Global dynamics of periodic infectious disease models with time-dependent delays

Li, Fuxiang (2020) Global dynamics of periodic infectious disease models with time-dependent delays. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

Many infectious diseases have seasonal trends and exhibit variable periods of peak seasonality. Understanding the population dynamics due to seasonal changes becomes very important for predicting and controlling disease transmission risks. For some directly transmitted and vector-borne diseases, the length of the incubation period strongly depends on the temperature. This thesis is devoted to the study of the global dynamics of some periodic epidemic models with periodic incubation periods. We start with a classical SEIRS epidemic model with a time-dependent latent period in Chapter 2. Moreover, vector-borne diseases, such as West Nile virus, bluetongue, and malaria, are always highly dependent on seasonal change, especially the temperature. To investigate the seasonal effects and temperature-dependent delays on West Nile virus, we present a periodic functional differential equations model with the vertical transmission, the periodic maturation delay, and the periodic extrinsic incubation period in Chapter 3. In Chapter 4, we propose a bluetongue model with seasonality and temperature-dependent incubation period, which describes the dynamics of bluetongue transmission via cattle and sheep as hosts and midges as vectors. To explore the effects of the spatial and temporal heterogeneity in hosts and vectors, and only vector movements on the spread of bluetongue, we develop a nonlocal periodic reaction-diffusion model of bluetongue disease with periodic time delays in Chapter 5. Based on the theory of the basic reproduction ratio, we derive and numerically compute the basic reproduction ratio for our models. By the theory of dynamical systems, we show that the basic reproduction ratio acts as a threshold parameter for the global dynamics for each model. Numerical simulations or case studies are carried out to illustrate the analytic results and help us provide some new findings. At the end of this thesis, we present a brief summary and some interesting future works.

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