This study used a system of delay differential equations (DDE) to construct a time-delayed vector-host dengue epidemic model that accounts for inhibitory impact rates, immunity loss rates, and partial immunity rates. The model's solution is investigated and is determined to be positive and bounded. Using the next-generation matrix technique, the reproduction number is utilized to assess the model's stability. The virus-free equilibrium points were found to be locally asymptotically unstable. The existence of endemic equilibrium stability with and without time delay was investigated; as a result, endemic equilibrium points were locally stable with delay under certain conditions. For dengue transmission sensitivity analysis, the epidemiological model was analyzed. Finally, our theoretical results are validated by numerical simulations.

Background: Mathematical models reflecting the epidemiological dynamics of dengue infection have been discovered dating back to 1970. The four serotypes (DENV-1 to DENV-4) that cause dengue fever are antigenically related but different viruses that are transmitted by mosquitoes. It is a significant global public health issue since 2.5 billion individuals are at risk of contracting the virus. Methods: The purpose of this study is to carefully examine the transmission of dengue with a time delay. A dengue transmission dynamic model with two delays, the standard incidence, loss of immunity, recovery from infectiousness, and partial protection of the human population was developed. Results: Both endemic equilibrium and illness-free equilibrium were examined in terms of the stability theory of delay differential equations. As long as the basic reproduction number (R0) is less than unity, the illness-free equilibrium is locally asymptotically stable; however, when R0 exceeds unity, the equilibrium becomes unstable. The existence of Hopf bifurcation with delay as a bifurcation parameter and the conditions for endemic equilibrium stability were examined. To validate the theoretical results, numerical simulations were done. Conclusions: The length of the time delay in the dengue transmission epidemic model has no effect on the stability of the illness-free equilibrium. Regardless, Hopf bifurcation may occur depending on how much the delay impacts the stability of the underlying equilibrium. This mathematical modelling is effective for providing qualitative evaluations for the recovery of a huge population of afflicted community members with a time delay.