Characterization of the mechanical properties and thermal conductivity of epoxy-silica functionally graded materials

1.   Introduction

Dengue fever, which is caused by the dengue virus, is known as one of the most important tropical diseases. There is a circle of dengue virus transmission that concerns humans and mosquitoes; therefore, dengue fever is also called a mosquito-borne disease. The spread of the virus occurs through the bite of an infected female Aedes Aegypti mosquito. So far, four serotypes have been detected for the dengue virus: DEN1, DEN2, DEN3, and DEN4 ^[1]. Humans can be immune to one serotype but can be reinfected by another serotype. Thus, humans living in areas endemic to dengue disease can be reinfected 3 or 4 times by viruses with different serotypes. Since the period of mosquito infection ends through death, the mosquitoes never recover from the infection ^[1]. The virus has an incubation period of 4 to 10 days for the mosquito, during which, it becomes an infected vector. Suppose the infected mosquito bites a human who is susceptible to the virus. In that case, the virus will enter the human body until it can be transmitted back to another mosquito. Infected humans can transmit the virus through the Aedes Aegypti mosquito for 4 to 5 days after their first symptoms appear ^[2].

Every year, approximately 100–400 million people are infected by the dengue virus ^[3]. The COVID-19 pandemic more or less changed the dynamics of dengue fever cases. Medical reports which focused on this outbreak showed that the dengue disease had lower reporting rates. Dengue fever cases experienced a decline between 2020–2022 (during the COVID-19 pandemic). As a result, in 2023, the dengue fever cases significantly increased globally. In particular, Indonesia has become one of the Southeast Asian countries, ranked among the world's 30 most susceptible to an endemic ^[4]. This condition becomes a challenge for researchers to reduce the endemicity of dengue fever that occurs year to year.

Understanding the transmission of disease from a mathematical perspective involves the use of mathematical models. Large-scale data-based simulations can be conducted using mathematical models to estimate the spread of epidemics ^[5,6]. However, it is also crucial to theoretically study epidemic models to understand their qualitative behavior. This understanding is valuable to inform the control policies. Disease epidemics have another point of view in memory and can have wide-ranging effects in the fractional models using integer order, as studied by ^[7,8]. In particular, mathematical modeling concerning the cycle of transmission of the dengue virus has been explored, such as in the host-vector model by the author in ^[9,10,11,12]. The author in ^[10] modified the model in ^[9] regarding the incubation period, that is, a percentage of infected humans who are incapable of transmitting the disease. Based on ^[10], the authors in ^[11] considered mathematical modeling to predict and control dengue transmission. Furthermore, the optimal control by ^[12] was performed to take precautions. Utilizing the optimal control theory to model the spread of disease is valuable for researchers and policymakers. This approach helps to explore evidence-based disease prevention and control strategies. The findings from this analysis can contribute to more impactful public health interventions and improved outcomes for populations affected by infectious diseases ^[13,14]. Optimal control within the host-vector dengue model is often used to assess the effectiveness of various intervention strategies ^[15,16].

Optimal control in dynamical systems is one of the mathematical theories that elaborate on determining the control policy to achieve certain optimality criteria ^[17]. The optimality procedures involve assigning manipulation system inputs or controls over time to achieve the intended behavior while minimizing or maximizing the objective function. Some notes that must be provided in the theory of optimal control in dynamic systems are control variables, objective functions, constraints, and optimality conditions. For continuous time problems, Pontryagin's Maximum Principle provides the necessary conditions for optimal solutions ^[18]. It involves Hamilton functions, adjoint variables, and control laws.

Different from the previous studies, in this study, we analyze the mathematical model of dengue virus transmission to control the infected class. Due to the high death rate of dengue fever, we modify the mathematical model in ^[11] by adding a death rate parameter. There is a possibility that someone will be reinfected with another serotype of the dengue virus; therefore, we enhance a parameter to represent this condition. Additionally, we develop the analysis of the model in ^[11] by discussing the global stability for both disease-free and endemic equilibrium points. The most commonly used methods for global stability analyses are the Lyapunov stability theorem and LaSalle's invariance principle (see, for example, ^[19,20]). However, it is often challenging to construct Lyapunov functions, since there is no general method available. Remarkable mathematical models can contribute to reducing both the infected host and the infected vector population. It is very promising to use host-vector models from a mathematical perspective to answer this.

