Abstract
ABSTRACTIn this paper, we establish a mathematical model with two delays to reflect the intrinsic and extrinsic incubation periods of virus in dengue transmission. The basic reproduction number of the model is defined. It is proved that the disease-free equilibrium is stable when and the positive equilibrium is stable when . Next, we derive an estimation formula for the reproduction number when the human population is partially susceptible to dengue. As an application, the values of dengue transmission in Singapore in the years 2013–2015 are estimated. Our estimation method can be applied to estimating of other infectious diseases, especially when the human population is not completely susceptible to the disease.
Highlights
The incidence of dengue has grown dramatically around the world in recent years
We shall investigate the threshold dynamics of the delayed model established in terms of the basic reproduction number R0, which is defined as the average number of secondary cases induced by a typical infectious individual in a completely susceptible population [1]
In view of the considerable amount of incubation time presents in both human and mosquito during dengue transmission, in Section 2, we have established a mathematical model with two delays that takes into account the extrinsic incubation time (τ1) and the intrinsic incubation time (τ2)
Summary
The incidence of dengue has grown dramatically around the world in recent years. The disease is endemic in more than 100 countries especially in tropical and subtropical regions. We shall investigate the threshold dynamics of the delayed model established in terms of the basic reproduction number R0, which is defined as the average number of secondary cases induced by a typical infectious individual in a completely susceptible population [1]. By regarding the number of infected cases periodically notified by surveillance organizations as the solution of an approximate mathematical model, we shall derive a formula to estimate the reproduction number Rt of dengue transmission in a partially susceptible population. Comparing with the model in [13], (i) we have introduced time delays τ1 and τ2, which are the average extrinsic incubation time and intrinsic incubation time, respectively; (ii) the natural immunity from dengue and the disease induced death of human are neglected; and (iii) the average biting rates of susceptible and infected vectors are assumed to be the same. (s∗H, i∗H, 1, i∗V ) when R0 > 1, where s∗H, i∗H and i∗V are given in (17) and (18)
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