MATHEMATICAL MODELS OF DENGUE TRANSMISSION DYNAMICS WITH VACCINATION AND WOLBACHIA PARAMETERS AND SEASONAL ASPECTS

Article History: The Aedes aegypti mosquito is the main carrier of dengue virus transmission to humans. In this study, a mathematical model for the transmission of the dengue virus is constructed using vaccination and Wolbachia parameters in an attempt to control the virus's spread. Furthermore, the fundamental reproduction number is set as a parameter of the infection threshold. Based on the stability of the equilibrium point analysis, it is found that the disease-free equilibrium point is locally asymptotically stable if 𝑅 0 < 1 . Then, a mathematical model of dengue was created by examining the seasonal aspect and adding a periodic term to the mosquito birth rate. Dengue virus transmission in mosquito populations is controlled by air temperature in addition to seasonal variables. In this study, three weather scenarios were simulated: scenario 1 for cold weather (air temperature 14 °C), scenario 2 for hot weather (air temperature 26 °C), and scenario 3 for moderate weather (air temperature between 14 and 26 °C).


INTRODUCTION
Dengue Hemorrhagic Fever (DHF) is an infectious disease caused by the dengue virus, which is transmitted by the mosquitoes Aedes aegypti and Aedes albopictus.Dengue fever is a major global health issue, particularly in tropical and subtropical regions.Every year, millions of people get infected with DHF, and thousands die as a result of the disease [1].Efforts to control DHF have been carried out through various approaches, including vector control and prevention of the spread of the virus [2].Common vector control methods involve using insecticides, fogging, eradicating mosquito breeding grounds, and vaccination [3].Vaccines have historically been regarded as a highly effective approach in combating diseases.Consequently, persistent endeavors have been undertaken to create a potent dengue vaccine ever since the initial occurrence of the disease [3].However, excessive use of pesticides can result in mosquito resistance to the chemicals, and other preventative measures frequently fail to effectively reduce mosquito populations [4].The use of Wolbachia bacteria, which infect the Aedes aegypti mosquito, is one method for controlling DHF [5].Wolbachia is a naturally occurring bacterium that can infect mosquitos and affect their capacity to transmit the dengue virus.Wolbachia can restrict the mosquito's ability to transmit the virus by suppressing virus replication in the mosquito's body, hence limiting disease transmission.Aedes aegypti carrying the symbiont Wolbachia and the introduction of natural predators of mosquito larvae are considered as biological control strategies [6], [7].
One of the important factors contributing to dengue transmission dynamics is the presence of mosquito populations, whose life cycle is influenced by climatic factors such as temperature and rainfall.In places with strong dry and rainy seasons, the mosquito population changes throughout the year, with more mosquitoes in the rainy season than in the dry season, resulting in an increase in the frequency of dengue infections during the rainy season [8], [9].For instance, in Indonesia, the rainy season, which runs from November to March, usually has an increase in dengue fever infections.This shows that seasonal variations in mosquito populations influence the dynamics of dengue transmission [10].
Mathematical models can be used to simulate various vector control strategies and analyze their impact on disease transmission.DHF tends to show seasonal variability in its distribution [11].Aedes mosquitoes tend to reproduce and become more active during particular seasons [8].The rainy season, with its more humid circumstances, frequently results in an increase in mosquito population, which can raise the risk of dengue transmission.Models with a seasonal component can help identify these seasonal patterns and provide more insight into the trend of dengue transmission over time.
Several researches have created seasonal mathematical models, including the malaria transmission model [8], the chikungunya periodic model [12], the dynamic model of Zika virus transmission [13], and dengue [14], [15].However, the dengue model constructed in [14] did not include Wolbachia as a variable controlling disease transmission.Hence, in this study, a mathematical model of the transmission of dengue was created, one that included vaccination, Wolbachia as a control variable, and seasonal aspects.The stability of the equilibrium point and the sensitivity of the basic reproduction number were then examined to determine which parameter has the most influence on the spread of the dengue virus.For seasonal variations in the dynamics of dengue virus transmission, numerical simulations were performed on dengue models with seasonal and non-seasonal aspects.This article consists of four sections.Section one explains the background of the problem.Section two gives the research methods for this work.Section three gives the results and discussion that consist of the proposed mathematical model of dengue, its stability, sensitivity analysis, and numerical analysis.And the last, Section 4, concludes all of this work.

RESEARCH METHODS
The method used in this research was as follow: 1. Develop a mathematical model of dengue without seasonal effect.The proposed model is based on host-vector model with human as a vector and mosquitos as a vector.
2. Calculate the equilibrium of the model by solving this equation: 5. Determine the sensitivity index of the model parameter by using basic reproduction number.A sensitivity analysis was performed on the basic reproduction number  0 , which is the threshold for disease spread.The normalized sensitivity index of variable  to parameter  is defined by [16] : 6. Develop a mathematical model of dengue with seasonal effect.
7. Simulate the dynamics of the model with and without seasonal effect.The solution of the proposed model is solved numerically using Runge-Kutta 4 th algorithm.

