My first crack at an age-structured disease model with multiple infections yielded the interesting result that one could observe differences in disease rates across age without any actual differences in the biology between old and young individuals. Today I’m trying to answer a different question analytically: does age structure matter for the course of the disease in the whole population?
Here’s the model. There are two minor changes: I no longer scale \(J\) and \(A\) with \(K\) in the third and fourth equations; there’s no reason the encounter rate should be tied to carrying capacity. Also, I’ve put stage-specific values for all the parameters (now including \(\alpha\), \(\mu\), and \(\lambda\)).
\[\begin{aligned} \frac{dJ}{dt} &= A f_A \left(1 - \frac{J+A}{K} \right) + J \left(f_J \left(1 - \frac{J+A}{K} \right) - d_J - g\right) - \alpha_J P_J \\ \frac{dA}{dt} &= J g - A d_A - \alpha_A P_A \\ \frac{dP_J}{dt} &= \lambda_J J (P_J + P_A) - P_J \left(d_J + \mu_J + g + \alpha_J \left(1 + \frac{P_J}{J} \right) \right) \\ \frac{dP_A}{dt} &= \lambda_A A (P_J + P_A) + P_J g - P_A \left(d_A + \mu_A + \alpha_A \left(1 + \frac{P_A}{A} \right) \right) \end{aligned}\]
To examine whether the age structure matters, I try to collapse this down to a model of just one aggregated population. Let
\[\begin{aligned} N = J + A, && P = P_J + P_A \end{aligned}\]
The parameters then become the population-weighted means of their components:
\[\begin{aligned} d = (J d_J + A d_A)/N, && f = (J f_J + A d_A)/N \\ \mu = (P_J \mu_J + P_A \mu_A)/P, && \alpha = (P_J \alpha_J + P_A \alpha_A)/P \end{aligned}\]
The first two equations sum easily:
\[\frac{dN}{dt} = Nf\left(1 - \frac{N}{K}\right) - dN - \alpha P\]
The second two equations don’t do quite the same thing:
\[\begin{aligned} \frac{dP}{dt} & = \lambda NP - \mu P - \left(d_J P_J + d_A P_A\right) - \alpha P - \left(\alpha_J \frac{P_J^2}{J} + \alpha_A \frac{P_A^2}{A}\right) \end{aligned}\]
To compare this equation to a model with no age structure, I define two terms, \(\psi\) and \(\phi\) which represent the differences between the terms in parentheses in the equation above and the equivalent term in a model without age structure. For ease of later calculations, I scale these by \(P\):
\[\begin{aligned} \psi P &= (d_J P_J + d_A P_A) - dP \\ &= (d_J P_J + d_A P_A) - \frac{d_J J + d_A A}{J + A} (P_J + P_A) \\ &= \frac{(d_J - d_A)P_J A + (d_A - d_J)P_A J}{J+A} \\ &= \frac{(d_J - d_A)(P_J A - P_A J)}{J+A} \end{aligned}\]
\[\begin{aligned} \phi P &= \left(\alpha_J \frac{P_J^2}{J} + \alpha_A \frac{P_A^2}{A}\right) - \alpha \frac{P^2}{N} \\ &= \left(\alpha_J \frac{P_J^2}{J} + \alpha_A \frac{P_A^2}{A}\right) - \frac{(\alpha_J P_J + \alpha_A P_A)(P_J + P_A)}{(J+A)} \\ &= \frac{(\alpha_J P_J A - \alpha_A P_A J)(P_J A - P_A J)}{JA(J+A)} \end{aligned}\]
I substitute these into \(\frac{dP}{dt}\) above to get
\[\frac{dP}{dt} = \lambda N P - (\mu + d) P - \alpha P \left(1 + \frac{P}{N}\right) - (\psi + \phi)P\]
How to \(\phi\) and \(\psi\) modify the dynamics? \(\psi\) is zero at \(d_A=d_J\) or \(\frac{P_J}{J} = \frac{P_A}{A}\). It will be positive if the class with higher infection rate also has a higher base mortality rate, and negative otherwise. A positive \(\psi\) value will decrease the growth of the parasite population, as more-infected individuals will die at a faster rate, removing more parasites from the population.
\(\phi\) is zero at at \(\frac{P_J}{J} = \frac{P_A}{A}\), but not at \(\alpha_J = \alpha_A\), meaning that dynamics should deviate from an a model without age structure even if there is no difference in disease-driven mortality between age classes. For \(\phi\) to be positive either both or neither of the terms in parenthesis must be greater than zero, or
\[\left(\frac{P_J}{J} > \frac{P_A}{A}\right) \Leftrightarrow \left(\alpha_J \frac{P_J}{J} > \alpha_A \frac{P_A}{A}\right)\]
Note that if \(\alpha_J = \alpha_A\), this is always true. So if the disease has an identical effect on both age classes, \(\phi\) will be positive. A positive value of \(\phi\) will slow growth of the parasite population by increasing the death rate of the more infected population, more rapidly removing parasites. The reverse will happen when \(\phi\) is negative. The more infected population will lose fewer individuals to disease because of a low relative \(\alpha\) value, and the rate of parasite growth will increase because these individuals will remain to reproduce. Essentially, the value of \(\phi\) represents whether the ratio \(\alpha_J/\alpha_A\) enhances or offsets the non-linear relationship between parasite load and parasite loss caused by aggregation of parasites.
Now the system of equations is:
\[ \begin{aligned} \frac{dN}{dt} &= Nf \left(1 - \frac{N}{K}\right) - dN - \alpha P \\ \frac{dP}{dt} &= \lambda N P - (\mu + d) P - \alpha P \left(1 + \frac{P}{N}\right) - P(\psi + \phi) \end{aligned} \]
Since all terms in \(\psi\) and \(\phi\) contain \(P_X\) values in the numerators but not denominators, as \(P \rightarrow 0\), so do \(\psi\) and \(\phi\). This means that the growth rate of the parasites is unaffected by the population structure when numbers are small. We can calculate this rate, which is the same as it would be with no population structure:
\[\frac{dP/dt}{P}\bigg|_{P\rightarrow 0} = \lambda N - \mu - d - \alpha\]
\[N_{\text{threshold}} = \frac{\mu + d + \alpha}{\lambda}\]
(A reminder here that reducing \(\alpha\) but not \(\lambda\) in the tanoak population is a losing strategy)
Setting \(dN/dt = 0\), we can solve for \(\frac{P}{N}\).
\[ \frac{P}{N} = \frac{f\left(1-\frac{N}{K}\right) -d }{\alpha} \]
Under disease-free conditions, this is easily solved for \(N\):
\[N_{eq}\big|_{P=0} = {K\left(1-\frac{d}{f}\right)}\]
If the population was unstructured, we would get:
\[N_{eq}\big|_{P_{eq} > 0} = \frac{\mu + \alpha + f}{\lambda + \frac{f}{K} } \]
With the age structure, \[ \begin{aligned} \frac{dP}{dt} &= 0 = P\left(\lambda N - \mu - d - \alpha - \alpha \frac{P}{N}\right) - P(\psi + \phi)\\ N &= \frac{\mu + \alpha + f + \psi + \phi}{\lambda + \frac{f}{K}} \end{aligned} \]
Positive values of \(\psi\) and \(\phi\) increase the overall host population, and following the expression for \(P/N\) above, reduce the mean infection level.