Eq. (5.1) gives N(di/dt) = bNi(N - Ni) - vNi and if we divide by N, and set 3 = bN we obtain d = 3i(1 - i)- vi (5.3)

as the equation for the dynamics of the infected fraction. Note that the parameter 3 has the units of a pure rate, whereas b has somewhat funny units: 1/time-individuals-infected, such as per-day-per-infected individual. I have more to say about this in the next section.

Now let us consider these disease dynamics from the perspective of the susceptible population. Furthermore, suppose that the initial number of infected individuals is 1. We can then ask, if the disease spreads in the population, how many new infections will occur as a result of contact with this one individual? Since the rate of new infections is b/S, the dynamics for S(t) are dS/dt =— b/S, which we will solve with the initial condition S(0) = N — 1, holding /(t) = 1. This will allow us to ask how many cases arise, approximately, from the one infected individual (you could think about why this is approximate). The solution for the dynamics of susceptibles under these circumstances is S(t) = (N — 1)exp(—bt). Recall that the recovery rate for infected individuals is v, so that 1/v is roughly the time during which the one infected individual is contagious. The number of susceptible individuals remaining at this time will be S(1/v) = (N — 1)exp(— b/v), so that the number of new cases caused by the one infected individual is S(0) — S(1/v) = (N — 1) — (N — 1)exp(—b/v) = (N — 1)(1 — exp(—b/v)). If we assume that the population is large, so that N — 1«N and we Taylor expand the exponential, writing exp(—b/v) ~ 1 —(b/v), we conclude that the number of new infections caused by one infected individual is approximately Nb/v. This value - the number of new infections caused by one infected individual entering a population of susceptible individuals - is called the basic reproductive rate of the disease and is usually denoted by R0. Note that R0 > 1 is the condition for the spread of the disease, and it is exactly the same condition that we arrived at by studying the Eq. (5.2) for the dynamics of infection. In this case, R0 tells us something interesting about the dynamics of the disease too, since we can rewrite Eq. (5.1) as (1/v)(d//dt) = (R0 — 1)/ — (b/v)/2; see Keeling and Grenfell (2000) for more on the basic reproductive rate.

0 0

Post a comment