There are five main findings from this work. First, we develop a new mathematical model that includes deaths due to the dengue virus and considers reinfection of the dengue virus to a person. Second, a global stability analysis is performed using the tricky Lyapunov function. Third, a sensitivity analysis is presented to understand the parameters that influence the spread of the virus. Fourth, the model is developed to provide control over the virus transmission. Fifth, through theoretical and numerical analyses, we can conclude that the presence of control makes the solution behavior move to a disease-free state more quickly.

2.   Formulation of the model

In this study, we divide the population into the host and the vector. The total host (human) population is denoted by $N_H$. The host population itself has three classes: susceptible, infected, and recovered at time $t$. Meanwhile, the vector population is divided into two classes: susceptible and infected at time $t$. We denote $N_V$ as the total vector (mosquito) population. The host population on the susceptible class at $t$ is denoted by $S_H (t)$, the host population that is infected at time $t$ is denoted by $I_V (t)$, and the host population that is recovered at time $t$ is denoted by $R_H(t)$. Furthermore, we denote $S_V(t)$ to represent the susceptible vector population at time $t$, and $I_V(t)$ to denote the infected vector population at time $t$.

We use the notation $\mu_H$ to represent the birth rate of the host population. In this model, we assume a closed host population; thus, the birth rate is equal to the natural death rate. Due to the high death rate due to dengue fever, we modify the mathematical model in ^[11] by adding the death rate of the host population caused by the virus, which is denoted by $\mu$. If the number of mosquito bites per day is denoted by $b$, then each day there will be $bN_V$ mosquito bites. Therefore, the host will receive $\frac{bN_V}{N_H}\frac{I_V}{N_V}$ infected mosquito bites per day. If $\beta_H$ denotes the transmission rate from the mosquito to the host, then $\beta_H\frac{bN_V}{N_H}\frac{I_V}{N_V}$ denotes the infection rate of the susceptible hosts. We use $\alpha$ to represent the susceptible host, who can be susceptible again after recovering from dengue fever. Furthermore, we denote $\gamma_H$ as the recovery rate.

The population of the vector has a growth rate as much as $A$, meanwhile $\mu_V$ denotes the mortality rate for the vector. Since a mosquito will bite $\frac{b}{N_H}$ per day per person, then one mosquito will get the virus by $\frac{b}{N_H} I_H$ per day. If the transmission rate from the host to the mosquito is denoted by $\beta_V$, then the infection rate per susceptible vector is $\beta_V \frac{b}{N_H} I_H$. Figure 1 portrays the mathematical modeling.

The mathematical model for dengue virus transmission is a complex system that takes various factors that contribute to the spread of the virus into account, which can be described as follows:

where ${{S}_{H}}+{{I}_{H}}+{{R}_{H}} = {{N}_{H}}$ and ${{S}_{V}}+{{I}_{V}} = {{N}_{V}}$.

To ensure that the model solution reflects the real-world population, it is important to determine the non-negative solutions of System (2.1). Because the negative populations are not feasible, we prove the non-negativity of System (2.1) solutions by Theorem 2.1.

Theorem 2.1. Solutions of System (2.1) in the set of a vector of a non-negative real number $\mathbb{R}^5_+$ are always non-negative for all time.

Proof. Let the solutions of System (2.1) rely on $\left[ 0, \tau \right)$, where $0 < \tau < \infty$. Based on the first equation of System (2.1), let ${{S}_{H}}(t) > 0$ for every $t\in \left[ 0, \tau \right)$. If it is not satisfied, then there exist ${{t}_{1}}\in \left(0, \tau \right)$ such that ${{S}_{H}}\left({{t}_{1}} \right)\le 0$; therefore, ${{\left. \frac{d{{S}_{H}}(t)}{dt} \right|}_{t = {{t}_{1}}}}\le 0$, and we have ${{S}_{H}}(t) > 0$ for every $t\in \left[ 0, {{t}_{1}} \right)$. We do the same for the second equation of System (2.1). Let ${{I}_{H}}(t)\ge 0$ for every $t\in \left[ 0, {{t}_{1}} \right)$. If it is not held, then there exist ${{t}_{2}}\in \left(0, {{t}_{1}} \right)$ such that ${{I}_{H}}\left({{t}_{2}} \right) < 0$; therefore, ${{\left. \frac{d{{I}_{H}}(t)}{dt} \right|}_{t = {{t}_{2}}}} < 0$, and we have ${{I}_{H}}(t)\ge 0$ for every $t\in \left[ 0, {{t}_{1}} \right)$.

