A Network Epidemic Model with Preventive Rewiring: Comparative Analysis of the Initial Phase

Stochastic SIRS models on networks: mean and variance of infection

Due to the heterogeneity of contact structure, it is more reasonable to model on networks for epidemics. Because of the stochastic nature of events and the discrete number of individuals, the spread of epidemics is more appropriately viewed as a Markov chain. Therefore, we establish stochastic SIRS models with vaccination on networks to study the mean and variance of the number of susceptible and infected individuals for large-scale populations. Using van Kampen's system-size expansion, we derive a high-dimensional deterministic system which describes the mean behaviour and a Fokker-Planck equation which characterizes the variance around deterministic trajectories. Utilizing the qualitative analysis technique and Lyapunov function, we demonstrate that the disease-free equilibrium of the deterministic system is globally asymptotically stable if the basic reproduction number R 0 < 1; and the endemic equilibrium is globally asymptotically stable if R 0 > 1. Through the analysis of the Fokker-Planck equation, we obtain the asymptotic expression for the variance of the number of susceptible and infected individuals around the endemic equilibrium, which can be approximated by the elements of principal diagonal of the solution of the corresponding Lyapunov equation. Here, the solution of Lyapunov equation is expressed by vectorization operator of matrices and Kronecker product. Finally, numerical simulations illustrate that vaccination can reduce infections and increase fluctuations of the number of infected individuals and show that individuals with greater degree are more easily infected.

Rewiring cattle movements to limit infection spread

Cattle tracing databases have become major resources for representing demographic processes of livestock and assessing potential risk of infections spreading by trade. The herds registered in these databases are nodes of a network of commercial movements, which can be altered to lower the risk of disease transmission. In this study, we develop an algorithm aimed at reducing the number of infected animals and herds, by rewiring specific movements responsible for trade flows from high- to low-prevalence herds. The algorithm is coupled with a generic computational model based on the French cattle movement tracing database (BDNI), and used to describe different scenarios for the spread of infection within and between herds from a recent outbreak (epidemic) or a five-year-old outbreak (endemic). Results show that rewiring successfully contains infections to a limited number of herds, especially if the outbreak is recent and the estimation of disease prevalence frequent, while the respective impact of the parameters of the algorithm depend on the infection parameters. Allowing any animal movement from high to low-prevalence herds reduces the effectiveness of the algorithm in epidemic settings, while frequent and fine-grained prevalence assessments improve the impact of the algorithm in endemic settings. Our approach focusing on a few commercial movements is expected to lead to substantial improvements in the control of a targeted disease, although changes in the network structure should be monitored for potential vulnerabilities to other diseases. This general algorithm could be applied to any network of controlled individual movements liable to spread disease.

Moment closure approximations of susceptible-infected-susceptible epidemics on adaptive networks

The influence of people's individual responses to the spread of contagious phenomena, like the COVID-19 pandemic, is still not well understood. We investigate the Markovian Generalized Adaptive Susceptible-Infected-Susceptible (G-ASIS) epidemic model. The G-ASIS model comprises many contagious phenomena on networks, ranging from epidemics and information diffusion to innovation spread and human brain interactions. The connections between nodes in the G-ASIS model change adaptively over time, because nodes make decisions to create or break links based on the health state of their neighbors. Our contribution is fourfold. First, we rigorously derive the first-order and second-order mean-field approximations from the continuous-time Markov chain. Second, we illustrate that the first-order mean-field approximation fails to approximate the epidemic threshold of the Markovian G-ASIS model accurately. Third, we show that the second-order mean-field approximation is a qualitative good approximation of the Markovian G-ASIS model. Finally, we discuss the Adaptive Information Diffusion (AID) model in detail, which is contained in the G-ASIS model. We show that, similar to most other instances of the G-ASIS model, the AID model possesses a unique steady state, but that in the AID model, the convergence time toward the steady state is very large. Our theoretical results are supported by numerical simulations.

Robustness of behaviorally induced oscillations in epidemic models under a low rate of imported cases

This paper is concerned with the robustness of the sustained oscillations predicted by an epidemic ODE model defined on contact networks. The model incorporates the spread of awareness among individuals and, moreover, a small inflow of imported cases. These cases prevent stochastic extinctions when we simulate the epidemics and, hence, they allow to check whether the average dynamics for the fraction of infected individuals are accurately predicted by the ODE model. Stochastic simulations confirm the existence of sustained oscillations for different types of random networks, with a sharp transition from a nonoscillatory asymptotic regime to a periodic one as the alerting rate of susceptible individuals increases from very small values. This abrupt transition to periodic epidemics of high amplitude is quite accurately predicted by the Hopf-bifurcation curve computed from the ODE model using the alerting rate and the infection transmission rate for aware individuals as tuning parameters.

