# How to model social structure in SIR models

I refer to J.H. Jones' Notes on R0.

More details in this question at Mathematics SE: How does the reproduction number depend on characteristics of the physical contact graph of a population?

The basic SIR model - as described in Jones' Notes - considers three factors that make up the reproduction number:

• $$\tau$$ = the transmissibility (i.e., probability of infection given contact between a susceptible and infected individual)

• $$\overline{c}$$ = the average rate of contact between susceptible and infected individuals

• $$d$$ = the duration of infectiousness

The (basic) reproduction number then is

$$R_0 = \tau \cdot \overline{c} \cdot d$$

I wonder how to model social structure in a SIR model, which might influence both transmissibilty $$\tau$$ and average rate of contact $$\overline{c}$$ and thus the reproduction number $$R_0$$, resp. the infection rate $$\beta = \tau \cdot \overline{c} = R_0 / d$$.

Specifically I wonder which "control parameters" appropriately describe the social structure of a population and may influence transmissibilty $$\tau$$ and average rate of contact $$\overline{c}$$ in a specific and comprehensible way.

Among these parameters may be:

• typical (mean) numbers of household members and acquainted persons

• typical ratio of shared acquaintances

• typical numbers of strangers one meets (e.g. per month)

• typical times of contact to household members, acquainted persons, and strangers

• the amount of persons with outstanding many social contacts, better: the standard deviation of the contact rate

• the accepted physical distance to family members, acquainted persons and strangers
(source)

I am looking for approaches that let go such parameters into the infection rate $$\beta = \tau \cdot \overline{c}$$ in a quantitative way.