Introducing the age compartment

Since the establishment of epidemiology as a science, a lot of attention has been devoted to the study of measles as its recurring patterns puzzled physicians and mathematicians alike. The distinguishing characteristic of measles epidemics is that they had a very regular temporal pattern with periodic outbreaks of the disease, as shown in figure3.2. As this disease affects specially the children and also conveys permanent immunity to those who have suffered it, analyzing the time evolution over large time-scales to obtain the patterns required the inclusion of age in the models. Nevertheless, with the basic model that we have analyzed we can already propose a plausible explanation for this behavior. Indeed, we know that if the amount of susceptibles in the population is below a given threshold, the epidemic cannot take place. Thus, it seems reasonable to think that once the epidemic fades-out, there is a period in which there are not enough susceptibles for it to appear again. Yet, when new children are born, the amount of susceptibles will increase, possibly going above the threshold and therefore allowing a new outbreak.

A similar explanation was already proposed by Soper in 1929 [179], although it only matched the observations qualitatively, not quantitatively. It was Barlett who, in 1957, finally provided a quantitative explanation of the phenomenon [180]. Besides the details that we have already discussed, in his proposal he added a new factor that we have not mentioned yet. He proposed that the problem of previous models was that they were deterministic, an approximation that is only valid in very large populations. However, it was observed that the periodicity of measles not only depended on the size of the city, but it was specially so in small towns. In physical terms, we would say that there were finite size effects, tearing down assumption 6 (see section 3.1). Thus, he proposed to use a stochastic model for which he could not obtain a closed form solution, so he had to resort to an “electronic computer”. Nowadays the use of stochastic computational simulations are much more common than the deterministic approach. The reasons why this approach is more favorable are out of the scope of this thesis (see, for instance [154,181,182,183] for a discussion) but we will leverage this opportunity to say that in the following sections we will mostly work with stochastic simulations, rather than deterministic approaches. Before concluding the discussion about Barlett’s paper, we find worth highlighting that it was presented during a meeting of the Royal Statistical Society, in 1956, after which a discussion followed. In said discussion, Norman T. J. Bailey said “One of the signs of the times is the use of an electronic computer to handle the Monte Carlo experiments. Provided they are not made an excuse for avoiding difficult mathematics, I think there is a great scope for such computers in biometrical work”. And indeed there was, as 25 years later Mr. Bailey was appointed Professor of Medical Informatics [184].

0 5000 10000 15000 1906 1909 1912 1915 1918 1921 1924 1927 1930 1933 1936 1939 1942 1945 1948 Time Measles cases

Figure 3.2: Measles epidemics in New York from 1906 to 1948. This figure represents the number of reported cases of measles in the city of New York from 1906 to 1948 with a biweekly resolution. There are some gaps due to missing reports. Data obtained from [185].

introduced age in his models in 1926 [186], one year before the publication of the full model that we have already explored. However, to introduce age we will use a slightly more modern formulation that will simplify the analysis. In particular, we need to revisit assumption 4, i.e., individuals are only distinguishable by their health status.

Let us state that individuals can now be identified both by their health status and their age. Hence, we have to add more compartments to the model, one for each age group and health status combination. In other words, rather than having three compartments, S, I, R, we now have 3 times the number of age brackets considered, i.e. Sa, Ia, Ra being a the

age bracket the individuals belong to (see2.6 for the definition of age bracket). Moreover, we will suppose that the disease dynamics is much faster than the demographic evolution of the population. The only thing left is to decide how to go from one compartment to another:

• For the rate of infectivity, we will define an auxiliary expression that will facilitate the discussion. By inspection of equation (3.9), we can define the force of infection [178] as

λ(t) ≡ φ(τ )I(t) = βI(t) , (3.15) which does not depend on any characteristic of the individual. Hence, we can simply incorporate age by modifying the force of infection so that

λ(t, a) =X

a0

φ(τ, a, a0)Ia0(t) . (3.16)

This way, both the age of the individual that is getting infected (a) and the age of all other individuals (P

a0) are taken into account. Furthermore, we can separate

φ(τ, a, a0) into two components: one accounting for the rate of contacts between individuals of age a and a0 and another one accounting for the likelihood that such contacts lead to an infection. Hence,

φ(τ, a, a0) ≡ C(a, a0)β(a, a0) . (3.17)

Recalling section 2.6, the term C(a, a0) can be obtained from the contact surveys that we have already studied. On the other hand, we will suppose that the likelihood of infection is independent of the age so that β(a, a0) = β.

• For the rate of recovery, we will assume that it is independent of the age of the individual, i.e. µ(a) = µ.

Under these assumptions, the homogeneous mixing model with age dependent contacts

reads _                   dSa(t) dt = − X a0 βC(a, a0)Ia0(t)S_a(t) dIa(t) dt = X a0 βC(a, a0)Ia0(t)S_a(t) − µI_A(t) dRa(t) dt = µIa(t)