Analysis of Vector-host SEIR-SEI Dengue Epidemiological Model

A BSTRACT . Approximately worldwide 50 nations are still infected with the deadly dengue virus. This mosquito-borne illness spreads rapidly. Epidemiological models can provide fundamental recommendations for public health professionals, allowing them to analyze variables impacting disease prevention and control efforts. In this paper, we present a host-vector mathematical model that depicts the Dengue virus transmission dynamics utilizing a susceptible-exposed-infected-recovered (SEIR) model for the human interacting with a susceptible-exposed-infected (SEI) model for the mosquito. Using the Next Generation Technique, the basic reproduction number of the model is calculated. The local stability shows that if R 0 <1 the system is asymptotically stable and the disease dies out, otherwise unstable. The Lyapunov function is also used to evaluate the global stability of disease-free and endemic equilibrium points. To analyze the effect of the crucial aspects of the disease's transmission and to validate the analytical findings, numerical simulations of a variety of compartments have been constructed using MATLAB. The sensitivity analysis of the epidemic model is performed to establish the relative significance of the model parameters to disease transmission.


Introduction
Dengue fever incidence has risen dramatically in the recent two decades [1]. Approximately half of the earth's population may be in danger of contracting one of the nearly 390 million new infections that are thought to arise annually. Dengue viral disease is transmitted from mosquitoes to humans by mosquitoes, spreading rapidly expanding over the world by its four serotypes. Prior to relatively recently, the female Aedes aegypti mosquito which identifies as a primary vector for dengue was mostly found in tropical and subtropical regions [2]. Due to the highly adapted ability in urban regions of the Aedes aegypti mosquito, dengue spread all over the world. However, the full extent of the disease's effects remains unclear, and new monitoring strategies are needed, according to issues with underreporting and case misidentification. Other arboviruses, chikungunya as well as zika have lately emerged, posing additional issues for surveillance and management, particularly in South Asia [3].
A wide range of factors (including people, mosquitoes, and the virus) interact with one another to spread the dengue virus in a diverse environment [4]. Studies on dengue transmission face several obstacles due to the space's inherent complexity. A variety of causes are related to the current epidemic. These included worldwide host and vector mobility (which accelerated viral circulation), urban congestion (which encouraged various transmissions through a single infected vector), and the loss of previously effective vector control measures. Temperature, precipitation, and humidity all affect vector development at all stages, from egg viability through adult longevity and dispersion, among other aspects of dengue transmission. Unplanned development, high inhabitants' density, and the instability of rubbish collection all of which support the growth of mosquito breeding sites, which lead to increasing dengue occurrence.
In recent years, epidemiology research and disease control have benefited greatly from the use of mathematical modeling. To understand the disease's nature as well as taking appropriate decisions regarding disease management strategies/interventions and processes, mathematical modeling has become a useful tool. Many scholars studied a deterministic model to study the influence of numerous biological parameters on disease dynamics. Prasad et al. [5] studied a systematic review of deterministic mathematical models for vector-borne viral infections. Bhuju et al. [6] described the fuzzy epidemic SEIR-SEI compartmental model with bed nets and fumigation intervention to simulate the transmission dynamics of dengue disease. Tay et al. [7] constructed a transmission model of SI-SIR dengue epidemiological characteristics model to control dengue in Malaysia. Abidemi et al. [8] analyzed the effect of single vaccine usage and its combination with treatment and adulticide measures on dengue population dynamics in Johor, Malaysia. Aleixo et al. [9] gave a clear explanation of a machine learning model that is used to predict the frequency of dengue outbreaks in Rio de Janeiro. Sow et al. [10] developed a computational Zika Dynamics model to examine the effects of vertical transmission between the vector population and the host population. Sweilam et al. [11] introduced a unique variable-order nonlinear model of the dengue virus that minimizes intervention dosage and duration through optimum bang-bang management. Abidemi et al. [12] developed and analyzed a two-strain deterministic dengue model based on the SIR modeling framework for the spread of the disease and its management in an area with two coexisting dengue virus serotypes. Asamoah et al. [13] investigated an ideal dengue infection control model with partly immune and asymptomatic patients. Linda et al. [14] examined the discrete-time versions of the SIS and SIR models that are stochastic in nature. To assess the influence of raising awareness through the press on the spread of vector-borne illnesses, a non-linear mathematical model was suggested by Misra et al. [15]. The dynamic SIR model with climatic parameters was discussed by Nur et al. [16] for the features of dengue disease transmission in a closed community.

