Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 13E / b

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- Solution b

Solution

With

S_{i} ≃ N_{i}

, the infective equations become

\dot{I}_{i} = β \sum_{j} \frac{N _{i} C _{ij}}{N _{j}} I_{j} - γ I_{i} = β \sum_{j} C_{ji} I_{j} - γ I_{i},

where reciprocity was used. For a column vector

I

,

\dot{I} = γ (L - 1) I, L = \frac{β}{γ} C^{T} .

Therefore

I (t) = e^{γ (L - 1) t} I (0) .

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