A previous model proposed a reaction and diffusion model to study global dynamics of Lyme disease. A reaction-diffusion model takes into account the interaction (reaction) of constituents within the system (in this case pathogens, susceptible hosts and infective hosts) and their change in density over time within their respective populations (diffusion).
The previously-proposed spatial model treats population densities in a continuous two-dimensional space, factoring in birth, death, infection and developmental advancement. However, the model does not account for seasonal patterns.
In this paper, the authors modify this previous model into a reaction and diffusion model in a periodic environment, which models seasonable variables (eg. temperature) as a periodic function. A periodic function is one that repeats its values at regular intervals.
Since seasonal variations are critical for tick development and their activities, which are strongly affected by temperatures, the authors assume that the development rates of ticks as well as their activity rates are time-dependent. The model is governed by a periodic reaction-diffusion system that factors in the densities of susceptible and pathogen-infected mice, densities of susceptible and infectious larvae and nymphs, and densities of uninfected and pathogen-infected adult ticks.
The authors introduce the basic reproduction number R0, which is the number of infectious cases one case of disease can generate on average over the course of the infectious period in a susceptible population. The authors
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Society for Industrial and Applied Mathematics