Impact of Diffusion on synchronization pattern of epidemics in nonidentical metapopulation networks

TL;DR

A reduced model is developed and the emergent clustering behavior is validated through comprehensive simulations that observes that nodes equipped with testkits and no testkits tend to segregate into two separate clusters when migration is low, but above a critical migration rate, they coalesce into one single cluster.

Abstract

In a prior study, a novel deterministic compartmental model known as the SEIHRK model was introduced, shedding light on the pivotal role of test kits as an intervention strategy for mitigating epidemics. Particularly in heterogeneous networks, it was empirically demonstrated that strategically distributing a limited number of test kits among nodes with higher degrees substantially diminishes the outbreak size. The network's dynamics were explored under varying values of infection rate. In this research, we expand upon these findings to investigate the influence of migration on infection dynamics within distinct communities of the network. Notably, we observe that nodes equipped with test kits and those without tend to segregate into two separate clusters when coupling strength is low, but beyond a critical threshold coupling coefficient, they coalesce into a unified cluster. Building on this clustering phenomenon, we develop a reduced equation model and rigorously validate its accuracy through comprehensive simulations. We show that this property is observed in both complete and random graphs.

Figures (8)

• Figure 1: An illustrative overview of our study depicts a meta-population model wherein each community shares an identical population size. Distinguished by colour, some communities possess access to test kits (depicted in green), while others lack this resource (depicted in red). Within each community, a subset of individuals are infected. In Case A, the network exhibits low coupling strength (represented by dotted arrows) between communities, while Case B portrays a scenario with increased coupling strength. In Case B, the infection trends of the communities cluster, forming distinct clusters under the influence of higher coupling strength.
• Figure 2: Figures A through I show snapshots of infection trends among different communities with changing coupling strengths. There are $20$ communities, with $5$ having access to test kits. Each community has a population of $10000$ people, and $E_0$ for each community is randomly sampled between [1, 750]. The plot parameters used were $\sigma = 0.1,\alpha_0 = 0.01,\ \alpha_1 = 0.0001,\ \gamma = 0.07,\ \zeta = 0.02,\ \chi = 0.1$ and $\beta = 0.03$.
• Figure 3: Plots A and C show the spread of $I$max values for different communities decreasing with increasing $\epsilon$. Plots B and D show the decrease in Synchronisation Error with increasing $\epsilon$. There are 100 communities each with a population of $10000$ people. The plot parameters used were $\sigma = 0.1,\ \alpha_0 = 0.01\, \alpha_1 = 0.0001,\ \gamma = 0.07,\ \zeta = 0.02,\ \chi = 0.1$ and $\beta = 0.03$. $p$ denotes the fraction of communities having test kits. We can observe the existence of a bicluster in Plots A and C and confirm that the GSE vanishes only after CSE for both test-kit and testkit-free communities vanishes in plots B and D.
• Figure 4: Here we take 51 communities, each with a population of $10000$ people. The plot parameters used were $\sigma = 0.1,\ \alpha_0 = 0.01,\ \alpha_1 = 0.0001,\ \gamma = 0.07,\ \zeta = 0.02,\ \chi = 0.1$ and $\beta = 0.03$. The 2D projection of Synchronisation errors (Global and Cluster-wise) in the $\mathcal{R}_0$ vs $\epsilon$ plane are shown from (A-C) and in the $p$ vs $\epsilon$ plane are shown from (D-F). The dashed line indicates the contour line when the CSE vanishes, and the solid line indicates the same for GSE. Additionally, the dotted line in plot (D) shows the same threshold obtained via the reduced equation model.
• Figure 5: Plotting $I$max obtained from simulation and reduced equation model together to see areas of synchronisation. There are $100$ communities with a population of $10,000$ people each. 0.2 and 0.8 fractions of communities get test kits. Initial conditions were randomised as a uniform distribution between 1 and 1000. The plot parameters used were $\sigma = 0.1,\ \alpha_0 = 0.01,\ \alpha_1 = 0.0001,\ \gamma = 0.07,\ \zeta = 0.02,\ \chi = 0.1$ and $\beta = 0.03$.