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Although it gives some nice predictions, there are some oversimplifications in the SIR model. One such oversimplification was made in the calculation of the number of newly infected people each day. We stated that the amount of new infections was proportional to the number of contacts, which was given by $SI$. This assumes the population mixes homogeneously. But this isn’t entirely correct. Children during schooldays, for example, make on average much more contact to others than elderly people. If a child becomes infected, the disease will therefore spread more rapidly than in the case of an infected elder. These differences in contacts in sub populations can be included in the SIR model.
To do this, we subdivide the whole population into age classes. We treat individuals in a particular class the same way. Instead of speaking of the amount of contacts between the whole population, we can speak of contacts between classes. Consider for simplicity the case of two subclasses: males and females. We can for example state that males make more contact with other males than with females. Consider 3 males (Bob, Rob and Zob) and 3 females (Julia, Victoria and Amelia) whom/which we have asked about their contacts with the others.
The table shows the contacts, one can see for example that Bob has been in contact with the other two males and with Victoria. We can now count the total amount of contacts between the two classes. There were 3 different male-male contacts, 2 male-female, the same 2 female-male and 2 female-female contacts. We insert this information in a matrix.
\[ \begin{pmatrix} 3 & 2 \\ 2 & 2 \end{pmatrix} \]
It is important to note that the numbers in the matrix needs to be scaled for them to give data which can be extrapolated to other sizes of population. If we have, for example, 30 males and 30 females, then we can estimate the contacts by the information we have on the 3 males and 3 females in the previous example.
Let’s explain this procedure. Take a person from the 30 females, let’s call her Sofia. When Sofia is placed in a group of two other womans and 3 males, we know by our data that she is likely to make 2/3 contacts with the males. Here we divided by the total amount of females to get an average of contacts of 1 woman in contact with 3 males. Now keep in mind that she was in touch with three males. In a simple model we can state that if she would have been placed with 30 other males her contacts would increase with a factor of 10. In the same way, the average amount of contacts of her with 1 male would then be 2/9. Let’s recall what we have done. First we averaged over the amount of females which participated in the contact-experiment. Second we averaged this further over all the male participants. The resulting number is independent of the population size and represents the total amount of contacts a female will make with another male. This can then be scaled back for more females and males. So in the previous example we would get a total of $ 2/9 \times 30 \times 30 = 200$ female-male contacts.
The number 2/9 we calculated previously, following from averaging over all the males and females is an important number. It doesn’t depend on the size of the population and contains all the information of the female-male contacts in such a population. We call such a number a mixing rate between two classes in a population. We can calculate the other mixing rates (male-male and female-female) in a similar way and get the mixing matrix
\[ C = \begin{pmatrix} 3/9 & 2/9 \\ 2/9 & 2/9 . \end{pmatrix} \]
In many situations we subdivide the populations into more than two subclasses. The number or rows and columns of $C$ is given by the number of subclasses. We usually denote the elements of $C$ by $\beta_{ij}$, the important thing to remember is that the amount of contacts between two subclasses $i$ and $j$ is given by \[\beta_{ij} N_i N_j, \] where $N_i$ and $N_j$ are the number of people in the corresponding subclass.
The matrix $C$ contains all intrinsic information of a population. However we made some oversimplifications, for example scaling doesn’t always work this way. Let’s say we put one woman in a group of 10 males and she makes 4 contacts a day. Now place that woman in a group of 1000 males. The previous method now states she is likely to make 400 contacts a day. This is an extraordinary amount of contacts for one person. In a real-life situation there will be a saturation, which we’re not going to consider.
The mixing matrix is important in the study of infectious diseases, we give some applications of it in the next page.