Individual preventive social distancing during an epidemic may have negative population-level outcomes

The outbreak of an infectious disease in a human population can lead to individuals responding with preventive measures in an attempt to avoid getting infected. This leads to changes in contact patterns. However, as we show in this paper, rational behaviour at the individual level, such as social distancing from infectious contacts, may not always be beneficial for the population as a whole. We use epidemic network models to demonstrate the potential negative consequences at the population level. We take into account the social structure of the population through several network models. As the epidemic evolves, susceptible individuals may distance themselves from their infectious contacts. Some individuals replace their lost social connections by seeking new ties. If social distancing occurs at a high rate at the beginning of an epidemic, then this can prevent an outbreak from occurring. However, we show that moderate social distancing can worsen the disease outcome, both in the initial phase of an outbreak and the final epidemic size. Moreover, the same negative effect can arise in real-world networks. Our results suggest that one needs to be careful when targeting behavioural changes as they could potentially worsen the epidemic outcome. Furthermore, network structure crucially influences the way that individual-level measures impact the epidemic at the population level. These findings highlight the importance of careful analysis of preventive measures in epidemic models.

Epidemic threshold in pairwise models for clustered networks: closures and fast correlations

The epidemic threshold is probably the most studied quantity in the modelling of epidemics on networks. For a large class of networks and dynamics, it is well studied and understood. However, it is less so for clustered networks where theoretical results are mostly limited to idealised networks. In this paper we focus on a class of models known as pairwise models where, to our knowledge, no analytical result for the epidemic threshold exists. We show that by exploiting the presence of fast variables and using some standard techniques from perturbation theory we are able to obtain the epidemic threshold analytically. We validate this new threshold by comparing it to the threshold based on the numerical solution of the full system. The agreement is found to be excellent over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. Interestingly, we find that the analytical form of the threshold depends on the choice of closure, highlighting the importance of model selection when dealing with real-world epidemics. Nevertheless, we expect that our method will extend to other systems in which fast variables are present.

A stochastic SIR network epidemic model with preventive dropping of edges

A Markovian Susceptible [Formula: see text] Infectious [Formula: see text] Recovered (SIR) model is considered for the spread of an epidemic on a configuration model network, in which susceptible individuals may take preventive measures by dropping edges to infectious neighbours. An effective degree formulation of the model is used in conjunction with the theory of density dependent population processes to obtain a law of large numbers and a functional central limit theorem for the epidemic as the population size [Formula: see text], assuming that the degrees of individuals are bounded. A central limit theorem is conjectured for the final size of the epidemic. The results are obtained for both the Molloy-Reed (in which the degrees of individuals are deterministic) and Newman-Strogatz-Watts (in which the degrees of individuals are independent and identically distributed) versions of the configuration model. The two versions yield the same limiting deterministic model but the asymptotic variances in the central limit theorems are greater in the Newman-Strogatz-Watts version. The basic reproduction number [Formula: see text] and the process of susceptible individuals in the limiting deterministic model, for the model with dropping of edges, are the same as for a corresponding SIR model without dropping of edges but an increased recovery rate, though, when [Formula: see text], the probability of a major outbreak is greater in the model with dropping of edges. The results are specialised to the model without dropping of edges to yield conjectured central limit theorems for the final size of Markovian SIR epidemics on configuration-model networks, and for the size of the giant components of those networks. The theory is illustrated by numerical studies, which demonstrate that the asymptotic approximations are good, even for moderate N.

Mathematical models to characterize early epidemic growth: A review

There is a long tradition of using mathematical models to generate insights into the transmission dynamics of infectious diseases and assess the potential impact of different intervention strategies. The increasing use of mathematical models for epidemic forecasting has highlighted the importance of designing reliable models that capture the baseline transmission characteristics of specific pathogens and social contexts. More refined models are needed however, in particular to account for variation in the early growth dynamics of real epidemics and to gain a better understanding of the mechanisms at play. Here, we review recent progress on modeling and characterizing early epidemic growth patterns from infectious disease outbreak data, and survey the types of mathematical formulations that are most useful for capturing a diversity of early epidemic growth profiles, ranging from sub-exponential to exponential growth dynamics. Specifically, we review mathematical models that incorporate spatial details or realistic population mixing structures, including meta-population models, individual-based network models, and simple SIR-type models that incorporate the effects of reactive behavior changes or inhomogeneous mixing. In this process, we also analyze simulation data stemming from detailed large-scale agent-based models previously designed and calibrated to study how realistic social networks and disease transmission characteristics shape early epidemic growth patterns, general transmission dynamics, and control of international disease emergencies such as the 2009 A/H1N1 influenza pandemic and the 2014-2015 Ebola epidemic in West Africa.