# Metapopulation SEIR model

Author: Constanze Ciavarella @ConniCia

Date: 2018-10-02

This code combines two deterministic metapopulation SEIR models as described in Lloyd & Jansen (2004).

### Cross-coupling between patches - Equations (8-10)

Cross-coupling is controlled through matrix $\beta$, describing the effective contact rates acting within and between patches.

Setting the off-diagonal elements of $\beta$ to zero, we switch off cross-coupling across patches.

### Migration between patches - Equations (11-13)

Matrix $C$ must be such that the elements on the diagonal, denoting outflow of each patch, are negative. Element $c_{ij}$ describes the flow from patch $i$ to patch $j$. For each row, the sum all elements on the row is 0.

Setting all elements of $C$ to zero, we switch off migration between patches.

### Model description

This model consists of many SEIR models connected through between-patch contact and/or migration of individuals between patches. The model has a constant total population size, which means that births and deaths correspond at each time step.

• $n$ = number of patches
• $S_1, …, S_n$ = susceptibles in patches $1, …, n$
• $E_1, …, E_n$ = exposed in patches $1, …, n$
• $I_1, …, I_n$ = infectious in patches $1, …, n$
• $R_1, …, R_n$ = recovered in patches $1, …, n$
• $\beta_{ij}$ = effective contact rate of infected individuals of patch $i$ to susceptible individuals of patch $j$
• $c_{ii}$ = outflow of patch $i$
• $c_{ij}, i \neq j$ = flow from patch $i$ to patch $j$
• $\sigma$ = rate of breakdown to active (and infectious) disease
• $\gamma$ = rate of recovery from active disease
• $\mu$ = background mortality/birth rate

The model will be written as %