RESULTS AND DISCUSSION
In this section, we will explain the development of mathematical model of dengue with and without seasonal effect.Then, the dynamics of the model will be analyzed, such as the equilibrium the basic reproduction number, and the stability analysis of the equilibrium.The sensitivity analysis of the basic reproduction number is carried out to determine the most influence model parameter.The numerical simulation will complete the study of the model's dynamics.

Mathematical model with non-seasonal aspect
In this sub-chapter, a mathematical model for the transmission of dengue fever will be developed, including control in the form of vaccine and Wolbachia but without taking into account the seasonal aspect.In this study, the total human population  ℎ was divided into four compartments, which were susceptible human  ℎ , dengue-infected human  ℎ , vaccinated human  ℎ , and immune human  ℎ .While the mosquito population is separated into two broad types, non-Wolbachia mosquitoes   and mosquitoes with Wolbachia   .The non-Wolbachia mosquito population was separated into four compartments: eggs of non-Wolbachia mosquitoes   , susceptible of non-Wolbachia mosquitoes   , mosquitoes infected with dengue   , and mosquitoes infected with Wolbachia   .Meanwhile, the Wolbachia mosquito population was separated into two compartments: mosquito eggs with Wolbachia   and susceptible mosquitoes with Wolbachia   .Table 1 includes comprehensive explanations of each compartment.
The mathematical model developed is a modification and combination of [14] and [17].In contrast to [17] , the compartment  ℎ was added in this study to accommodate the number of persons who had been vaccinated.Vaccination is considered to be transitory, allowing the  ℎ compartment to lose its immunity and revert to the susceptible compartment  ℎ .Unlike [14], humans in the  ℎ compartment shift to the  ℎ compartment at a proportion equal to the vaccine's effectiveness.  ,  , and   compartments were also included to examine the impact of Wolbachia spread on the dynamics of Aedes aegypti mosquitoes.
The model is based on assumptions about the dynamics of dengue virus transmission in real-world scenarios.The following assumptions are made in this study: i) the total human population and the total mosquito population are constant; ii) immunity from vaccination and recovery is temporary, so immune humans will become vulnerable again when they lose their immunity; (iii) humans infected with dengue can spread the virus, causing mosquitos to become infected if they interact with infected humans; iv) mosquitos infected with dengue will remain infected for the rest of their lives; (v) there is no vertical transmission from parent to child; (vi) a Wolbachia-infected mosquito bite does not result in dengue infection.Susceptible humans will increase with a recruitment rate of Λ. Susceptible humans will become infected if bitten by a mosquito carrying the dengue virus, with a virus transmission rate of . Infected humans will recover with a recovery rate of  ℎ after receiving treatment.Vaccination was administered at a rate of  ℎ to the susceptible human compartment.This model examines the imperfect vaccine component, specifically conditions where the vaccine's efficiency has not reached 100%.As a result, humans who have been vaccinated have a risk of being infected with dengue at a rate of (1 − ) . The  ℎ compartment consists of immunity from medication and vaccination.As soon as immunity starts to decrease, compartment  ℎ will switch to compartment  ℎ .Each of the  ℎ ,  ℎ ,  ℎ , and  ℎ compartments will decrease at a rate of  ℎ death on its own.
In the mosquito population, it is hypothesized that Aedes aegypti mosquitoes carrying Wolbachia are discharged into the environment and mate with non-Wolbachia Aedes aegypti mosquitoes.Wolbachia bacteria in Wolbachia-infected mosquitos will spread to other mosquitos, weakening the dengue virus in mosquitos.As a result, these mosquitoes will be unable to spread the dengue virus.2. Based on the explanation above, a mathematical model for the distribution of dengue without seasonal aspects is obtained in the following Equation ( 5):

𝜇 𝑤
The death rate of Wolbachia-infected mosquitoes

𝜃 𝑣
The recruitment rate of non-Wolbachia mosquitoes     [12] Mosquito.Day -1 The death rate of non-Wolbachia mosquitoes

