Threshold Dynamics in a Model for Zika Virus Disease with Seasonality

Abstract

We present a compartmental population model for the spread of Zika virus disease including sexual and vectorial transmission as well as asymptomatic carriers. We apply a non-autonomous model with time-dependent mosquito birth, death and biting rates to integrate the impact of the periodicity of weather on the spread of Zika. We define the basic reproduction number $${\mathscr {R}}_{0}$$ R 0 as the spectral radius of a linear integral operator and show that the global dynamics is determined by this threshold parameter: If $${\mathscr {R}}_0 < 1,$$ R 0 < 1 , then the disease-free periodic solution is globally asymptotically stable, while if $${\mathscr {R}}_0 > 1,$$ R 0 > 1 , then the disease persists. We show numerical examples to study what kind of parameter changes might lead to a periodic recurrence of Zika.

Threshold dynamics in mathematical models for mosquito- and rodent-borne diseases with seasonality

“…The calculations show that the time-averaged reproduction number [R 0 ] is less than the reproduction ratio R 0 , suggesting that the time-averaged reproduction number provides an underestimation of the risk of disease transmission. From this aspect, our results are similar to the those in [103] and [54]. We note that various papers present different results on under-and overestimation of the average basic reproduction number.…”

“…The content of this chapter has been published in [54] M. A. Ibrahim and A. Dénes. Threshold dynamics in a model for Zika virus disease with seasonality.…”

“…To study the dynamics of our time-periodic model, we will apply the theory initiated in [10,11,90,103,113] and detailed in Chapter 3, later applied in several periodic epidemic models (see, e.g. [53,54,61,62,63,78,89,102]). Here we adapt these methods to our system with human-to-human and rodent-to-human transmission with a logistic growth of rodents.…”

“…Numerical examples to study what kind of parameter changes might lead to a periodic recurrence of Zika are shown. The results of this chapter were published in Ibrahim and Dénes [54].…”

Impact of Infection on Honeybee Population Dynamics in a Seasonal Environment

“…Several sophisticated mathematical models predicting the Honeybee population dynamics have been proposed [1][2][3]. Since the seasonality of infectious diseases is very repetitive [4], several mathematical models of infectious diseases that take into account of the seasonality were proposed [5][6][7][8][9][10][11][12][13].…”