A mathematical model of dengue transmission with memory

We propose and analyze a new compartmental model of dengue transmission with memory between human-to-mosquito and mosquito-to-human. The memory is incorporated in the model by using a fractional differential operator. A threshold quantity R0, similar to the basic reproduction number, is worked out. We determine the stability condition of the disease-free equilibrium (DFE) E0 with respect to the order of the fractional derivative α and R0. We determine α dependent threshold values for R0, below which DFE (E0) is always stable, above which DFE is always unstable, and at which the system exhibits a Hopf-type bifurcation. It is shown that even though R0 is less than unity, the DFE may not be always stable, and the system exhibits a Hopf-type bifurcation. Thus, making R0<1 for controlling the disease is no longer a sufficient condition. This result is synergistic with the concept of backward bifurcation in dengue ODE models. It is also shown that R0>1 may not be a sufficient condition for the persistence of the disease. For a special case, when α=12, we analytically verify our findings and determine the critical value of R0 in terms of some important model parameters. Finally, we discuss about some dengue control strategies in light of the threshold quantity R0.