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A mathematical model of dengue transmission with memory. (English) Zbl 1329.92140
Summary: 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 \(\alpha = \frac{1}{2}\), 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 \(R_0\).

92D30 Epidemiology
Full Text: DOI
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