Furthermore, for the third equation of System (2.1), we claim that ${{R}_{H}}(t)\ge 0$ for every $t\in \left[ 0, {{t}_{2}} \right)$ if ${{\left. \frac{d{{R}_{H}}(t)}{dt} \right|}_{t = {{t}_{3}}}} < 0$. To prove this statement, we use reductio ad absurdum. Suppose that there exist ${{t}_{3}}\in \left[ 0, {{t}_{2}} \right)$ such that $R_H(t_3) < 0$. Then, we obtain the following:

since $R_H(t_3) < 0$, $\mu_H > 0$, and $\alpha > 0$. It is a contradiction with the fact that ${{\left. \frac{d{{R}_{H}}(t)}{dt} \right|}_{t = {{t}_{3}}}} < 0$. It implies that if ${{\left. \frac{d{{R}_{H}}(t)}{dt} \right|}_{t = {{t}_{3}}}} < 0$, then ${{R}_{H}}(t)\ge 0$ for every $t\in \left[ 0, {{t}_{2}} \right)$.

In a similar way, we can observe that $S_V > 0$ and $I_V\ge0$. Now, we come to the first equation of System (2.1); if ${{\left. \frac{d{{S}_{H}}(t)}{dt} \right|}_{t = {{t}_{1}}}} < 0$, then ${{S}_{H}}(t) > 0$ for every $t\in \left[ 0, \tau \right)$. Suppose that there exist ${{t}_{1}}\in \left(0, \tau \right)$ such that ${{S}_{H}}\left({{t}_{1}} \right)\le 0$,

It is a contradiction to our sufficient condition that ${{\left. \frac{d{{S}_{H}}(t)}{dt} \right|}_{t = {{t}_{1}}}} < 0$; therefore, our supposition should be ${{S}_{H}}(t) > 0$ for every $t\in \left[ 0, \tau \right)$.

Using the same manner, we obtain that $I_H\ge0$. Hence, all of the solutions of System (2.1) are non-negative for all the time $t$ on $\left[ 0, \tau \right)$, where $0 < \tau < \infty$. □

The intuition of the boundedness of the solution can be explained as follows. We note that ${{N}_{H}}$ remains constant. Furthermore, since ${{\mu }_{V}}$ denotes the per capita death rate of the vector population, then ${{\mu }_{V}}{{N}_{V}}$ is the total deaths of the vector. The rate of change of the vector population is represented by the following:

As $t$ approaches infinity, ${{N}_{V}}$ approaches $\frac{A}{{{\mu }_{V}}}$. Thus, ${{S}_{V}}+{{I}_{V}} = \frac{A}{{{\mu }_{V}}}$. Therefore, for ${{N}_{H}}$ and ${{N}_{V}}$, which are constant, and for convenience, we introduce the following transformation:

Additionally, we use the relation ${{r}_{H}} = 1-\left({{s}_{H}}+{{i}_{H}} \right)$ and ${{s}_{V}} = 1-{{i}_{V}}$. Using transformation (2.2), System (2.1) can be written as follows:

In the next section, we examine the equilibrium points of System (2.3) and their stability. Generally, we obtain two types of equilibrium: disease-free and endemic. The local stability of the equilibrium can be studied through the value of the basic reproduction number. Meanwhile, La-Salle Lyapunov can be applied to analyze the global stability of the equilibrium point.

3.   Equilibrium points and their stability

The equilibrium points of System (2.3) will be explained in the following discussion. For convenience, we use the relation (3.1) for our analysis. Let

3.1.   Disease-free equilibrium point

The equilibrium solutions are determined by setting each equation in System (2.3) equal to zero. Based on (2.3a), (2.3c), and (3.1), we obtain the following:

By substituting Eq (3.2) into (2.3b), then we have that ${{i}_{H}}$ must satisfy the following quadratic equation:

Therefore, the disease-free equilibrium point can be summarized by the following lemma.