Dengue Transmission Model
The Ross-Macdonald model, which was first designed for malaria, is a classic mathematical model for vector-borne illnesses that monitors infections in humans as well as mosquitos. In this research, we present a compartmental host-vector mathematical model [17] that depicts the Dengue virus transmission dynamics utilizing a susceptible-exposed-infected-recovered (SEIR) model for the human interacting with a susceptible-exposed-infected (SEI) model for the mosquito. The host-vector mathematical model categorizes the overall human (host) population into four classes: susceptible ( ℎ ), exposed ( ℎ ), infectious ( ℎ ), and recovered ( ℎ ), whereas the mosquito (vector) population is divided into three classes: susceptible ( ), exposed ( ), and infectious ( ). Thus, the total human(host) population denoted by ℎ is given as And total mosquitoes'(vector) population is given by:

Fig 1. Dengue Virus Transmission Dynamics in Different Population Stages
In our suggested model, we attempt to provide a fresh direction by taking panic, tension, or anxiety into account in the susceptible, exposed, and infected classes to host population. The influence of panic as well as stress, or anxiety on these clusters is discussed in this work. Panic, stress, and anxiety are all harmful to human's health. Anxiety may raise insulin levels, which can have an impact on heart health, diabetes, and blood pressure. At the same time, stress can have a negative impact on human immune system. Extreme stress can impair immunity as well as chronic stress might jeopardize a major health condition. People suffering from panic attacks are more likely to get infected, and the death rate among infected people rises. Therefore, we anticipate that the amount of susceptible, exposed, and infected, is decreasing, i.e., moving to death due to panic, stress, or anxiety. Figure 1 depicts the suggested model's flow diagram as well as the nonlinear system of differential equations that represents the dynamics of host-vector dengue disease, which is represented by: with the initial conditions where the biological descriptions of parameters is presented in Table 1.

Positivity and boundedness of solutions
The positivity and boundedness of the solutions are crucial features of an epidemiological model. As a result, it is critical to demonstrate that all variables are non-negative for all time ≥ 0, implying that any solution with positive beginning values will remain positive for all time ≥ 0. So, positivity indicates that the population will survive for a long period.
The dynamical model of the transmission shall be investigated into the biologically feasible regions The feasible region is positively invariant for the model (1) with the initial condition defined by Θ⊂R 7 Thus, ℎ ( ) converges for all non-negative time as t approaches infinity, and the results of the system (1) stay in Θ with starting conditions.
Thus, N m (t) converges for all non-negative time as t approaches infinity, and the results of the system (1) stay in Θ with starting conditions. Therefore, the feasible region Θ is positively invariant, attracting all solutions in ℝ 7 + .
Theorem 3.2. The solution of the system (1) is positive and bounded for all Proof. To demonstrate the solution's positivity, we need to show that on any hyperplane enclosing the positive vector space ℝ 7 + from the system (1), we have So, the system (1) solution is positive.

Qualitative analysis of model
In this section qualitative analysis of the dengue system (1) by calculating disease free equilibrium (DFE) and the endemic equilibrium (EE) with help of basic reproduction number (E0).

