V

R0 > 1. Thus the heuristic analysis and the phase plane analysis lead to the same conclusion. This remarkable result is called the Kermack-McKendrick epidemic theorem. Note that once again, the threshold depends upon the number of susceptible individuals, not the number of infected individuals.

We can actually do more by noting that dI/dS = (d//dt)/(dS/dt) from which we conclude

If we think of I as a function of S, then I will takes its maximum when dI/dS = 0; this occurs when S = b/v. We already know this from the phase plane, but Eq. (5.7) allows us to find an explicit representation for I(t) and S(t).

Separate the variables in Eq. (5.7) to show that

I (t)+ S(t)— b log(S (t)) = I (0)+ S(0)— b log(S(0)) (5.8)

Note that this equation allows us to find the relationship between I(t) and S(t) at any time in terms of their initial values.

How about computation of trajectories? That involves the solution of Eq. (5.6.) We might work with the variables S(t) and I(t) themselves, which could involve dealing with relatively large numbers. For those who want to write their own iterations by treating the differential equation as a difference equation, I remind you of the warning that we had in Chapter 2 on the logistic equation. The following observation is helpful. If we set S(t + dt) = S(t)exp(—bI(t)dt), then in the limit that dt! 0, we get back the first line of Eq. (5.6) (if this is unclear to you, Taylor expand the exponential, subtract S(t) from both sides, divide by dt and take the limit). This reformulation also provides a handy interpretation: exp(—bI(t)dt) < 1 and can be interpreted as the fraction of susceptible individuals who escape infection in the interval (t, t + dt) when the number of infected individuals is I(t).

However, because the population is closed and R(t) = N — S(t) — I(t), we can focus on fraction of susceptible and infected individuals, rather than absolute numbers. That is, if we set S(t) = s(t)N, I(t) = i(t)N and P = bN as in Eq. (5.3), the first two lines of Eq. (5.6) become

— —Bis — vi dt to which we append initial conditions s(0) = s0 and i(0) = i0. Note that the critical susceptible fraction for the spread of the epidemic is now v/3. These equations can be solved by direct Euler iteration or by more complicated methods, or by software packages such as MATLAB.

Solve Eqs. (5.9) for the case in which the critical susceptible fraction is 0.4, for values of s(0) less than or greater than this and for i(0) = 0.1 or 0.2.

Kermack and McKendrick, who did not have the ability to compute easily, obtained an approximate solution of the equations characterizing the epidemic. To do this, they began by noting that since the population is closed we have dR/dt = v/ = v(N — S — R), which at first appears to be unhelpful. But we can find an equation for S in terms of R by noting the following dR=~(b)s (5-)

and so we see that S, as a function of R, declines exponentially with R; that is S(R) = S(0)exp(—(b/v)R). When we use this in the equation for R, we thus obtain dR

to which we add the condition R(0) = N — S0 — /0 and from which we would like to find R(t), after which we compute S(t) = S(0)exp(—(b/v) R(t)) and from that /(t) = N — S(t) — R(t). However, Eq. (5.11) cannot be solved either. In order to make progress, Kermack and McKendrick (1927) assumed that bR ^ v (how do you interpret this condition?), so that the exponential could be Taylor expanded. Keeping up to terms of second order in the expansion, we obtain dR

and this equation can be solved (Davis 1962). In Figure 5.3, I have reprinted a figure from Kermack and McKendrick's original paper, showing the general agreement between this theory and the observed data, the solution of Eq. (5.12) (although their notation is slightly different than ours), and their comments on the solution.

To close this section, and give a prelude to what will come later in the chapter, let us ask what will happen to the dynamics of the disease if individuals can either recover or die. Thus, let us suppose that the

5 10 15 20 25 30 eeks

Figure 1. Deaths from plague in the island of ombay over the period 17 December 1905 to 21 uly 1906. The ordinate represents the number of deaths per week and the abscissa denotes the time in weeks. s at least 80-90% of the cases reported terminate fatally the ordinate may be taken as approximately representing dzd t as a function of t. The calculated curve is drawn from the formula y = ^ = 890 sech2(0.2t - 3.4) dt e are in fact assuming that plague in man is a reflection of plague in rats and that with respect to the rat (1) the uninfected population was uniformly susceptible (2) that all susceptible rats in the island had an equal chance of being infected (3) that the infectivity recovery and death rates were of constant value throughout the course of sickness of each rat (4) that all cases ended fatally or became immune (5) that the flea population was so large that the condition approximated to one of contact infection. one of these assumptions are strictly fulfilled and consequently the numerical equation can only be a very rough approximation. close fit is not to be expected and deductions as to the actual values of the various constants should not be drawn. t may be said however that the calculated curve which implies that the rates did not vary during the period of epidemic conforms roughly to the observed figures. 5 10 15 20 25 30 eeks

Figure 5.3. Reproduction of Figure 1 from Kermack and McKendrick (1927), showing the solution of Eq. (5.12) and a comparison with the number of deaths from the plague in Bombay. Reprinted with permission.

mortality rate for the disease is m. The dynamics of susceptible and infected individuals are now

and the basic reproductive rate of the disease is now R0 = bS0/(v + m). How might the mortality from the disease, m, be connected to the rate at which the disease is transmitted, b? We will call m the virulence or the b(m ) Contagiousness b(m ) Contagiousness m ir ulence

Figure 5.4. The assumed relationship between contagion or infectiousness, b(m) and virulence or infectedness, m.

m ir ulence

Figure 5.4. The assumed relationship between contagion or infectiousness, b(m) and virulence or infectedness, m.

infectedness and assume that the contagiousness or infectiousness is a function b(m) with shape shown in Figure 5.4. The easiest way to think about a justification for this form is to think of m and b(m) as a function of the number of copies of the disease organism in an infected individual. When the number of copies is small, the chance of new infection is small, and the mortality from the disease is small. As the number of copies rises, the virulence also rises, but the contagion begins to level off because, for example, the disease organism is saturating the exhaled air of an infected individual.

If we accept this trade-off, the question then becomes what is the optimal level of virulence? To answer this question, which we will do later, we need to decide the factors that will determine the optimal level, and then figure out a way to find the optimal level. For example, is making m as large as possible optimal for the disease organism? I leave this question for now, but you might want to continue to think about it.

In this section, we considered a disease that is epidemic: it enters a population, and runs it course, after which there are no infected individuals in the population. We now turn to a case in which the disease is endemic -there is a steady state number of infected individuals in the population.

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