Lemma 3.1. If $i_H = 0$, then System (2.3) has a disease free equilibrium point, ${{E}_{0}} = \left(1, 0, 0 \right)$.

Proof. One of the solutions of (3.3) is $i_H = 0$. By taking the right-hand side of (2.3c) to zero and substituting $i_H = 0$, we come to have $i_V = 0$. Moreover, since $i_H = 0$ and $i_V = 0$, then we obtain $s_H = 1$ by taking zero to the right-hand side of (2.3a). □

Now, we will discuss the basic reproduction number that indicates the value of secondary infections resulting from the first infection. This value is determined using the next-generation matrix method ^[21]. In this study, matrix $F_1$ is defined as the matrix whose entries consist of the first derivative of all terms that represent the rate of change of all factors contributing to the infectious class at the disease-free equilibrium $E_0$. Therefore,

We create the matrix $F_2$ as a matrix of loss terms for each class, which is evaluated at the disease-free equilibrium as follows:

The inverse form of (3.5) is as follows:

Based on Matrices (3.4) and (3.6), we obtain Matrix $P$, which is a product of $F_1$ and $F_2^{-1}$; then,

The largest eigenvalue of the matrix $P$ is as follows:

If $R_0$ is a notation for this value, then, in this study, $R_0$ is referred to as the basic reproduction number of System (2.3). Furthermore, if

then

This can be interpreted as the average number of secondary cases in the susceptible population caused by a single case. In a detailed biological interpretation, this can be seen as follows: an infected host that exists in the susceptible population is bitten during his/her infective period by $\frac{b(A/{\mu_V})}{N_H (\mu_H + \mu + \gamma_H)}$ susceptible vectors. This results in a portion $\frac{\beta_V b(A/{\mu_V})}{N_H (\mu_H + \mu + \gamma_H)}$ of the vectors becoming infected. On the other hand, an infected vector administers $\frac{b}{\mu_V}$ of bites to the susceptible hosts. These bites lead to a portion $\frac{\beta_H b}{\mu_V}$ of new infections to the host population.

The next section will describe the stability near the disease-free equilibrium point.

3.2.   Stability analysis of disease-free equilibrium point

We describe the stability analysis of the disease-free equilibrium in Theorem 3.1.

Theorem 3.1. Given $R_0$ in (3.9):

$i.$ If $\hat{R}_0 > 1$, then $E_0$ is unstable.

$ii.$ If $\hat{R}_0 < 1$, then $E_0$ is locally asymptotically stable.

Proof. The Jacobian matrix at ${{E}_{0}}$ is as follows:

If $\lambda$ denotes the eigenvalue and $I$ denotes the identity matrix, then the eigenvalues of the matrix (3.10) are determined by solving the characteristic equation $\left| J\left({{E}_{0}} \right)-\lambda I \right| = 0$. It implies that

Therefore, ${{\lambda }_{1}} = -\left({{\mu }_{H}}+\alpha \right)$ and

where $\hat{R}_0$ is written as in (3.8). Thus,

$i.$ If ${{\hat{R}}_{0}} > 1$, then one of the eigenvalue in (3.11) is positive. Since we have a positive eigenvalue, then ${{E}_{0}}$ is unstable.

$ii.$ If ${{\hat{R}}_{0}} < 1$, then the eigenvalues in (3.11) are negative. Since all the eigenvalues are negative, then ${{E}_{0}}$ is locally asymptotically stable.

□

The stability of the disease-free equilibrium point can be interpreted whenever ${{\hat{R}}_{0}} < 1$; then, there is no virus transmission in the population, and all humans are free of a dengue virus infection. Next, we analyze the global stability of the disease-free equilibrium points using La Salle Lyapunov ^[22]. In this study, we consider the Lyapunov function $V:\mathbb{R}_{+, 0}^3 \to \mathbb{R}^3$, where $\mathbb{R}_{+, 0}^3 = \left\{(s_H, i_H, i_V)\in \mathbb{R}^3:s_H > 0, i_H, i_V\ge0\right\}$, and

It is easy to show that $V$ and its partial derivatives are continuous in $\mathbb{R}^3$. Additionally, $V$ is positive definite. The orbital derivative of function (3.12) is determined by the following:

Therefore, we achieve the global stability of the disease-free equilibrium point in Theorem 3.2.

