Applying a Probabilistic Infection Model for studying contagion processes in contact networks

Highlights

• Developing the Probabilistic Infection Model (PIM) which can assign multiple disease states to each individual, based on their probability of being in that state.

• Validating the simulation results of PIM by comparing with those obtained from the Monte-Carlo method.

• Studying the spread of three disease (measles, and two strains of influenza) on the overall contact network as well as at individual levels

• Studying the relation of network structure to probability of infection.

• Studying how varying transmissibility factors affects the spread of COVID in contact networks.

Abstract

Modeling the spread of infectious diseases is central to the field of computational epidemiology. Two prominent approaches to modeling the contagion process include (i) simulating the spread in contact networks through Monte-Carlo processes and (ii) tracking the disease dynamics using meta-population models. In both cases, the individuals are explicitly (contact networks) or implicitly (meta-population) assumed to belong to exactly one disease state (e.g., susceptible, infected, etc.).

In reality, the disease states of individuals are rarely so cleanly compartmentalized. A particular agent can exist in multiple disease states (such as infected and exposed) concurrently with varying probability. To model this stochasticity, we present a new method, that we term as the Probabilistic Infection Model (PIM). Unlike traditional models that assign exactly one state to each agent at each time step, the PIM computes the probability of each agent being in each of the infectious states.

Our proposed PIM provides a more layered understanding of the dynamics of the outbreak at individual levels, by allowing the users to (i) estimate the value of R₀ at individual vertices and (ii) instead of an all or none value, provides the probability of each infected state of an agent. Additionally, using our probabilistic approach the overall trajectories of the outbreaks can be computed in one simulation, as opposed to the numerous (order of hundreds) repeated simulations required for the Monte Carlo process.

We demonstrate the efficacy of PIM by comparing the results of the PIM simulations with those obtained by simulating stochastic SEIR models, as well as the time required for the simulations. We present results at the system and at the individual levels for three diseases; measles and two strains of influenza. We demonstrate how the PIM can be used to study the effect of varying the transimissibility of COVID-19 on its outbreak.

This paper is an extended version of a manuscript published in the proceedings of the 2020 International Conference on Computational Science (ICCS)[30]. These extensions are primarily within Sections 4 (Relationship between graph structure and probability of infection) and 5 (Effect of varying COVID-19 transmissibility on outbreak dynamics).

Introduction

A primary component of computational epidemics is modeling and simulating how infections spread in a population. Two main approaches to simulating the spread of disease are (i) stochastic agent-based modelling; and (ii) deterministic meta-population models [1], [14].

Both models assume that the individuals are in exactly one disease state. For example, the SEIR model, which we simulate in this paper, the states are Susceptible, Exposed, Infected, and Recovered. This framework is ¹ is depicted in Fig. 1. S, E, I, and R represent the number of individuals in Susceptible, Exposed, Infected, and Recovered states respectively. The total population is then given by N = S + E + I + R. Parameter β is the proportion of contacts between members of S and members of E that lead to disease transmission. Parameter σ is the rate at which the exposed become infected. Parameter γ is the recovery rate at which the infected transition to the recovered state.

In stochastic agent-based models each individual (or group of individuals) in a population is represented as agents. Dyadic interactions between agents are governed by functions of the agents’ characteristics, or their environment. These interactions can be used to form a contact network. The infection spreads through the connections in this network. Meta-population models use a system of differential equations to approximate the rate of change of the number of individuals in each disease state (e.g., susceptible, infected, etc.). Here the specific connections between the individuals is not modeled.

Both these models exhibit competing benefits and drawbacks. The advantage of the stochastic agent-based approach is that it can model population heterogeneity including variations in the numbers of contacts of each individual, as well as by varying the infection parameters, such as γ, σ, per individual. The disadvantage is that due to the reliance on stochastic processes, a single run of an outbreak simulation is not representative of an expected outcome. Hundreds of repeated executions per unique set of parameters are needed in order to adequately estimate trends in the data.

In an almost exact reverse, meta-population disease models are computationally efficient due to their deterministic nature. Further, closed form approximations of significant epidemiological parameters such as the basic reproduction number R₀ (i.e. the expected number of secondary cases resulting from a single infectious individual in a completely susceptible population) can be derived analytically using meta-population models. But these models do not represent the diversity of the individuals, and assume a homogeneous mixing rate within a homogeneous population. Motivated by these trade-offs, our goal is to combine the advantages of these two popular epidemiological models. We introduce the Probabilistic Infection Model (PIM), which combines the heterogeneity of the stochastic models with the computational efficiency and deterministic nature of the meta-population models. The key idea of PIM is to calculate the probabilities of the four SEIR states associated with that vertex for each vertex in a contact network.

To compute the probability function, we leverage the research conducted in escape probabilities by Thomas and Weber [32]. The probabilities for each state and each vertex are compounded over windows of time corresponding to the latent and infectious periods of the given disease. This allows for probabilistic values of different states over time at the individual levels and also provides the expected values of the sizes of the SEIR sub-populations corresponding to each state. As an added advantage, our proposed PIM allows us to compute an expression for $R_0(v_0)$, which yields the value of R₀ for specific single infectious individuals in an otherwise susceptible contact network. In Table 1 we provide a comparison between the stochastic model, the meta-population model and our proposed PIM.

We applied our model to a contact network created from class enrollment data from the University of North Texas, as well as on two other contact networks that are available online; (i) on a network of friendship of students in high school and (ii) a network of students living in a residential hall. We conducted our experiments by simulating the following epidemics; two varieties of influenza, measles, and Covid-19. We compared simulation results as well as the timing of the PIM with those produced by the stochastic models. Our results demonstrate that the PIM simulations are similar to those produced by averaging trials from Monte Carlo models. This similarity is most notable when simulating diseases that are highly infectious, such as measles.