Population mobility induced phase separation in SIS epidemic and social dynamics

Abstract

Understanding the impact of behavior dependent mobility in the spread of epidemics and social disorders is an outstanding problem in computational epidemiology. We present a modelling approach for the study of mobility that adapts dynamically according to individual state, epidemic/social-contagion state and network topology in accordance with limited data and/or common behavioral models. We demonstrate that even for simple compartmental network processes, our approach leads to complex spatial patterns of infection in the endemic state dependent on individual behavior. Specifically, we characterize the resulting phenomena in terms of phase separation, highlighting phase transitions between distinct spatial states and determining the systems' phase diagram. The existence of such phases implies that small changes in the populations' perceptions could lead to drastic changes in the spatial extent and morphology of the epidemic/social phenomena.

Figure 1

Dominant mixing preferences within each quadrant of the ${\alpha }^I,{\alpha }^S$ phase space. The larger the magnitude of ${\alpha }^I$ or ${\alpha }^S$ the more dominant the mixing behaviour. The mixing of each quadrant from top right to bottom right counter-clockwise may be used to describe: typical epidemic behaviour where all individuals avoid infectious individuals (top right), Schelling style segregation where individuals actively seek similar individuals (top left), spreading of social myths where individuals who are aware of the myth continue to associate with those who do (bottom left) and rioting behaviours (bottom right).

Figure 2

Simulation snapshots after 20000 timesteps with Δ$t=0.001$ for $\beta =10$, $\gamma =5$ $\omega =1$ on a $50\times 50$ lattice demonstrating qualitatively different phases across a phase space defined by ${\alpha }^S$ and ${\alpha }^I$. Region (i), the unseparated regime, is coloured green, region (ii), the connected regime, is coloured red, region (iii), the isolated regime is coloured blue whilst region (iv) the anti-aligned regime is coloured in orange. Each snapshot is renormalised by the infection mean and standard deviation to emphasise key spatial differences. Darker areas represent regions of higher relative infection.

Figure 3

Heatmaps of each of the order parameters (a) standard deviation σ, (b) staggered magnetisation $M_N$ and (c) percolation order parameter $\langle \hat{r}\rangle$ for $\beta =10$, $\gamma =5$, $\omega =1$ on a $50\times 50$ lattice averaged across 50 runs. A vanishing standard deviation (subfigure (a)) is associated with the unseparated regime whilst positive standard deviations are associated with phase separation (the connected, isolated and anti-aligned regimes).We note the largest standard deviations occur for $\left|{\alpha }^S\right|\approx 0$ when the movement of the susceptible population occurs irrespective of the presence of infection. The staggered magnetisation (subfigure (b)) distinguishes the anti-aligned regime (yellow) from the other phase separated regimes (light blue) and the unseparated regimes (dark blue). $\langle \hat{r}\rangle$ (subfigure (c)) can be used to distinguish two distinct phases in the upper left quadrant previously referred to as the connected regime (yellow region) and the isolated regime (blue region). $\langle \hat{r}\rangle$ is undefined in the white regions (the unseparated regime).

Figure 4

Normalized mean cluster size of a randomly selected location $\langle \hat{r}\rangle$ as a function of ${\alpha }^S$ for ${\alpha }^I=27.5$ with $\beta =10$, $\gamma =5$, $\omega =1$, on three different lattice sizes. This slice across ${\alpha }^I=-27.5$ intersects the connected and isolated regimes. Inset are simulation snapshots on a $50\times 50$ lattice after 20,000 timesteps with Δ$t=0.001$ for three values of ${\alpha }^S$ which demonstrate examples of the spatial patterns observed in regions (a) ${\alpha }^S<20$ (the isolated regime), (b) $20<{\alpha }^S<30$ (near criticality), (c) ${\alpha }^S>30$ (the connected regime).

Figure 5

Fisher information of $\hat{r}$ with respect to ${\alpha }^S$, the probability that a randomly selected site belongs to a cluster of normalized area $\hat{r}$. We compute $F_{\hat{r}}\left({\alpha }^S\right)$ for ${\alpha }^I=-27.5$, $\beta =10$, $\gamma =5$, $\omega =1$ on three different lattice widths $L$. The distribution for calculating the Fisher Information is an aggregate of 5000 time series from randomized initial conditions with an identical number of bins. We observe that the maximal Fisher information corresponds with the maximal growth in $\langle \hat{r}\rangle$ shown in Fig. 4.

Figure 6

Phase diagrams for multiple values of $\beta$ for $\gamma =5$, $\omega =1$ on a $50\times 50$ lattice. The region in grey corresponds to the unseparated regime below the critical epidemic threshold. Regions in green correspond to unseparated regime, with endemic infection levels. Yellow regions correspond to the anti-aligned phase. Blue regions correspond to the isolated regime and red regions correspond to the connected regime. The majority of the phase separating behaviour in the assortative mixing case when ${\alpha }^S>0$ and ${\alpha }^I<0$.