Complex dynamics induced by delayed adaptive behavior during epidemics

Ronan F. Arthur; James H. Jones; Matthew H. Bonds; Marcus W. Feldman

We note that the five distinct kinds of dynamical trajectories (Fig. 3 ) seen in these computational experiments come from a simple set of purely deterministic equations. This means that oscillations and even erratic, near-chaotic dynamics and collapse in an epidemic may not necessarily be due to seasonality, complex agent-based interactions, changing or stochastic parameter values, demographic change, host immunity, or socio-cultural idiosyncracies. This dynamical behavior in number of infecteds can result from mathematical properties of a simple deterministic system with homogeneous endogenous behavior change, similar to complex population dynamics of biological organisms [40] . The mathematical consistency with population dynamics suggests a parallel in ecology, that the indifference point for human behavior functions in a similar way to a carrying capacity in ecology, below which a population will tend to grow and above which a population will tend to shrink. For example, the Ricker Equation [41] , commonly used in population dynamics to describe the growth of fish populations, exhibits similar complex dynamics and qualitative state thresholds. The existence of a non-zero, stable equilibrium in our model is consistent with economic epidemiology theory: if individuals are incentivized to change their behavior to protect themselves, they will, and they will cease to do this when they are not [10] . If we couple this with delayed information, the results can lead to limit-cycle dynamics, consis-tent with other negative feedback mechanisms with time delays [42, 43] . This is because the system is reacting to conditions that were true in the past, but not necessarily true in the present.