality system is formed by combining a system of states in equations (53) and adjoint of (50) with the characterization of the optimal control including initial and transversality conditions which is defined as:

$$\begin{aligned} \left\{ \begin{array}{cl} \frac{dS_{h}}{dt}=& \Lambda _{h}-\frac{\alpha b(1-u_{1})I_{m}(t) S_{h}(t)}{N_{h}}-\mu _{h} S_{h}(t)\\ \frac{dE_{h}}{dt}=& \frac{\alpha b(1-u_{1})I_{m}(t) S_{h}(t)}{N_{h}}-(\mu _{h}+ \delta )E_{h}(t)\\ \frac{dI_{h}}{dt}=& \delta E_{h}(t)-(d(1-u_{3})+u_{3}\phi +\mu _{h}) I_{h}(t)\\ \frac{dR_{h}}{dt}=& \phi I_{h}(t)-\mu _{h} R_{h}(t)\\ \frac{dS_{m}}{dt}=& \theta (1-u_{2})-\frac{\beta b(1-u_{1})I_{h}(t) S_{m}(t)}{N_{h}}-(\mu _{m}+u_{2}\tau )S_{m}(t)\\ \frac{dE_{m}}{dt}=& \frac{\beta b(1-u_{1})I_{h}(t) S_{m}(t)}{N_{h}}-(\mu _{m} +\rho (1-u_{2})) E_{m}(t)\\ \frac{dI_{m}}{dt}=& \rho (1-u_{2})E_{m}(t) -\mu _{m} I_{m}(t)\\ \frac{d\lambda _{{1}} (t) }{{ dt}}=& \frac{( -I_{{m}} ( E_{h}(t ) +I_{h}( t) +R_{h}( t) ) \left( -1+u_{1} \right) b +\mu _{h}\,{N_{h}}^{2}) \lambda _{{1}}(t)}{N_{h}^2} +A_1\\ \frac{d\lambda _{{2}}( t) }{{ dt}}=& \frac{ \delta \,{N_h}^{2}+\mu _h\,( S_h( t)) ^{2}+ ( 2\,\mu _h\,I_h( t) +2\,\mu _h\,E_h ( t) +2\,\mu _{h}\,R_{h}( t) -bE_{{m}}( t) ( -1+u_{1})) S_{h}( t)}{N_{h}^2}+A_2\\ \frac{d\lambda _{{3}}(t) }{{ dt}}=& -{ w_{1}}+{\frac{\lambda _{{1}}(t) b( -1+u_{1}) S_{h}(t) I_{{m}}(t)}{{N_{h}}^{2}}}+A_3+\frac{\lambda _{6}(t) b(-1+u_{1}) S_{{m}}(t)(S_{h}(t) +E_{h}(t) +R_{h}(t))}{{N_{h}}^{2}} \\ \frac{d\lambda _{{4}}(t) }{{ dt}}=& {\frac{\lambda _{{4}}(t) \mu \,{N_{h}}^{2}+( -1+u_{1}) b(E_{{m}}(t)(\lambda _{{1}}(t) -\lambda _{{2}}( t)) S_{h}(t) +I_{h}( t) S_{{m}}(t)( \lambda _{{5}}(t) -\lambda _{{6}}(t)))}{{N_{h}}^{2}}}\\ \frac{d\lambda _{5}(t)}{{ dt}}=& -\lambda _{5}(t)(-{\frac{b(1-u_{1}) I_{h}(t)}{N_{h}}}-\tau \,u_{2}-\mu _{{m}})+{\frac{\lambda _{6}(t) b(-1+u_{1}) I_{h}(t) }{N_{h}}}\\ \frac{d\lambda _{6}(t)}{{ dt}}=& -{ w_{2}}-{\frac{\lambda _{{1}}(t) b( -1+u_{1}) S_{h}(t) }{N_{h}}}+{\frac{\lambda _{{2}}(t) b( -1+u_{1}) S_{h}(t) }{N_{h}}}+A_4\\ \frac{d\lambda _{{7}}(t)}{{ dt}}=& \mu _{{m}}\lambda _{{7}}(t) \end{array}\right. \end{aligned}$$

(54)

