Consider an epidemic spreading in a population that has been aggregated by age into groups numbered $i=1, \ldots, M$. The $i$ th age group has size $N_{i}$ and the numbers of susceptible, infective and recovered individuals in this group are, respectively, $S_{i}, I_{i}$ and $R_{i}$. The spread of the infection is governed by the equations

$$\begin{aligned} \frac{d S_{i}}{d t} &=-\lambda_{i}(t) S_{i} \\ \frac{d I_{i}}{d t} &=\lambda_{i}(t) S_{i}-\gamma I_{i} \\ \frac{d R_{i}}{d t} &=\gamma I_{i} \end{aligned}$$

where

$$\lambda_{i}(t)=\beta \sum_{j=1}^{M} C_{i j} \frac{I_{j}}{N_{j}},$$

and $C_{i j}$ is a matrix satisfying $N_{i} C_{i j}=N_{j} C_{j i}$, for $i, j=1, \ldots, M$.

(a) Describe the biological meaning of the terms in equations (1) and (2), of the matrix $C_{i j}$ and the condition it satisfies, and of the lack of dependence of $\beta$ and $\gamma$ on $i$.

State the condition on the matrix $C_{i j}$ that would ensure the absence of any transmission of infection between age groups.

(b) In the early stages of an epidemic, $S_{i} \approx N_{i}$ and $I_{i} \ll N_{i}$. Use this information to linearise the dynamics appropriately, and show that the linearised system predicts

$$\mathbf{I}(t)=\exp [\gamma(\mathbf{L}-\mathbf{1}) t] \mathbf{I}(0),$$

where $\mathbf{I}(t)=\left[I_{1}(t), \ldots, I_{M}(t)\right]$ is the vector of infectives at time $t, \mathbf{1}$ is the $M \times M$ identity matrix and $\mathbf{L}$ is a matrix that should be determined.

(c) Deduce a condition on the eigenvalues of the matrix $\mathbf{C}$ that allows the epidemic to grow.