Disease dynamics on a network game: a little empathy goes a long way

Individuals change their behavior during an epidemic in response to whether they and/or those they interact with are healthy or sick. Healthy individuals are concerned about contracting a disease from their sick contacts and may utilize protective measures. Sick individuals may be concerned with spreading the disease to their healthy contacts and adopt preemptive measures. Yet, in practice both protective and preemptive changes in behavior come with costs. This paper proposes a stochastic network disease game model that captures the self-interests of individuals during the spread of a susceptible-infected-susceptible (SIS) disease where individuals react to current risk of disease spread, and their reactions together with the current state of the disease stochastically determine the next stage of the disease. We show that there is a critical level of concern, i.e., empathy, by the sick individuals above which disease is eradicated fast. Furthermore, we find that if the network and disease parameters are above the epidemic threshold, the risk averse behavior by the healthy individuals cannot eradicate the disease without the preemptive measures of the sick individuals. This imbalance in the role played by the response of the infected versus the susceptible individuals in disease eradication affords critical policy insights. Infectious diseases change the social interaction patterns in the society they impact. During the Ebola outbreak, many studies pointed to changes in social customs, e.g., switch to safe burial methods from traditional ceremonial burials, playing a critical role in impeding disease spread [1]. Similar behavioral responses played important roles in modifying disease spread in other pandemics, e.g., wearing protective masks during the SARS pandemic [2–4], decrease in unprotected sex when STD is at high levels [5, 6] or covering one’s own cough and staying at home if sick during a flu pandemic [7–9]. These responses to prevalence of the disease are efforts to preempt disease spread by the infected and the susceptible individuals in the population. In many infectious diseases infected and susceptible individuals have to be in contact for disease transmission. Accordingly, there has been a surge of interest on disease spread models in which a contact network determines the subset of individuals that can be infected by an infected individual [10–15]. These studies were influential in relating network structural properties to epidemic thresholds and in revealing the limits to inferences made by models that assume homogeneous mixing. The rate at which individuals meet with their contacts changes depending on the individual preemptive measures during the course of a disease [16]. Consequently, a number of dynamic models have been developed to assess the effects individual preemptive measures have on infectious disease spread over networks [17–20]. These models couple behavior and disease dynamics. That is, the state of the disease and the contact network determine the preemptive measures of the individuals which then affect the disease spread. Preemptive measures in these models, which are in the form of social distancing or rewiring of transmissive links, are results of simple heuristics that ∗School of Biology, Georgia Institute of Technology, Atlanta, GA †School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA ‡Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST) Thuwal, Saudi Arabia §School of Physics, Georgia Institute of Technology, Atlanta, GA ¶This work is supported by Army Research Office grant #W911NF-14-1-0402, and supported in part by KAUST. The authors thank K. Paarporn (Georgia Inst. Tech.) and J. W. Glasser (Center for Disease Control (CDC)) for their comments. ‖To whom correspondence should be addressed. E-mail: [email protected], [email protected] 1 ar X iv :1 60 4. 03 24 0v 2 [ cs .S I] 1 5 A pr 2 01 6 approximate the decision-making of healthy individuals. These heuristic based decision-making algorithms are intended to be approximations of decisions made by self-interested individuals. When individuals act according to their selfish interests, they would compare the inherent costs of these preemptive measures with the risks of disease contraction. However, the actions of other individuals also affect the risk of infection. Game theory provides a means to consider how individuals make rational decisions by reasoning strategically about the decisions of others. Recent epidemiological models with game theoretic individual decisionmaking either consider one-shot rational decisions of all susceptible individuals at the beginning, e.g., vaccinate or not, social distance or not, that determines the course of the disease [21–26], or use bounded rational heuristics for repeated decision-making [27–30] (see [31] for a recent extensive review). Here, we consider individuals—susceptible and infected— making daily rational decisions on preemptive measures, e.g., social distancing, staying home from school/work, wearing protective masks, based on the current risks of disease spread for a susceptible-infected-susceptible (SIS) infection. In particular, a healthy individual compares the cost of protection measures with its current risk of infection. This means a healthy individual can forgo any protective measure (free-ride) if it perceives its sick contacts are taking the utmost preemptive measures. However, at the same time, a sick individual compares the cost of preemptive measures with the current risk of spreading the disease to its healthy neighbors. This means sick individuals have to reason strategically about the decisions of their healthy neighbors who reason about the decisions of their sick neighbors. This sets up a daily game among healthy and sick individuals. The daily rational measures taken by both the healthy and the sick as a result of the network disease game set the probabilities of disease contractions which in turn stochastically determine the status of the disease in the following day. Using this model, we explore the interrelationship among contact network structure, rational daily decisions and SIS disease dynamics. Specifically, we provide analytical bounds for the initial spread of the disease from a single infected individual based on selection of the initial infected individual, network degree distribution, disease infection and healing rates, and the relative weight (empathy) that sick individuals have on disease spread versus cost of preemptive measures. These bounds show that increasing empathy of the sick individuals stops the initial disease spread. We also show that these bounds are good indicators of disease eradication starting from any initial level of infection, not just a single infected individual. Moreover, we confirm past results in [19, 20] by showing that susceptible individuals cannot by themselves quickly eradicate the disease no matter how risk averse they are if the empathy factor is zero. Yet, for any positive empathy by the sick individuals, we can find a critical risk averseness factor value that can help eradicate the disease. These results imply that a little preemptive action by the sick individuals is crucial in fast eradication of an epidemic. 1 Model We consider stochastic SIS disease dynamics where an individual i in the population N := {1, . . . , n} is either susceptible (si(t) = 0) or infected (si(t) = 1) at any given time t = 1, 2, . . . . If the individual is susceptible at time t, it gets infected at time t + 1 with probability p01(t) := P[si(t + 1) = 1|si(t) = 0]. If the individual is sick at time t, it becomes susceptible at time t+ 1 with probability p10(t) := P[si(t+ 1) = 0|si(t) = 1]. These transition probabilities define a Markov chain for the disease dynamics of individual i ∈ N as follows si(t+ 1) =  1 with prob. p01(t) if si(t) = 0 0 with prob. 1− p01(t) if si(t) = 0 1 with prob. 1− p10(t) if si(t) = 1 0 with prob. p10(t) if si(t) = 1. (1) A susceptible individual can only contract the disease in the next time step if in contact with an infected individual. We define the set of contacts of each individual by a contact network G with node set N and edge set E – see Fig. 1(A) for an example. The contact neighborhood of individual i is Ni := {j : (i, j) ∈ E}. The chance of a susceptible individual (si(t) = 0) contracting the disease from a neighboring infected contact (sj(t) = 1) depends on the infection probability of the disease β ∈ (0, 1), action of i ai(t) belonging to the unit interval [0, 1], and the