Stability Analysis of Equilibrium Points
The model equilibrium point is reached when there is no change in the dynamics of the system, or when The non-endemic equilibrium point is reached when there is no disease transmission in the population ( ℎ =   = 0).Equation (1) has the following equilibrium point  0 :  0 = ( ℎ0 , 0,  ℎ0 ,  0 ,  0 ,  0 , 0,  0 ,  0 ,  0 ) where  +) .
Before analyzing the stability of the equilibrium point, the basic reproduction number of the system Equation (1) must first be calculated.The basic reproduction number is the number of secondary infections resulting from the primary infection to which the entire population is susceptible [18].The basic reproduction number  0 is the threshold for the spread of the disease.If  0 < 1 the infection will become extinct, but if  0 > 1 it will result in a pandemic situation.The amount of  0 is determined using the Next Generation Matrix (NGM) method [18].First, create a vector with the compartments that may induce virus transmission,  = ( ℎ ,   )  .Then decompose  into ℱ −  to generate the Jacobi matrix of ℱ and  which are evaluated at the non-endemic equilibrium points  0 ,  = ).Furthermore, the Next Generation Matrix (NGM) can be formed using the formula  =  −1 as follows: where , and .
The basic reproduction number is obtained from the largest eigenvalue modulus of the NGM matrix, 0 = ( −1 ) = max|  |, as follows: Theorem 1.The non-endemic equilibrium point  0 is locally asymptotically stable if  0 < 1.
Proof.The Jacobi matrix of system (1) which is evaluated at the equilibrium point  0 as follows: .( 9) Furthermore, the eigenvalues obtained from the Jacobi matrix  0 are  1 = −  ,  2 = −(  +  + ),  3 = − ℎ ,  4 = −( ℎ + ),  5 = −  ,  6 = −(  + ),  7 = −( +  ℎ ),  8 = −( 1 +   ), and  9,10 are the roots of Equation ( 6) below: where In other words, if  0 2 < 1 ⇔  0 < 1, the eigenvalue  9,10 will be negative.Therefore, the non-endemic equilibrium point  0 is asymptotically stable if  0 < 1. ∎ Based on Theorem 1, it is concluded that the non-endemic equilibrium point is asymptotically stable if  0 < 1.This means that disease-free conditions (the spread of dengue will stop or even become extinct) will be achieved if the threshold  0 in equation ( 8) can be reduced so that the value reaches  0 < 1.It is possible to achieve it by decreasing parameters that have a positive impact on  0 and increasing parameters that have a negative impact on  0 .The basic reproduction number  0 in equation ( 8) has a complicated form.Therefore, in subsection 3.3, a sensitivity analysis is provided to find out which parameters have the most influence on changes in the  0 value.
The proposed dengue mathematical model is very complicated, though, which makes it hard to figure out the endemic equilibrium point analytically.This results in difficulty in analyzing the stability of the endemic equilibrium point analytically.Therefore, the stability analysis of the endemic equilibrium point  1 can be carried out numerically by using the phase field with the help of the parameter values in Table 2.

Sensitivity Analysis
The sensitivity index is calculated using parameter sensitivity analysis, which gives a sense of the magnitude of the influence of changes in parameter values on the dynamics of the model.The parameters with an index value of one or a negative one has the largest influence on the model change rate.A sensitivity analysis was performed on the basic reproduction number  0 , which is the threshold for disease spread.The normalized sensitivity index of variable  to parameter  is defined by [16]: Thus, to obtain the sensitivity index of the  parameter to the basic reproduction number  0 it is calculated using the formula: Parameters with a positive (negative) index will be directly proportional (inversely) to the basic reproduction number  0 .In other words, an increase (reduction) in the positive index parameter will accelerate (slow down) the rate of disease transmission.In contrast, an increase (reduction) in the parameter with a negative index will halt (accelerate) the spread of the disease.By applying the formula in Equation (7) to the basic reproduction number  0 in Equation ( 4), the parameters {Λ,  ℎ ,   ,   } each have a sensitivity index of 0.5.While the other parameters have a complex sensitivity index.Therefore, the parameter values in Table 2 are used to obtain the sensitivity index values.In Table 3, all of the parameter sensitivity index values are listed.According to Table 3, there are parameters with positive indices such as {Λ,  ℎ ,   ,   , , , } and negative indices such as { ℎ , ,   , ,  1 ,  0 }.If the infection rate  ℎ increases (decreases) by 10%, then the value of  0 will increase (decrease) by 5%.In the opposite case, if the effectiveness of the  vaccine increases (reduces) by 10%, the  0 value decreases (increases) by 2.045%.

Mathematical Models Of Dengue Transmission Dynamics With Seasonal Aspect
In this sub-chapter, mathematical models of dengue transmission in the Equation ( 1) is modified by including the seasonal aspect.It is assumed that the birth rate of mosquitoes varies according to season using the sine function with a one-year period, as shown below: where  is the strength of seasonality, 0 ≤  ≤ 1.Thus, the mathematical model of dengue transmission dynamics with seasonal aspect can be stated as follows:  1 and Table 2.

