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

Abstract

• A mathematical model of the dengue transmission process with memory and relapse between host-to-vector and vice-versa is presented. • The dengue model contains a major parameter namely relapse and memory. • The fractional order α shows the memory behaviour. Taking α = 0.5 gives threshold value R 0 < 1 which is no longer a sufficient condition but necessary for die out of the disease. • It is found the transmission memory increases α approaches to zero, the chance of dengue fever persistence in the host will go on the increase. 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 R 0 is investigated and calculated by next-generation technique. It’s also shows that if basic reproduction number R 0 < 1 , the disease-free equilibrium(DFE) is locally asymptotically stable(LAS) and if R 0 > 1 then, the DFE is unstable. It’s also found that the fractional-order α also depends upon R 0 . Therefore, if fractional-order α = 1 and R 0 > 1 , then dengue fever model doesn’t show Hopf-type bifurcation. Further, it’s also worth mentioning that although R 0 < 1 , the DFE E 0 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.