Theorem 3.2. If ${{\hat{R}}_{0}} < 1$, then the disease-free equilibrium point is globally asymptotically stable.

Proof. If ${{\hat{R}}_{0}} < 1$, then we have $\dot{V} < 0$. We also note that $\dot{V}\left({{E}_{0}} \right) = 0$. Therefore, based on La Salle Lyapunov, the disease-free equilibrium $E_0$ is globally asymptotically stable. □

3.3.   Existence and local stability analysis of endemic equilibrium point

Recall the solution of Eq (3.3). We obtained $i_H = 0$, as the solution coincides with the disease-free equilibrium point; therefore, the other solution, $i_H\ne 0$, corresponds to the endemic equilibrium point. Based on (3.3), we obtain the following:

The solution ${{i}_{H}} > 0$ in (3.13) exists if ${\hat{R}}_{0} > 1$. Consequently, the existence of the endemic equilibrium point can be concluded through the following Lemma 3.2.

Lemma 3.2. If ${\hat{R}}_{0} > 1$, then the endemic equilibrium point exists.

Proof. Note that ${\hat{R}}_{0} > 1$. The solution in (3.13) can be substituted into (3.2); then, we acquire the endemic equilibrium point, ${{E}_{1}} = \left(s_{H}^{*}, i_{H}^{*}, i_{V}^{*} \right)$, where

□

Furthermore, a stability analysis of the endemic equilibrium point can be explained by Theorem 3.3.

Theorem 3.3. If ${\hat{R}}_{0} > 1$, then the endemic equilibrium point $E_1$ is locally asymptotically stable.

Proof. We consider that $\hat{R}_0 > 1$. Based on this and (3.1), we derive the Jacobian matrix at $E_1$ as follows:

where

The eigenvalues of matrix (3.14) are determined by solving the characteristic equation $\left| J\left({{E}_{1}} \right)-\lambda I \right| = 0$, where $\lambda$ denotes the eigenvalues; we solve $\lambda$ in (3.15) as follows:

Solving (3.15) means that we solve

where

In this case, if ${{\hat{R}}_{0}} > 1$, then it implies ${{Q}_{1}}, {{Q}_{2}}, {{Q}_{3}} > 0$. By manipulating the algebra, we also obtain that ${{Q}_{1}}, {{Q}_{2}} > {{Q}_{3}}$. Based on the Routh Hurwitz criterion, the endemic equilibrium point ${{E}_{1}}$ is locally asymptotically stable. □

3.4.   Global stability of the endemic equilibrium point

The discussion in this article continues regarding the global stability around the endemic equilibrium point, particularly when $\alpha = 0$ represents a population with permanent immunity. The dynamic model for the spread of dengue fever is as follows:

The endemic equilibrium $\left(s_{H}^{**}, i_{H}^{**}, i_{V}^{**} \right)$ satisfies the following:

Let the function

and note that $g\left(x \right) = x-1-\ln x\ge g\left(1 \right) = 0$ for $x > 0$. We introduce ${{y}_{1}} = \frac{{{s}_{H}}}{s_{H}^{**}}, {{y}_{2}} = \frac{{{i}_{H}}}{i_{H}^{**}}$, and ${{y}_{3}} = \frac{{{i}_{V}}}{i_{V}^{**}}$. The derivative of $W$ along the solution (3.16), and using (3.17), we obtain the following:

If (3.19)–(3.21) are substituted into (3.18), then we have the following:

where we use the inequality $1-x\le -\ln x$ and $g\left(x \right)\ge 0$. Therefore, we come to the global stability of the endemic equilibrium point in Theorem 3.4.

Theorem 3.4. If ${{\hat{R}}_{0}} > 1$, then the endemic equilibrium point is globally asymptotically stable.

Proof. Based on Lemma 3.2, the endemic equilibrium exists if ${\hat{R}}_{0} > 1$. Moreover, based on (3.22), we have $\dot{W}\le 0$. Furthermore, $\dot{W} = 0$ is satisfied at $\left(s_{H}^{**}, i_{H}^{**}, i_{V}^{**} \right)$. The claim follows by LaSalle's Invariance Principle ^[22]. □

4.   Sensitivity analysis

A sensitivity analysis explains the impact of changes in the parameter values on the system behavior. We study the sensitivity of the basic reproduction number to determine the robustness of the model to the parameter values measured by the sensitivity index. Next, the sensitivity index is defined as follows.