Where $A_1=\frac{( -1+u_{1}) b(( E_{{m}}( t) \lambda _{{2}} ( t) +( \lambda _{{5}}(t) -\lambda _{{6}}( t)) S_{{m}}( t)) I_{h}( t) +E_{{m}}( t) \lambda _{{2}}(t)( E_{h}( t) +R_{h}( t)) ) }{{N_{h}}^{2}}$,

$A_2=\frac{\mu _{h}\,(E_{h}( t) +I_{h}( t) +R_{h}( t)) ^{2} \lambda _{{2}} (t)-\lambda _{{3}}( t) \delta \,{N_{h}}^{2}+( -1+u_{1}) b( E_{{m}}( t) \lambda _{{1}} ( t) S_{h}( t)}{N_{h}^2}+ \frac{I_{h}(t) S_{{m}}(t) ( \lambda _{{5}}(t) -\lambda _{{6}}(t))) }{{N_{h}}^{2}}$,

$A_3={-\frac{\lambda _{{2}}(t) b( -1+u_{1}) S_{h}(t) I_{{m}}(t)- (( d-\phi ) u_{3}-d-\mu _{h}) \lambda _{{3}}(t) -\lambda _{{4}}(t) \phi }{{N_{h}}^{2}}} -\frac{\lambda _{{5}}(t) b( -1+u_{1}) S_{{m}}(t)(S_{h}(t) +E_{h}(t) +R_{h}(t)) }{{N_{h}}^{2}}$,

$A_4=\lambda _{{6}}(t)( \mu _{h}+ ( 1-u_{2}) \rho )+\lambda _{{7}}(t)( -1+u_{2}) \rho$ and $\lambda _{i}(t_{f})= 0,\, i=1, 2, 3, 4, 5, 6, 7$; with the characterization. Note that, this optimality system has initial value problem (system of state) with initial condition and boundary value problem (system of co-state) with transversality condition which is stated in Eqs. (50) and (53) respectively. Moreover, it has unique solution for final time $t_{f}$, as optimality system holds Liptshitz conditions and bounded. Thus, being $t\in [0, t_{f}]$ also ensures its uniqueness.

Strategies

The strategies that we employed were prevention(pesticide treated bed-net ($u_{1}$), and pesticide-spray($u_{2}$)), and treatment ($u_{3}$). To examine the impact of each control on eradication of dengue disease, we used the following strategy:

• a) Applying first prevention and treatment ($u_{1}$ and $u_{3}$) with infected class as interventions,

• b) Implementing treatment ($u_{3}$) with infected human and second prevention ($u_{2}$) with mosquito class as interventions,

• c) Applying first($u_{1}$) and second ($u_{2}$) preventions control with mosquito class as interventions,

• d) Using all controls, (i.e $u_{1}$ and $u_{2}$ are to kill mosquito) whereas $u_{3}$ to treat infected human.

Cost-effectiveness analysis

In this section, we identified a plan of action which is effective to use by all communities compared to other strategies in-terms of cost. To obtain this result, by using incremental cost-effectiveness ratio which is denoted by (ICER) is computed by dividing the difference of costs between two strategies to the difference of the total number of their infectious averted. We estimated the total number of dengue infectious averted for each strategy by subtracting total infectious with control from the total infectious without control.

To analyze cost effectiveness, we used world health organization recommended costs for intervention and coverages to obtain values in Table 3 in addition to the following formulae. Since the strategy is pairwise, its costs also computed as follow.

$$\begin{aligned} ICER(strategy)= \frac{\text {Difference in cost of strategies}}{\text {Difference in infection averted between strategies}} \end{aligned}$$

(55)

To estimate the variables in the Table 3, we assume baseline infections $10^4$ and calculate infections averted using combined intervention efficiencies. Costs are based on per-person intervention expenses, and cost averted considers net savings (averted infection costs minus intervention costs).

Table 3 Number of infectious averted and total cost of each strategies.

