Last modified: 2014-03-31

#### Abstract

Realistic epidemic models take into account some spatial heterogeneity. The approach of complex networks can be

suitable to deal with a set of local populations pairwise-connected by diffusion/migration of individuals. We have extended our

previous works, [1] and [2], to a non-linear diffusion and demographic turnover. The mean-field type model is a system of ordinary differential equations which combines the (random, memoryless) movement of individuals among patches (nodes) with a local SIS-epidemics (with infection and recovery $\beta, \mu$, and equal birth and death rates $\delta$) within each

patch. The state variables refer to the

average number of susceptible and infected individuals in patches of degree $k$ (connections). The total number of individuals is conserved at the metapopulation level, i.e. $N \sum_{k} p(k) (\rho_{S,k}(t) + \rho_{I,k}(t))= N \rho^0$ with $p(k)$ being the degree distribution and $N$ the number of nodes.

The system includes a density-dependent contact rate $c(\rho_k)$, defined as the number of contacts an

individual makes in a local population of size $\rho_k= \rho_{S,k}+\rho_{I,k}$, and we consider two cases: limited

or non-limited transmission. In addition, the present model also includes density-dependent diffusion rates $D_S( \rho_{k} )$ and $D_I( \rho_{k} )$ in order to deal with demographic effects on the migration process. For instance, the model can tackle the human mobility from rural areas to big cities to look for job opportunities, or the other way round, from crowded areas to small villages to get rid of stress.

The spatial structure is described by the Connectivity Matrix $\mathcal{C}= (\frac{k}{k'} P(k'|k))$ with $P(k'|k)$ being the conditional probability that a patch of degree $k$ has one connection (link) to a patch of degree $k'$. We study both the correlated and uncorrelated cases.

Under suitable assumptions on the diffusion rates, i.e. the function $D( \rho ) \rho$ is increasing for both types of individuals, the disease-free equilibrium is determined by the migration process ($\mathcal{C}-Id$) although is independent of the degree-correlation. Here, each local population size is a function of the degree of the patch which is not necessarily linear as in the case of constant diffusion rates ([1] and [2]). Moreover, its (un)stability is analysed according to both the Jacobian Matrix and the Next Generation Matrix, at the disease-free equilibrium.