Results and Discussion of Dengue Model with Temperature Effects in Interval Environment

We have investigated a dengue model with temperature effects under interval uncertainty in this work. This study observed the Aedes aegypti temperature-dependent entomological parameters that affect dengue illness transmission dynamics in Taiwan's subtropical zone. A vector-host transmission model was used to examine how temperature fluctuations influence the development of pre-adult mosquitoes, their egg-laying rates, adult mortality, and the incubation rate of viruses within them. This study showed that although estimations of entomological parameters were positively correlated with slow temperature rises, no such correlation was detected with mosquito mortality or maturation rates, underscoring the slow rate of maturation of pre-adult mosquitoes. The findings suggest that the dynamic modeling of vector-host interactions is significantly influenced by temperature. Additionally, our modeling indicates that a temperature range of about 32°C is ideal for dengue transmission. In the future, control measure modeling and cost-effectiveness assessments may benefit from these insights.

Given the ongoing concern about global warming, understanding how temperature impacts the spread of dengue is crucial.This topic is a major focus in fields like epidemiology, computational biology, public health, and environmental science.A recent review [9] used a meta-analysis to explore how both rainfall and temperature influence dengue outbreaks.
For public health, it's vital to develop effective strategies for predicting, managing, and treating such epidemics.Over the years, many researchers have delved into this issue, contributing to both biology and mathematics.Deterministic mathematical models for dengue transmission, like the SEI (susceptible-exposedinfectious) for mosquitoes and the SIR (susceptible-infectious-recovery) for humans, are well covered in the literature [10], with more research in [11].
To tackle uncertainties in these models, researchers have used methods like interval approaches and stochastic methods [12 -17].The interval approach uses interval-valued functions to describe unknown parameters.Professor Zadeh pioneered fuzzy set theory [18] and proposed fuzzy differential equations as a way to model systems with probabilistic uncertainties [19].Additionally, Sadhukhan et al. [20] explored optimal strategies for managing a food chain model under fuzzy conditions, using a fuzzy instantaneous annual discount rate.

|Model Formulation
Kaohsiung [19], is a tropical city, with an average annual temperature of 25°C, a maximum temperature of 32°C, and a minimum temperature of 18°C, according to statistics from 2001 to 2010 [7].The population is divided into three groups: pre-adult vectors, adult female mosquitoes, and human hosts.For the A. aegypti pre-adult female mosquito population, two parameters-eggs and larvae-are defined within both the susceptible (Se) and infected (Ie) populations.Sv, Ev, and Iv, which stand for the number of susceptible, infected but not yet infectious, and infectious female mosquitoes at time t, respectively, were specified as the three parameters for the adult vector (mosquito) population.Similarly, three parameters were found for the host population (humans): Sh, Ih, and Rh.With this, integer order dynamics is now used to develop the dengue model.
These parameters were selected based on the availability of experimental or observed values (Table 1) from the literature review.

|Model in Imprecise Environment
The coefficients in the suggested model ( 1) can be treated as interval numbers in the following way to modify it in the uncertain environment:

 The Basic Reproduction Number of the System
Using the next-generation matrix approach [21,22] we have, .

|Numerical Study
We used the math programs Matlab (2018) and Matcont to numerically estimate the solution of our model system.Using the model parameter values listed in Table 2, we simulate the system (3) in this scenario and change the value of parameter "ρ" to four different levels (ρ = 0, 0.4, 0.6, 1).The model system (3)'s time series plot, displayed in Figure 1, for various values of parameter ρ, clearly illustrates the stability of the disease-free equilibrium point throughout the time range [0, 100].

|Conclusion
We have investigated a dengue model with temperature effects in the presence of interval uncertainty.The behavior of the model in an uncertain environment is examined in this analysis.The temperature has an impact on dengue and other illnesses spread by mosquitoes; these effects are anticipated to vary over time and between geographical areas.Predictions regarding how these temperature effects may alter in response to climate change may also become more complex due to potential interactions with other environmental elements.Our analysis of the available data on dengue disease and temperature lends credence to the hypothesis that temperature has a nonlinear effect on ecological processes.In particular, we anticipate reduced or negative effects at extremely low and high temperatures and considerable favorable effects within certain middle.However because the information on the species that transmits disease differed throughout research, we were unable to test this notion.Compared to the simpler models that assumed a constant or linear relationship between average temperature and dengue, the flexible quadratic temperature model and the a priori marginal temperature suitability model fared much better.The authors want to use mathematical modeling methodologies and more ecological modeling with neutrosophic environments in future studies to offer new medical research models.

Table 1 .
Significance of the relevant parameters Parameter Significance Vertical infection rate Pre-adult mosquito maturation rate  Transmission biting rate  Rate of incubation  Human recovery rate

Table 2 .
Shows the values of the several parameters that are used to model the system (3).
̂ [0.87, 0.97] Figure 1.For various values of the parameter , depicts the stable nature of the disease-free equilibrium point.