First we compared the cost effectiveness of strategy (1),

$$\begin{aligned} ICER(1)=\frac{10,901,345,000}{6,500}=1.67713\times 10^{6}\$. \end{aligned}$$

Now we compute strategy (1) and (2) as,

$$\begin{aligned} ICER(2)=\frac{145,340,000-10,901,345,000}{7,500-6,500}=-1.0756005\times 10^{7}\$. \end{aligned}$$

This indicates that, strategy (2) is cheaper than strategy (1), which means strategy (2) is cheaper and more effective in infection avert. Therefore, we exclude strategy (1) and continue to compare strategy (2) and (3).

$$\begin{aligned} ICER(3)=\frac{12,926,387,500-145,340,000}{7,200-7,500}=-4.2603492\times 10^{7}\$. \end{aligned}$$

In this case, the total cost of strategy (3) is expensive but less effective in in infection avert, hence we reject this strategy and proceed with strategy (2).

$$\begin{aligned} ICER(4)=\frac{12,968,215,000-145,340,000}{8,400-7,500}=1.4247639\times 10^{7}\$ \end{aligned}$$

Lastly, the comparison results indicate that Strategy (2) is the most cost-effective option. Therefore, Strategy (2) which combines second prevention with treatment emerges as the best choice, given its low cost and health benefits, as our calculations demonstrate in comparison to all other strategies. While the combination of all three interventions is effective, it proves to be more expensive, as indicated by its ICER value. Thus, utilizing the strategy that incorporates second prevention and treatment is the most effective approach among all the combined intervention strategies.

Numerical simulation

Simulations were carried out using an Intel dual-core processor, 32GB of RAM, and Windows 10. MATLAB R2013a, employing the RK45 method,were used to visualize the stated objective those that mentioned theoretically with good resolution. The detail results of graphical representations with its theoretical descriptions were discussed as follow.

The phase portrait of the dengue model of Fig. 5 illustrates the dynamic relationship between the number of infected humans and mosquitoes over time. The trajectory indicates that initially, as human infections increase, mosquito infections also rise, reflecting the transmission cycle. The loop in the phase plane suggests the presence of oscillations or cyclical behavior, which could correspond to seasonal variations or other factors influencing disease spread. Overall, the model highlights the interconnected and cyclical nature of dengue transmission between humans and mosquitoes.

Fig. 5

The global solution of infected class of the model.

Figure 6a and 6b show the application of seasonal change to the model, amplifying infected mosquitoes followed by infected humans, and how fluctuations in season potentially affect the dengue transmission rate, respectively.

Fig. 6

Impact Seasonal variation on transmission rate of dengue disease.

Figure 7a shows that for $R_0 < 1$, the trajectories of infected individuals with different $R_0$ values all decay to zero over time. Conversely, the number of susceptible individuals increases with time, as revealed in Fig. 7b. These observations confirm that $E_0$ is globally stable.

Fig. 7

Illustration of global stability of DFE for different $R_0 < 1$.

Figure 8a illustrates that for $R_0> 1$, the trajectories of infected individuals with different $R_0$ values all tend to some point (EE) as time increases. Conversely, the number of susceptible individuals decreases substantially due to the presence of infection, as shown in Fig. 8b. Ultimately, all trajectories converge to a positive constant(EE). This implies that EE is globally stable.

Fig. 8

Demonstration of global stability of EE for different $R_0> 1$.

Results

This study highlights how climate change enhances mosquito populations and virus development, thereby increasing dengue transmission risk. Key parameters influencing the epidemic include the infection probability from humans to mosquitoes ($\beta$), from mosquitoes to humans ($\alpha$), and mosquito recruitment rate ($\theta$), with the biting rate (b) being the most critical. Higher temperatures (21-$30^{\circ }\text {C}$) boost mosquito activity, breeding, and development, peaking at $30^{\circ }\text {C}$, while rainfall up to 30 mm improves mosquito survival; heavier rains can reduce populations through flushing. Temperature also raises transmission probabilities and shortens the extrinsic incubation period, accelerating virus spread. Moreover, seasonal change also potentially elevate the disease transmission rate. Regional differences affect transmission dynamics, with Dire Dawa showing higher mosquito growth potential followed by Somali, and Afar exhibiting higher biting rate and incubation rates but also higher mosquito mortality. Associating the overall climate-based output with $R_0$, we conclude that Somali has the most conducive conditions for dengue virus transmission. Overall, climate factors significantly influence mosquito ecology and dengue risk, and targeted interventions like pesticide-treated bed nets combined with treating infected individuals are identified as cost-effective control strategies.

