Discussion on Vector Control Dengue Epidemic Model for Stability Analysis and Numerical Simulations

Despite advancements in medicine and vaccination, the global death toll from dengue disease continues to rise, especially in developing countries, where modern healthcare access and prevention are limited. In the twenty-first century, dengue disease has emerged as a severe global challenge for the health sector because it severely affects more than 100 countries and causes thousands of deaths annually. This study designs an SEIR-SEI mathematical model of the dengue virus by introducing a control parameter within the mosquito population. The inclusion of this parameter enables us to determine the impact of control measures on the transmission dynamics of the dengue virus. Firstly, we ensure that all state variables are bounded and remain non-negative throughout the study. Then, we calculate the two equilibrium points, the dengue-free equilibrium (DFE) and dengue-endemic equilibrium point (DEE), of the model for further analysis. We also calculate the reproductive number \(\mathfrak {R}\), which is a crucial threshold parameter in epidemiology. The qualitative analysis shows that the model possesses local and global stability at the DFE and DEE points if \(\mathfrak {R}<1\) and \(\mathfrak {R}>1\), respectively. We also conduct a sensitivity analysis to identify the parameter with the most significant impact on the transmission dynamics of the dengue virus. This insight assists health policy makers in optimizing their struggles to lower the infection rates between susceptible humans and infected mosquito populations. We apply the two numerical schemes, the non-standard finite difference scheme (NSFD) and the Runga-kutta (RK4) scheme, to validate the theoretical and numerical results of the proposed SEIR-SEI dengue epidemic model. Numerical simulations are also provided in support of these results.