Definition 4.1. ^[23,24] The sensitivity index of variable $y$ is the relative change ratio of $y$ to the relative change of parameter $s$, and is defined as follows:

We analyze the sensitivity index of $\hat{R}_0$ using the formula (4.1), especially for the parameters $\beta_V$, $\beta_H$, $b$, $\mu$, and $\alpha$ that influence $R_0$. The sensitivity index of $\hat{R}_0$ can either depend on the values of other parameters or be a constant that does not depend on the parameter values. Based on Definition 4.1 and Eq (3.9), we obtain the following:

Based on the signs of the sensitivity index, the value of $\hat{R}_0$ increases by $\beta_V$, $\beta_H$, and $b$. On the contrary, it decreases by $\mu$. Moreover, $\alpha$ does not impact the value of $\hat{R}_0$. We note that the most sensitive parameter corresponds to the largest sensitivity index. In this case, $b$ is the most sensitive parameter; an increase of $b$ by $10\%$ increases $\hat{R}_0$ by $10\%$. Recall that in this case, $b$ represents the number of mosquito bites per day. Based on the sensitivity index, the value of $\hat{R}_0$ decreases if the value of $b$ decreases. Biologically, the value of $b$ can be a target to reduce as the intervention of dengue transmission. We can reduce the value of $b$ by reducing the presence of mosquitoes. In addition, increasing the value of $\mu$ decreases $\hat{R}_0$. Changing the value of an individual who can be re-infected cannot alter the value of $\hat{R}_0$.

5.   Optimal control

This section will delve deeper into the dengue virus transmission and discuss how it can be controlled using the prevention control variable. We denote ${{u}_{1}}$, where $0\le {{u}_{1}}\le 1$, as a control variable to reduce the contact between the host and the vector. We have derived the following model:

To investigate the optimality of these efforts, we utilize the objective function $G$, which minimizes the infected host class and the cost of implementing the ${{u}_{1}}$ control. In this study, if $B$ is the weight for the infected hosts class, and $C$ is the weight for prevention efforts, then the solution of (5.1) minimizes the number of infected hosts with a minimal control cost ${{u}_{1}}\left(t \right)$ such that

To further elaborate, the Hamiltonian of this study is as follows:

where ${{\delta }_{1}}, {{\delta }_{2}}, ..., {{\delta }_{5}}$ are costate variables. We use the Maximum Pontryagin principle to determine the existence of an optimal control ^{[17,18,24,25]}. The following theorem provides an idea.

Theorem 5.1. If $u_{1}^{*}$ is the optimal control that minimizes $G\left({{u}_{1}} \right)$ with System (5.1) as the constraint, then there exist costate variables ${{\delta }_{1}}, {{\delta }_{2}}, ..., {{\delta }_{5}}$ such that

with transversality conditions ${{\delta }_{i}}\left(T \right) = 0, i = 1, ..., 5$, and the $u_{1}^{*}$ satisfies the optimality condition

Proof. The costate equations are obtained by differentiating the Hamiltonian function, evaluated at the optimal control, as follows:

The optimal control at the solutions $S_{H}^{*}, I_{H}^{*}, R_{H}^{*}, S_{V}^{*}, I_{V}^{*}$ at System (5.1) is obtained by the following:

and

□

6.   Numerical simulation

To illustrate the analytical results, in this simulation, we select parameter values based on prior research, see ^[9,11] and the references cited therein. We use the following parameters: $\alpha = 0, 9619$; ${{\mu }_{V}} = 0, 071428$; $b = 0, 5; {{\beta }_{V}} = 1; {{\beta }_{H}} = 0, 75$; ${{\mu }_{H}} = 0, 3\text{91389}; \mu = 0, 8$; and ${{\gamma }_{H}} = 0, 1428$. Using these parameter values, using $A = 400$, and ${{N}_{H}} = 1000$, we obtain ${{\hat{R}}_{0}}\approx 22,036 > 1$. On the other hand, if we use $A = 40;{{N}_{H}} = 10000$, then we have ${{\hat{R}}_{0}}\approx 0, 22 < 1$.

