Numerical solution of hybrid mathematical model of dengue transmission with relapse and memory via Ada–Bashforth–Moulton predictor-corrector scheme

In this paper, a novel hybrid compartmental model of the dengue transmission process is proposed and studied with memory and relapse between host-to-vector and vice versa. The memory and correlated learning system in the dengue models by using the fractional differential operators such as Riemann–Liouville and Caputo has been a fascinating area of research. A threshold parameter which is called basic reproduction number R₀ is investigated and calculated by next-generation technique. It's also shows that if basic reproduction number R₀<1, the disease-free equilibrium(DFE) is locally asymptotically stable(LAS) and if R₀>1 then, the DFE is unstable. It's also found that the fractional-order α also depends upon R₀. Therefore, if fractional-order α=1 and R₀>1, then dengue fever model doesn't show Hopf-type bifurcation. Further, it's also worth mentioning that although R₀<1, the DFE E₀ may not be always stable but it's necessary and the model shows a Hopf-type bifurcation. We employed the scheme of Adams–Bashforth–Moulton predictor-corrector to find an approximate the solution of the dengue model. The numerical simulation is carried out to validate the analytic solution.