On the Interplay of Clustering and Evolution in the Emergence of Epidemic Outbreaks

TL;DR

This work is the first to jointly analyze the impact of clustering and evolution on the emergence of epidemic outbreaks, and derives a mathematical framework to quantify the epidemic characteristics of a contagion that evolves as it spreads with the structure of the underlying network as given via arbitrary degree distributions.

Abstract

In an increasingly interconnected world, a key scientific challenge is to examine mechanisms that lead to the widespread propagation of contagions, such as misinformation and pathogens, and identify risk factors that can trigger large-scale outbreaks. Underlying both the spread of disease and misinformation epidemics is the evolution of the contagion as it propagates, leading to the emergence of different strains, e.g., through genetic mutations in pathogens and alterations in the information content. Recent studies have revealed that models that do not account for heterogeneity in transmission risks associated with different strains of the circulating contagion can lead to inaccurate predictions. However, existing results on multi-strain spreading assume that the network has a vanishingly small clustering coefficient, whereas clustering is widely known to be a fundamental property of real-world social networks. In this work, we investigate spreading processes that entail evolutionary adaptations on random graphs with tunable clustering and arbitrary degree distributions. We derive a mathematical framework to quantify the epidemic characteristics of a contagion that evolves as it spreads, with the structure of the underlying network as given via arbitrary {\em joint} degree distributions of single-edges and triangles. To the best of our knowledge, our work is the first to jointly analyze the impact of clustering and evolution on the emergence of epidemic outbreaks. We supplement our theoretical finding with numerical simulations and case studies, shedding light on the impact of clustering on contagion spread.

Figures (8)

• Figure 1: An illustration of the multi-strain spreading on a clustered random network. The contact network comprises a clustered network where the number of single-edges and triangles attached to each node is separately specified through the joint degree distribution for the configuration model. We consider the propagation of two strains (strain-$i$, $i=1,2$) of a contagion indicated in blue and red. The spreading proceeds as follows: i) Initially all nodes are in the susceptible state and an arbitrarily chosen seed node acquires strain-$i$. ii) The seed node independently infects its susceptible neighbors with a probability corresponding to the strain it carries ($T_i$). iii) After transmission, the contagion mutates to strain-$j$ within the newly infected hosts independently with probability $\mu_{ij}$. iv) The process continues recursively and terminates when no further infections are possible.
• Figure 2: An illustration of the modified configuration model NewmanRandomClustering with a given realization for the degree sequence of the single-edges and triangle-edges. We assign single stubs and triangle corners as per the degree sequence, e.g., node $v_1$ admits no single stubs and only one triangle corner. In order to create the network, we choose pairs of single stubs uniformly at random and join them to make a complete edge between two nodes, and we choose triplets of triangle corners uniformly at random and join them to form a triangle. The graph generation algorithm enables generating networks with tuneable clustering coefficients.
• Figure 3: Different possible configurations for a triangle emanating from a parent node carrying strain-$1$. Nodes that acquire strain-$1$ (resp., strain-$2$) after mutation are indicated in blue (resp., red). The configurations are based on whether the node at either endpoint of the triangle gets infected and the resulting strain it acquires after mutation.
• Figure 4: The probability of emergence on contact networks with doubly Poisson distribution \ref{['eq:doublypoisson']}, with the distribution for single-edges and triangles, respectively parameterized by $\lambda_s$ and $\lambda_t$. For $\lambda_s = \lambda_t = \lambda$, we vary $\lambda$ in the interval $(0,10)$. The analytically predicted probability (indicated as $P_E$(pred.)) and epidemic threshold (indicated by the vertical dashed line corresponding to $\rho( \pmb{J})=1$) are derived from Theorems \ref{['thm:prob']} and \ref{['thm:threshold']}, respectively, while the mean epidemic size ($S$(pred.)) computed by the proposed transformation to a single-strain model using \ref{['eq:single-strain-tx']}. The empirical probability of emergence ($P_E$(exp.)) and the conditional mean epidemic size ($S$(exp.)), given that it occurs averaged over $1.5\times10^4$ independent experiments. For each data point, the network size $n$ is $2 \times 10^5$, the number of independent experiments is $1.5\times10^4$, and we set $T_1=0.2, T_2=0.5$ and $\mu_{11}=\mu_{22}=0.75$.
• Figure 5: We plot the probability of emergence and expected epidemic size. We consider doubly Poisson distribution \ref{['eq:doublypoisson']} with the distribution for single-edges and triangles, respectively parameterized by $\lambda_s$ and $\lambda_t$, such that $\lambda_s = \lambda_t = \lambda$. The network size $n=2 \times 10^5$, and we report the empirical probability of emergence ($P_E$ (exp.)) and mean epidemic size ($S$ (exp.)) as averaged over $1.5\times10^4$ independent experiments. The analytically predicted probability ($P_E$ (pred.)) is derived from Theorems \ref{['thm:prob']} while the mean epidemic size ($S$ (pred.)) is computed by transforming to a single-strain model using \ref{['eq:single-strain-tx']}. We let $T_2$ be the more transmissible strain with $T_2=0.5,~T_1=0.2$. Using the parameters in Fig. \ref{['fig:sim1']}, as a baseline, through sub-figures (a), (b), and (c), we vary $\mu_{11}\in\{0.25,0.5,0.9\}$ while holding $\mu_{22}$ constant. Conversely, in sub-figures (d), (e), and (f), we hold $\mu_{22}=0.75$ constant and vary $\mu_{22}\in\{0.25,0.5,0.9\}$. We observe good agreement between our predictions for the probability of emergence and mean epidemic size and the corresponding observed values in the simulations.

Theorems & Definitions (3)

• Theorem 3.1: Probability of Emergence
• Theorem 3.2: Epidemic Threshold
• Lemma 3.3