Figure 2 shows the illustration of Theorems 3.1 and 3.3. Using different initial values in Figure 3 (i.e., (0, 45;0, 22;0, 3), (0, 5;0, 25;0,165), and (0, 4;0, 24;0, 2) shows us that the solutions tend to the disease-free equilibrium point. On the other hand, we aim to reduce the number of infected humans and infected vectors in the population. Therefore, the endemicity should be controllable. Figure 4 showcases a dynamic tornado plot which illustrates the parameter sensitivity index. We calculate the sensitivity index using the given values and obtain the following values: $\beta_V = \beta_H = 1$, $b = 2$, $\mu = -0, 59962$, and $\alpha = 0$. In this case, the number of mosquito bites per day is the most crucial parameter to reduce the spread of the dengue virus.

Introducing control enables the solution to reach the disease-free equilibrium point more rapidly. Figures 5–7 illustrate the dynamics of the system solutions, both without control and with control. The proportion of susceptible humans in the controlled system is higher due to lower transmission rates from control measures. Conversely, the proportion of infected humans in the controlled system is lower throughout the simulation period, highlighting the success of the implemented control strategies in restricting disease transmission. Moreover, a significant decrease in infected vectors indicates that the intervention suppresses the spread within the vector population.

In Figure 8, we present the profiles of infected humans and mosquitoes using several control values. It can be explained that for a system without control, the virus transmission rate is high enough to cause dengue fever to spread rapidly, so that the number of infected humans increases in a short time. This condition is a cycle; the high number of infected humans also causes the high number of infected mosquitoes, and vice versa.

The impact of the control measures aims to reduce the infected host population (i.e., minimize $G(u_1)$). These measures include providing vaccinations, community education regarding the importance of cleanliness to prevent the proliferation of mosquitoes, and promoting awareness of dengue fever symptoms. Therefore, we present the numerical solution of the optimal control $u_1^*$ in Figure 9 to understand the control behavior. It shows that the control measured significantly reduced the infected hosts at day 30.

Meanwhile, in a system with control, strategic intervention in the prevention efforts is optimally implemented to control the spread of disease and to reduce costs. This treatment makes the behavior of the system solution in infected humans and infected mosquitoes decrease more quickly toward a disease-free condition.

7.   Conclusions

This paper presents a detailed mathematical model of dengue virus spread in the form of a host-vector that considers the role of humans as hosts and mosquitoes as vectors. The model analysis is generally divided into two, namely the model analysis without control and the model with vector control. Based on the results of the model analysis without control, the population will be free from dengue virus infection if the basic reproduction number is less than one. Mathematically, this is proven by La-Salle Lyapunov in Theorem 3.2. On the contrary, the endemicity holds if the basic reproduction number is more than one, see Theorem 3.4.

The model is further enriched by providing control strategies through the effective prevention of the dengue virus spread. Mathematically, we use the Pontryagin Maximum Principle in Theorem 5.1, which is a powerful optimization tool in the field of control theory. Through a numerical analysis, we showed that the control strategy can significantly reduce the population of infectious classes for both the hosts and the vectors. This is a significant contribution to developing effective strategies to control the spread of dengue fever, which is a major public health problem in many parts of the world.

Our research focused on the potential for a person to be re-infected without considering the different susceptible classes. In fact, a person can be re-infected by various types of dengue viruses. It is important to remember that there are four types of viruses, and being infected with one kind of virus does not confer immunity to the other. Therefore, it can be considered that susceptible host populations to have different dengue virus infections. In terms of memory effects, utilizing a fractional model can significantly advance the research in understanding the behavior of dengue spread ^[7,8,15]. In future research, we can further develop the mathematical model for dengue by including compartments such as Exposed ^[26], Quarantined ^[27], and Treated ^[28]. Control measures, such as vaccination strategies ^[26], and Wolbachia bacterium ^[29] are considerable to future studies.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The first author would like to thank Universitas Negeri Yogyakarta for the financial support. We also thank the Mathematical Biology team at Universitas Negeri Yogyakarta for the discussion.

Conflict of interest

Authors declare that they have no conflicts of interest.