Disease-free equilibrium
To calculate the disease-free equilibrium (DFE) E0 of the dengue system (1), we set the right-hand side of equals to zero and obtain the following expression's

Basic reproduction number
For the purpose of assessing an infectious disease, a crucial threshold parameter is the basic reproduction number R0. It decides whether the disease will disappear or stay in the community throughout time. R0 is the secondary infections number which may be caused by a single primary infection whereas the population is susceptible. Assume R0 > 1, and one primary infection can generate in several secondary infections. Therefore, the disease-free equilibrium (DFE) is unstable, also an epidemic occurs. The reproduction number for the Dengue system is calculated utilizing the next generation matrix approach [18]. We look at the F * and V * matrices, are designated for the new infections' development and classified migration of infective partitions.

Endemic equilibrium
The endemic equilibrium point of the dengue dynamical system (1)

Global stability around equilibrium point
In this segment, we will evaluate equilibrium points E0 and E1 stability. The next two theorems show the results of the stability analysis of these equilibrium sites. On solving further get: Using the equilibrium condition (μ 1 +α 1 )S h 0 =Λ 1 and μ 2 S m 0 =Λ 2 into the above equation The above equation shows that ′ ( ) ≤ 0 . So, the largest invariance set is the singleton set { 0 }. Therefore, by using the principle of LaSalle's invariance the disease-free equilibrium ( 0 ) is globally asymptotically stable.
Proof. We consider the Lyapunov function of the form in Differentiating with respect to time t, we get:   6. Sensitivity analysis of the system Sensitivity analysis identifies the most effective model parameters that have effects on the Dengue model system's basic reproduction number [19]. Epidemiologists may forecast the important factors that play a significant part in virus-spreading dynamics using such analyses [20].
To avoid or manage the disease's effect, we must first identify the sensitivity induce values, which will give us an idea of which parameters for model would be maintained or monitored.
In  On basic reproduction number 0 , the parameters higher sensitivity index indicates the more influence sensitive parameter. The system parameter's sensitivity index with positive sign suggests that the basic reproduction number 0 increases when the parameter increases, and vice versa. In Table 2 and fig. 2, we applied a sensitive index to R0 in relation to each parameter.
According to our research, the most important model parameters are recruitment rates of human population ( 1 ), recruitment rates of vector population ( 2 ), rate of infectious from vector to host ( 1 ), natural death rate of human ( 1 ), panic/tension/anxiety rate of human ( 1 ), recovery rate of infected human ( 3 ), infection rate from human to vector ( 4 ), extrinsic incubation of vector ( 5 ), and natural death rate of vector population ( 2 ). The most significant sensitivity index of the system is the natural death rate of vector population 2 . In the numerical part, the suggested model simulation is performed with the assist of MATLAB software. For numerical simulation, the parameter for the system (1) are given in Table 3.  [21] .8500 day -1 [22] .6794 day -1 [23] .003468 day -1 [24] .000123 day -1 assumed .5555 day -1 [22] .0034 day -1 [25] .7186 day -1 [26] .0062 day -1 [8] . vector to host ( 1 ) and the rate of panic/tension/anxiety in humans ( 1 ). Fig. 3(A) describes that there is no effects on different values of 1 between the early stage 0 to 5 days. Also, it indicates that the susceptible humans decrease with an increase in transmission the infectious rate from vector to host ( 1 ) and vise-verse. In fig. 3(B) indicates that panic/tension/anxiety rate in humans ( 1 ) has an impact between 5 to 75 days.

Conclusion
The epidemic vector-borne disease has devastated many nations. Form which, the focus of this article was to analyze dynamic dengue fever. We developed a dynamical mathematical model that would represent them and incorporate the impact of panic, tension, or anxiety on the human population. Model's qualitative analysis was calculated, including illness free equilibrium, endemic equilibrium, and basic reproduction number. Numerical simulation through various parameter settings showed the progression of epidemics, the system's behaviors, and support theoretical results. The maximum sensitivity index was obtained for the vector death rate in the sensitivity study, and this parameter was regarded the most sensitive. It was discovered that increasing the rate of 2 results in the greatest decrease in reproduction number, and no other parameter had the same effect on reducing infection. Such model analysis can give vital information to policy makers and health specialists who may be confronted the infectious disease reality.