Numerical simulation
Air temperature has the potential to influence dengue virus transmission in mosquito populations.Therefore, three weather scenarios are included in this simulation: scenario 1 for cold weather (air temperature 14 °C), scenario 2 for hot weather (air temperature 26 °C), and scenario 3 for moderate weather (air temperature between 14 and 26 °C) [8] .Table 4 shows the parameter scores in the three scenarios.Then three variations of the seasonal strength factor  are used, namely  = 0.1,  = 0.3, and  = 0.8 [14].The simulation was carried out using the parameter values listed in Table 2 at a simulation time of  = 365 days.The initial values for each compartment were ( ℎ0 ,  ℎ0 ,  ℎ0 ,  ℎ0 ,  0 ,  0 ,  0 ,  0 ,  0 ,  0 ) = (83297, 233, 0, 90, 4500, 2500, 1500, 100, 800, 200 ).The solution of the proposed model is solved numerically using Runge-Kutta 4 th algorithm.Figure 2 shows the simulation results for Scenario 1 (cold climate).Based on Figure 2, there aren't many changes between the population dynamics   and  ℎ when there are seasonal effects and when there aren't.The simulation results for scenario 2 (hot climate) are shown in Figure 3. Figure 3 (c) shows that seasonal factors have a significant impact on the dengue infected human population  ℎ .The peak of infection in this scenario is the highest of the three, reaching 11 thousand people.The simulation results for scenario 3 (mild climate) are presented in Figure 4.According to Figure 4, there are very slight variations in population dynamics between   and  ℎ under seasonal and non-seasonal.Even after reaching a high peak of infection, the total human population infected with dengue ℎ may decrease to zero in all three scenarios.This is due to the parameters of vaccination and the release of Wolbachia into the wild.
Meanwhile, based on the simulation results of the three scenarios in Figure 2 -Figure (4), it can be seen that the population dynamics of susceptible-Wolbachia mosquitoes (  ) are strongly influenced by seasonal effects.The greater the seasonality, the greater the fluctuation in the population of susceptible-Wolbachia mosquitoes.This means that in certain periods, the population of susceptible-Wolbachia mosquitoes will boom.And conversely, in other periods, the population of susceptible-Wolbachia mosquitoes will decrease quite drastically.4)also displays the dynamics of the vaccinated human population ( ℎ ) based on the application of the three scenarios.In Figure 2 (scenario 1) and Figure 4 (scenario 3), the  ℎ population is not significantly affected by seasonal factors applied to mosquito population dynamics.However, in Figure 3 (Scenario 2), the  ℎ population is quite affected by seasonal factors.Based on Figure 3, the greater the seasonal effect, the greater the influence on the population size ( ℎ ).Overall, seasonal effects influence the dynamics of the entire population in the dengue model that has been prepared.

CONCLUSIONS
A mathematical model for the spread of dengue has been developed by evaluating the seasonal variables that influence the dynamics of the mosquito population.The periodic term on the mosquito birth rate represents the seasonal aspect.Based on the stability of the equilibrium point analysis, it is discovered that the equilibrium point free of disease is locally asymptotically stable if  0 < 1.Then, using three different scenarios based on air temperature changes, a numerical simulation was performed.Based on numerical simulations, it was discovered that seasonal variables provide mosquito dynamics with an annual recurring pattern.The largest peak of infection in humans is produced by numerical simulation in scenario 2 (air temperature 26 °C), with 11 thousand infected persons.Following the model used which involves vaccination and Wolbachia release, the peak of the infection may reduce to zero infections.

Figure 1 .Figure 1
Figure 1.Dengue Virus Transmission Diagram (a) in Human Population (b) in Mosquitos Population Figure 1 shows the dengue virus transmission diagram.A complete description of the model parameters is given in Table2.Based on the explanation above, a mathematical model for the distribution of dengue without seasonal aspects is obtained in the following Equation (5): The dengue model in the system (1) has two equilibrium points, the non-endemic equilibrium points  0 and the endemic equilibrium point  1 .

Figure 2 -
Figure 2 -Figure (4)also displays the dynamics of the vaccinated human population ( ℎ ) based on the application of the three scenarios.In Figure2(scenario 1) and Figure4(scenario 3), the  ℎ population is not significantly affected by seasonal factors applied to mosquito population dynamics.However, in Figure3(Scenario 2), the  ℎ population is quite affected by seasonal factors.Based on Figure3, the greater the seasonal effect, the greater the influence on the population size ( ℎ ).Overall, seasonal effects influence the dynamics of the entire population in the dengue model that has been prepared.

Figure 2 .Figure 3 .Figure 4 .
Figure 2. The Dengue Model Dynamic of (a)  (b)  (c)  (d)  with Seasonal and Non-Seasonal Aspects for Scenario 1 3. Determine the basic reproduction number  0 using the Next Generation Matrix (NGM).First, create a vector with the compartments that may induce virus transmission,  = ( ℎ ,   )  .Then decompose  into ℱ −  to generate the Jacobi matrix of ℱ and  which are evaluated at the nonendemic equilibrium points  0 ,  =   ,   ,   ,   ,   }.