Evolution and emergence of infectious diseases in theoretical and real-world networks

Abstract

One of the most important advancements in theoretical epidemiology has been the development of methods that account for realistic host population structure. The central finding is that heterogeneity in contact networks, such as the presence of 'superspreaders', accelerates infectious disease spread in real epidemics. Disease control is also complicated by the continuous evolution of pathogens in response to changing environments and medical interventions. It remains unclear, however, how population structure influences these adaptive processes. Here we examine the evolution of infectious disease in empirical and theoretical networks. We show that the heterogeneity in contact structure, which facilitates the spread of a single disease, surprisingly renders a resident strain more resilient to invasion by new variants. Our results suggest that many host contact structures suppress invasion of new strains and may slow disease adaptation. These findings are important to the natural history of disease evolution and the spread of drug-resistant strains.

Figure 1. Model of disease evolution on networks.

Individuals are represented by nodes in a network (shapes) and connections between individuals through which the disease can spread are represented by edges (grey lines). (a) In a population of initially susceptible individuals (green circles), a single individual becomes infected (red hexagons). (b) The infection spreads throughout the population, and eventually reaches a dynamic equilibrium (becomes ‘endemic’), where the number of new infections is balanced by the number of recoveries. (c) The pathogen in a single individual gains a beneficial mutation, creating a new pathogen strain (blue octagon). We are interested in the probability that this new strain fixes in the population, reaching endemic equilibrium and causing the resident strain to go extinct.

Figure 2. Network structure influences the evolution of diseases on real and theoretical contact networks.

(a,b) Graphical representation of the networks. Large red or blue circles represent nodes with a high degree, small purple or green circles represent nodes with a low degree for the empirical and theoretical networks, respectively. (c,e) The probability that a single disease causes an epidemic (the emergence probability), P_emerge, versus the scaled transmissibility τ=β(‹k›−1)/γ. β is varied and τ represents the expression for the basic reproductive ratio for the uniform network. The thick grey line indicates the emergence probability for a well-mixed network, P_emerge=1−1/R₀. The transmissibility value at which P_emerge becomes non-zero (that is, the epidemic threshold) depends on the network. (d,f) Dynamics of new disease variants. The probability of fixation, P_fix, versus the selective advantage, r=β₂/β₁, of a new disease variant is strongly influenced by the population structure, but is not predicted by P_emerge. The thick grey line indicates the fixation probability in a well-mixed network, . (g) The selection exponent, α, is calculated by fitting P_fix versus r to equation (27). Lower values mean that selection is suppressed compared to the uniform network. For a uniform or well-mixed population we expect α=1. Fits are shown by the solid lines in panels (d) and (f).

Figure 3. Dynamics of disease evolution in heterogeneous networks and small-world networks.

(a–d) Heterogeneous networks; (e–g) small-world networks. (a) Degree distribution for upper panels is specified by a discrete gamma distribution (see Methods) with constant mean ‹k›=4 but tunable variance σ². (b) The probability that a single disease causes an epidemic (the emergence probability), P_emerge, versus the scaled transmissibility τ=β(‹k›−1)/γ. β is varied and τ represents the expression for the basic reproductive ratio for the uniform network. The multi-type branching process approximation (equation (9), solid lines) is in excellent agreement with the simulations (dots). (c) The probability of being infected at endemic equilibrium versus degree k. Predictions using pair-wise approximations (equation (17), lines) are in excellent agreement with simulations (bars). (d) The probability of fixation versus the selective advantage (r=β₂/β₁) of a new disease variant decreases for networks with larger variance in degree. Calculations from a new combined analytical technique (equation (18), solid lines) match well with simulations (dots). (e) For the lower panels, a set of small-world networks was created with constant homogeneous degree k=4 but varying clustering coefficient φ. (f) The probability of emergence for the first strain as a function of scaled transmissibility depends on clustering. (g) The fixation probability of the second strain as a function of the selective advantage, r=β₂/β₁, is independent of local clustering. For small-world networks, lines are simply connections between points to guide the eye.

Figure 4. Mechanics of disease spread in theoretical networks.

(a) Star graphs, or networks of interconnected stars, are an example of networks with large variance in degree distribution. Hubs facilitate the spread of the first disease in a susceptible population. (b) New disease strains are likely to appear in the leaves. Hubs are likely to be already infected, hindering invasion. (c) Susceptible hubs are likely to be quickly reinfected by leaves infected with the resident strain. Therefore, the fixation probability of new strains on star-like graphs is low. (d) Small-world networks are made up of mainly local connections with variable rewiring to create shortcuts. (e) The initially infected individual can potentially infect all its neighbours, while those subsequently infected have more limited options. (f) Shortcuts allow the disease to jump to fully susceptible areas of the network, facilitating spread. They are less important for the second disease, as all parts of the network are already infected. Therefore, fixation probability of new strains on uniform, locally connected networks does not depend on the rewiring probability. (g,h) The degree distribution of the full network (green; left) is compared with the residual network (red; right) of susceptible individuals remaining at a particular time point during endemic equilibrium. (g) Uniform network (equivalent to gamma-distributed network with σ=0). (h) Gamma network with σ=3. Dotted lines represent the mean of the distributions.