A Few Bad Apples: A Model of Disease Influenced Agent Behaviour in a Heterogeneous Contact Environment

For diseases that infect humans or livestock, transmission dynamics are at least partially dependent on human activity and therefore human behaviour. However, the impact of human behaviour on disease transmission is relatively understudied, especially in the context of heterogeneous contact structures such as described by a social network. Here, we use a strategic game, coupled with a simple disease model, to investigate how strategic agent choices impact the spread of disease over a contact network. Using beliefs that are based on disease status and that build up over time, agents choose actions that stochastically determine disease spread on the network. An agent’s disease status is therefore a function of both his own and his neighbours actions. The effect of disease on agents is modelled by a heterogeneous payoff structure. We find that the combination of network shape and distribution of payoffs has a non-trivial impact on disease prevalence, even if the mean payoff remains the same. An important scenario occurs when a small percentage (called noncooperators) have little incentive to avoid disease. For diseases that are easily acquired when taking a risk, then even when good behavior can lead to disease eradication, a small increase in the percentage of noncooperators (less than 5%) can yield a large (up to 25%) increase in prevalence.


Initial estimates of payoffs
Here we describe the initial estimates of payoffs that farms hold. These are only used until the farm has direct experience of an action and are typically only used at the very beginning of a simulation.
Let d sus be the susceptible disease state, and d inf the infected disease state. If farm f is currently susceptible, then the estimated payoff next turn for farm f taking action a ∈ A from disease state d sus with currently infected neighbours in set F is: (1) If farm f is currently infected, then the estimated payoff next turn for farm f taking action a ∈ A from disease state d sus with currently infected neighbours in set F is:

Transition to Lower Prevalence
The transition to lower prevalence in Figure 3 happens at about 22% probability of bringing in disease with the risky action over a variety of percentages of non-cooperators, including the situation with no noncooperators! In fact, it occurs in the same probability of bringing in disease with the risky action in a system of isolated farms with no fence line neighbours. This suggests that the location of this transition is determined only by disease threat from outside, and not from geographic neighbours.
We provide an analytical explanation. Consider a farm with no infection threat from neighbours. Then that farm's choices should take into account only the payoffs and probabilities of bringing in disease with the risky action. At very low additional risk from the risky action, the cost of the action will prevent farmers taking the safe action.
Consider, for example, a disease that would cost a farm £1000. Say the safe action gives 0% chance of bringing in the disease, and the risky action a 1% chance of bringing in the disease, but the safe action costs £500 more than the risky action. It is not worth a farm taking the safe action for only a 1% decrease in the probability of bringing in the disease. However, if the safe action cost only a very small amount, or the risky action increased the probability of bringing in the disease by larger amount, then it would, in expectation, be worth taking the safe action.
The point at which we expect mass switching to the safe action is the point at which the expected payoffs from the safe and risky action are the same to a currently uninfected farm. Then we proceed as below (note that we omit the f in all subscripts for succinctness, and because there is only one farm involved): In our simulation, we intended the disease to have a cost and taking the safe action to have cost such that these costs combined additively. That is, (Y (a0,sue) − Y (a0,inf ) = (Y (a1,sus) − Y (a1,inf ) ) and (Y (a1,inf ) − Y (a0,inf ) ) = (Y (a1,sus) − Y (a0,sus)) . We introduce notation for these two costs, saying that: Then continuing from Statement 3: −(P a0 δ state − P a1 δ state ) = Y (a1,sus) − Y (a0,sus) (9) P a1 δ state − P a0 δ state = δ act (10) P a1 − P a0 = δ act /δ state In our non-cooperator simulations, P a0 = 0, δ act = 0.1, δ state = 0.45 giving a threshold of P a1 = 0.22. This is consistent with our experimental results.