A mathematical model of dengue transmission with memory

作者：Sardar Tridip; Rana Sourav; Chattopadhyay Joydev^*

来源：Communications in Nonlinear Science & Numerical Simulation, 2015, 22(1-3): 511-525.

DOI：10.1016/j.cnsns.2014.08.009

• Dengue transmission
• Mathematical model
• Fractional order differential equations

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 R-0, similar to the basic reproduction number, is worked out. We determine the stability condition of the disease-free equilibrium (DFE) E-0 with respect to the order of the fractional derivative alpha and R-0. We determine alpha dependent threshold values for R-0, below which DFE (E-0) 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 R-0 is less than unity, the DFE may not be always stable, and the system exhibits a Hopf-type bifurcation. Thus, making R-0 < 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 R-0 > 1 may not be a sufficient condition for the persistence of the disease. For a special case, when alpha = 1/2, we analytically verify our findings and determine the critical value of R-0 in terms of some important model parameters. Finally, we discuss about some dengue control strategies in light of the threshold quantity R-0.

• 出版日期2015-5