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Winter is coming: Pathogen emergence in seasonal environments

Fig 2

The winter is coming effect.

Pathogen birth rate (i.e. transmission rate) λ(t) is assumed to vary periodically following a square wave (A and B). During a portion 1 − γ of the year transmission is maximal (γ = 0.7 in this figure) and λ(t) = λ₀. In the final portion of the year λ(t) drops (low transmission season in gray). In A λ(t) varies between λ₀ = 2.5 and 1.5 and, in B λ(t) varies between λ₀ = 2.5 and 0. Pathogen death rate μ(t) (a function of recovery and death rates of the infected host) is assumed to be constant and equal to 1 in this figure. When the net growth rate of the pathogen remains positive in the low transmission season (λ(t) > μ(t), A, C and E) the probability of emergence of a pathogen introduced at time t₀ can be well approximated by Eq (6): (dashed line in E and F) if the duration of the infection is short relative to the period T of the fluctuation (E). In contrast, if the low transmission season is more severe (λ(t) < μ(t), B, D and F), the negative growth rate φ(t) of the pathogen population during this period creates a demographic trap and reduces the probability of emergence at the end of the high transmision season. This winter is coming effect is indicated with black arrow in (D) and with the light gray shading in (D) and (F). This effect is particularly pronounced when the period of the fluctuations of the environment is large relative to the duration of the infection (i.e., when T is large, F). When the period T of the fluctuation is small relative to the duration of the infection, the probability of emergence is well approximated by Eq (5): whatever the time of pathogen introduction (in A, R₀ = 2.2 and p_e ≃ 0.55; in B, R₀ = 1.75 and p_e ≃ 0.43).

doi: https://doi.org/10.1371/journal.pcbi.1007954.g002