Discussion

This section explores how climate variables influence dengue transmission dynamics, utilizing an advanced deterministic model that integrates climate-dependent entomological parameters. Dengue transmission is highly sensitive to weather fluctuations such as temperature and rainfall³³. These climatic factors critically affect the development of the dengue pathogen and the life cycle of its mosquito vector, thereby elevating the risk for human populations in contact with infected mosquitoes. Communities experiencing variable weather patterns may face challenges in accessing essential infrastructure like piped water, often leading to temporary settlements and suboptimal environments⁵.

Our sensitivity analysis highlights several key parameters that significantly impact dengue spread, including the probability of a bitten individual becoming infected ($\alpha$), the likelihood that a mosquito biting an infected person becomes infectious ($\beta$), and the mosquito recruitment rate ($\theta$). These factors collectively account for roughly 50% of the epidemic acceleration. Moreover, the biting rate (b) which depends on interactions between female mosquitoes and humans, as well as the success rate of bites emerges as the most influential parameter in transmission dynamics. Parameters such as $\mu _m$, $\Lambda _h$, and $\phi$ exhibit inverse relationships: increasing these parameters reduces disease prevalence. Conversely, parameters like $\delta$, $\rho$, and d demonstrate lower sensitivity. Our findings suggest that, aside from climatic factors, an increase in the biting rate correlates with a higher incidence of new infections among susceptible individuals. Importantly, the disease-free equilibrium remains locally stable when the basic reproduction number ($R_0$) is below 1, provided effective mosquito control measures are in place; otherwise, stability is lost. These theoretical insights are supported by detailed simulation results.

In order to validate the model, we fit the model into existing dengue cases, which is depicted in Fig. 4. As a result some crucial statistical tools were computed. As illustrated in Fig. 9b, increasing temperatures between $21^o\hbox {C}$ and $30^o\hbox {C}$ correlate with heightened mosquito biting rates. As noted by³⁴ elevated temperatures accelerate mosquito development, shorten growth cycles, and increase adult mosquito density, compelling female mosquitoes to seek more frequent blood meals, thus heightening transmission risks. At $30^o\hbox {C}$, we observe peak mosquito populations, ideal for breeding. This surge in biting rates aligns with findings from³⁵. Additionally, elevated temperatures expedite egg hatching, pathogen proliferation, and increased biting activity, as shown in Figs. 9a and 9b.

Fig. 9

Impact of temperature on hatching success of eggs and biting rate of adult mosquito.

Figures 10a and 10b demonstrate that as rainfall increases, the hatching success of eggs as well as the survival rates of both larvae and pupae also rise. However, when rainfall exceeds 30 mm, there is a slight decline in these populations due to heavy rain flushing out the immature stages. Similarly, Figs. 11a and 11b highlight stochastical rainfall’s impact on mosquito larvae and pupae survival, which are crucial for adult female mosquito populations and directly influence dengue transmission dynamics.

Fig. 10

Impact of rainfall on hatching success, survival rate of larva and pupa.

Fig. 11

Simulation of rainfall via mosquito’s eggs and pupae.

Figure 12a shows that as temperature increases from $12.5^o\hbox {C}$ to $26.5^o\hbox {C}$, the probabilities of dengue transmission, denoted as $\alpha$ and $\beta$, also increase. Notably, the probability of transmission from mosquito to human is particularly high within the temperature range of $26.5^o\hbox {C}$ to $32^o\hbox {C}$. Conversely, as the biting rate of infected mosquitoes rises, the number of infected humans also increases, as illustrated in Fig. 12b.

Fig. 12

Simulation of mosquito biting rate and disease transmission probabilities.

Table 2 presents values of the parameters $H_s$, $S_p$, $S_l$, and $\theta$ for three regions: Dire Dawa, Afar, and Somali. Dire Dawa exhibits $H_s=0.607$, $S_p=0.329$, and $S_l=0.0334$, resulting in $\theta =0.151$. In contrast, Afar shows lower values: $H_s=0.0877$, $S_p=0.473$, and $S_l=0.04702$, with the lowest $\theta =0.045$. Somali demonstrates intermediate values: $H_s=0.1297$, $S_p=0.549$, and $S_l=0.05767$, with $\theta =0.090$, indicating that Dire Dawa possesses the most favorable conditions based on this metric. Variations in these parameters reflect differing environmental efficiencies across regions.

