Time new infections is also b(N — I)I; this is often called the force of infection. We assume that individuals lose infectiousness at rate v, so that the rate of loss of infected individuals is vl. Combining these, we obtain an equation for the dynamics of infection:

If we combine the linear terms together we have d = I (bN — v) — bI2 (5.2)

and we see from this equation that if bN< v, the number of infecteds will decline from its initial value. However, if bN> v, then Eq. (5.2) is the logistic equation, written in a slightly different format (what would the r and K of the logistic equation be in terms of the parameters in Eq. (5.2)?). The resulting dynamics are shown in Figure 5.1. If bN< v, the disease will not spread in the population, but if it does spread, the growth will be logistic - an epidemic will occur, leading to a steady level of infection in the population I = (bN — v)/b. Furthermore, whether the disease spreads or not can be determined by evaluating bN/v without having to evaluate the parameters individually. Pybus et al. (2001) fit this model to a number of different sets of data on hepatitis C virus.

Since the population is closed, we could also work with the fraction of the population that is infected, i(t) = I(t)/N. Setting I(t) = Ni(t) in

Characterizing the transmission between susceptible and infected individuals

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