## Adding demography to SIR or SIRS models

Until now, we have ignored all other biological processes that might occur concomitantly with the disease. One possibility is population growth and mortality that is independent of the disease. There are many different ways that one may add demographic processes to the S/R or S/RS models. Here, I pick an especially simple case, to illustrate how this can be done and how the conclusions of the previous sections might change.

When adding demography, we need to be careful and explicit about the assumptions. Let us assume that (1) only susceptible individuals reproduce, and do so at a density-independent rate r, (2) all individuals experience mortality h that is independent of the disease with r > h, and (3) there is no disease-dependent mortality. In that case, the S/R equations (5.6) become ddt = -b/S +(r - h)S

The term representing demographic process of net reproduction is (r - h)S. Other choices are possible; for example we might assume that both susceptible and recovered individuals could reproduce, that all individuals can reproduce (still with no vertical transmission) or that birth rate is simply a constant (e.g. proportional to N). Each of these could be justified by a different biological situation and may lead to different insights than using Eqs. (5.18); h/ and hR are demographic sources of mortality. If one particularly appeals to you, I encourage you to redo the analysis that follows with the assumption that you find most attractive.

We proceed to find the steady states by setting the left hand side of Eqs. (5.18) equal to 0. When we do this, we obtain (from dS/dt = d//dt = dR/dt = 0 respectively)

We learn an enormous amount just from the steady states. First, recall that for the S/R model without demography, the only steady state is / = 0. However, from Eqs. (5.19), we conclude that in the presence of demographic factors, a disease that would be epidemic becomes endemic. Second, we see that the steady state levels of susceptible, infected, and recovered individuals depends upon a mixture of demographic and disease parameters. Third, and perhaps most unexpected, note that the steady state level of susceptibles is independent of r! (You should think about the assumptions and results for a while and explain the biology that underlies it.) It is helpful to summarize the various versions of the S/R model in a single figure (Figure 5.6). Here I show the S/R model for an epidemic (panel a), the S/RS model for an endemic disease (which approaches the steady state in an oscillatory fashion) (panel b), and the S/R model with demography (panel c). Note the progression of increasing dynamic complexity (also see Connections).

Equations (5.19) beg at least two more questions: first, what is the nature of this steady state; second, what happens if there is more

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Figure 5.6. Solutions of various forms of the SIR model. (a) The basic SIR model for an epidemic (b = 0.005, v = 0.3; true for panels b and c); (b) the SIRS model for an endemic disease (f = 0.05); and (c) the SIR model with demography (f = 0, r = 0.1, ^ = 0.05).

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Figure 5.6. Solutions of various forms of the SIR model. (a) The basic SIR model for an epidemic (b = 0.005, v = 0.3; true for panels b and c); (b) the SIRS model for an endemic disease (f = 0.05); and (c) the SIR model with demography (f = 0, r = 0.1, ^ = 0.05).

complicated demography? These are good questions, but since I want to move on to other topics, I will leave them as exercises.

Conduct an eigenvalue analysis of the steady state in Eqs. (5.19). Note that there will be three eigenvalues. How are they to be interpreted?

How do Eqs. (5.19) change if we assume logistic growth rather than exponential growth as the demographic term. That is, what happens if we replace (r — m)S by rS(1 — (S/K))?

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