Analysis of mosquito mortality rates indicates that Afar experiences the highest death rate (0.023/day), followed closely by Dire Dawa (0.022/day), while Somali has the lowest (0.021/day). Correspondingly, the average lifespan of Aedes mosquitoes in these regions is approximately 45.45, 43.48, and 47.62 days, respectively. These differences suggest ecological factors influencing mortality vary regionally, which is critical for designing targeted control strategies.

Afar exhibits the highest biting rate (0.3389), with Dire Dawa (0.3269) and Somali (0.3168) following. Despite all regions being vulnerable to dengue, Afar’s elevated biting rate indicates a higher transmission potential during peak periods. The incubation completion rate is highest in Afar (about 0.186), implying an extrinsic incubation period of approximately 5.378 days, which is shorter compared to other regions. This aligns with³⁶, where increased temperatures are shown to shorten the virus’s extrinsic incubation period, thereby enhancing transmission.

The probability of dengue transmission from humans to mosquitoes varies regionally: $\alpha (T)$ is 0.967 in Dire Dawa, 0.957 in Afar, and 0.972 in Somali. Meanwhile, the probability of transmission from mosquitoes to humans remains at 1 within the temperature range of $26.1^o\hbox {C}$ to $32.5^o\hbox {C}$ across all regions.

To curb the spread of dengue influenced by climate factors, optimal control strategies are implemented as illustrated below. Figures 13a and 13b demonstrate the impact of control measures on mosquito populations. Similarly, Figs. 14a and 14b show the effectiveness of insecticide-treated bed nets in combination with treatment. On the other hand, Fig. 15a indicates the effects of applying prevention measures alone, while Fig. 15b illustrates the application of all control strategies to achieve a dengue-free community.

Fig. 13

Effect of control on mosquito populations.

Fig. 14

Impact of Preventions on infected dengue transmission.

Fig. 15

Impact of control on Dengue Transmission.

Overall, our findings show that optimal temperatures and rainfall greatly influence mosquito populations. Somali provides ideal conditions for larval and pupal survival and hatching success. Afar exhibits higher temperature-dependent parameters like biting and development rates. Rising temperatures shorten the extrinsic incubation period, boosting viral transmission. However, increased temperatures can also raise mosquito mortality, reducing lifespan. Somali’s average mosquito lifespan is about 47.62 days. Climate change may worsen mosquito-borne disease spread, especially in tropical regions. Rainfall between 30-40 mm favors pupal survival; proper drainage and removal of standing water are key control measures.

Conclusion

In conclusion, our comprehensive analysis underscores the crucial role of climate variables, particularly temperature and rainfall, in shaping the dynamics of dengue transmission. By employing an advanced deterministic model, we illuminated how these environmental factors influence the life cycles of both the dengue virus and its mosquito vectors. The findings indicate that regions like Somali exhibit optimal conditions for mosquito survival and proliferation, while Afar presents significant potential for high transmission rates due to its favorable biting patterns. Furthermore, our results reveal that as temperatures rise, the extrinsic incubation period of the virus shortens, enhancing dengue transmission risks a midst varying rainfall patterns. The proposed model was also fitted to existing dengue cases to validate the model and its parameters, ensuring the accuracy of the results. These insights highlight the need for targeted public health strategies that incorporate environmental monitoring and proactive mosquito control measures. Therefore, our optimal control strategy suggests managing transmission effectively, and recommends the use of combined pesticide-treated bed nets with treatment as a cost-effective solution. By addressing the impacts of climate change on mosquito-borne diseases, we can better mitigate the public health challenges posed by dengue and improve the resilience of affected communities. This work is limited to focusing solely on temperature and rainfall as determinant climate factors. However, there are other factors that we did not consider. For instance, the influence of natural events such as volcanic eruptions, decomposition of organic matter, and animal respiration, as well as human activities like deforestation and the operation of factories that release carbon dioxide and sulfur dioxide, which contribute to the growth of mosquito populations and the development of pathogens. Hence, a multidisciplinary approach that engages environmental scientists, epidemiologists, and public health officials will be essential to combat dengue fever effectively in an increasingly variable climate.