Which population will benefit most from vaccinating children?

Question

This is from Vynnycky and White's An Introduction to Infectious Disease Modelling.

In Population A, children have more contacts than do adults and most of the contacts of children are with other children.

In Population B, children have as many contacts as do adults and mainly contact adults, whereas adults mainly contact other children.

In the question posed in the image, "stronger effect" is equivalent to asking "in which population would the subsequent overall incidence be the lowest"?

The authors do not give an answer, but they give us a good start:

One intuitive answer to this question might be population A, given that the children in population A have more contacts than do children in population B. On the other hand, more of the contacts of children in population A than in population B are with other children. Thus, adults in population A will benefit less from the reduced incidence among children in their population resulting from the introduction of vaccination, than will adults in population B, in which there is much contact between children and adults.

Now, I am guessing that the answer depends on the ratio of adults and children in each populations. And I think that not every contact leads to a disease, but we can use probability and work out an average. I am also guessing that it is OK for us to assume that the contact that leads to disease does not differ whether it is a child contacting an adult or the other way around.

My question is, if the answer does depend on the ratio of adults and children in each population, can we find the ratio for which the children's vaccine will have a larger effect on a population if we are above the ratio? (Maybe the answer is always one population and does not depend on the ratio. Or maybe I am missing something besides the ratio.)

Answer 1

Suppose that in each round, every infected person contacts random people with the shown contact pattern (e.g. in population B each child contacts 1 other child and 3 adults), and causes each such contact to be infected with probability $p$ if the contact is not vaccinated and $p·r$ if the contact is vaccinated (i.e. $p$ is the transmission rate and $1-r$ is the immunization rate), and then recovers (i.e. becomes no longer infected). Here one round corresponds roughly to the average duration of the disease from onset to the end of the contagious period. Let $I(k)$ be the vector for the expected numbers of infected children and of infected adults just before round $k$. Then $I(k+1) ≤ M∘I(k)$ where $M$ is the linear transformation representing the contact pattern combined with the infection probabilities. This inequality is tight when $I(k)$ is very small compared to the total population size vector, since the random contacts are likely to be mostly unique. We are primarily interested in this situation, because the other situation is when the outbreak has swamped the whole population.

Specifically, if the outbreak always remains small compared to the whole population, then we have $I(k) ≈ M∘I(k)$ and hence $I(k) ≈ M^k∘I(0)$. If also $M = A∘D∘A^{-1}$ for some invertible $A$ and diagonal $D$ (i.e. $D$ is a scale transformation), then $I(k) ≈ A∘D^k∘A^{-1}∘I(0)$, and the asymptotic growth of $I(k)$ as $k→∞$ would be dominated by $D^k$. In this situation, the largest eigenvalue of $M$ (i.e. the largest scale factor of $D$) must be less than $1$. We can find $A,D$ by diagonalization if they exist.

In population A with only children vaccinated we have $M = p·\pmatrix{ 4r & r \\ 2 & 2 }$ with largest eigenvalue $p·(1+\sqrt{1-2r+4r^2}+2r) ≈ p·(2+r)$ as $r→0$.

In population B with only children vaccinated we have $M = p·\pmatrix{ r & 3r \\ 3 & 1 }$ with largest eigenvalue $p·\frac12(1+\sqrt{1+34r+r^2}+r) ≈ p·(1+9r)$ as $r→0$.

Therefore if the vaccination has high immunization rate, then it is more effective in population B. But if the vaccination has low immunization rate (i.e. $r > \frac18$ roughly) then it is more effective in population A.

Actually, it is easy to observe the results for $r→0$ directly from the contact patterns; if all children are immune then clearly in population B the virus will eventually die off whereas in population A every adult passes to about $2p$ others.

Anyway note that the assumption of random contacts is actually very bad. In real life there is a rather constant connectivity graph and the contacts are chosen based on this connectivity graph and not randomly from the entire population. But that makes it much harder to mathematically analyze, and it is unclear what the contact patterns shown even mean in this model.