Consider the case of a disease that can only spread during the proportion T of the year that temperature is above a threshold level, where 0 ≤T ≤1. During that proportion
of the year, infection occurs at a rate β, such that the average infection rate across the
full year is Tβ. Then there are two ways to “leave” the infected class. First, during
the infectious proportion of the year T, the infected individuals experience disease-
induced mortality at a rate μ. Second, during the non-infectious proportion of the
year 1 −T, infected individuals can recover at a rate γ. There is no immunity, so
recovered individuals become susceptible again. The disease dynamics are very rapid
in comparison to the host populations’ typical (uninfected) lifespan. The dynamics of
the susceptible S and infected I individuals are then:
dS/dt = −TβS (I/S + I) + (1 −T)γI
dI/dt = TβS (I/S + I) −TμI−(1 −T)γI= TβS (I/S + I)−(Tμ + (1 −T)γ) I

TβS (I/S + I) is Infection.
TμI is Mortality
−(1 −T)γI is Recovery

(a) What are the units of all components of this model: β, γ, μ, T, S, I, and
t? (If a value has no dimensions, say “none”.)