Adaptive social contact rates induce complex dynamics during epidemics

Abstract

Epidemics may pose a significant dilemma for governments and individuals. The personal or public health consequences of inaction may be catastrophic; but the economic consequences of drastic response may likewise be catastrophic. In the face of these trade-offs, governments and individuals must therefore strike a balance between the economic and personal health costs of reducing social contacts and the public health costs of neglecting to do so. As risk of infection increases, potentially infectious contact between people is deliberately reduced either individually or by decree. This must be balanced against the social and economic costs of having fewer people in contact, and therefore active in the labor force or enrolled in school. Although the importance of adaptive social contact on epidemic outcomes has become increasingly recognized, the most important properties of coupled human-natural epidemic systems are still not well understood. We develop a theoretical model for adaptive, optimal control of the effective social contact rate using traditional epidemic modeling tools and a utility function with delayed information. This utility function trades off the population-wide contact rate with the expected cost and risk of increasing infections. Our analytical and computational analysis of this simple discrete-time deterministic strategic model reveals the existence of an endemic equilibrium, oscillatory dynamics around this equilibrium under some parametric conditions, and complex dynamic regimes that shift under small parameter perturbations. These results support the supposition that infectious disease dynamics under adaptive behavior change may have an indifference point, may produce oscillatory dynamics without other forcing, and constitute complex adaptive systems with associated dynamics. Implications for any epidemic in which adaptive behavior influences infectious disease dynamics include an expectation of fluctuations, for a considerable time, around a quasi-equilibrium that balances public health and economic priorities, that shows multiple peaks and surges in some scenarios, and that implies a high degree of uncertainty in mathematical projections.

Fig 1. Discrete-time SIS (blue) and continuous-time SIS (orange) dynamics for delays Δ = 0 to Δ = 5.

N = 10, 000, b₀ = 0.05, γ = 0.08, $\widehat{c}=0.0015$, α = 0.375, and I₀ = 1. Here the epidemic equilibrium is $\widehat{I}=35.72$.

Fig 2. Effect of initial number of infecteds I₀ on the dynamics for delay Δ = 2.

Discrete- and continuous-time results are in blue and orange, respectively. Other parameters as in Fig 1. As in Fig 1, $\widehat{I}=35.72$.

Fig 3. Effect of baseline contact rate b₀ on dynamics with delay Δ = 3.

Other parameters as in Fig 1 with I₀ = 1. Discrete- and continuous-time results are in blue and orange, respectively. Note that α changes with b₀ as α = b₀ α₂/2α₁: (A) α = 0.0375; (B) α = 0.075; (C) α = 0.225; (D) α = 0.3; (E) α = 0.375; (F) α = 0.75.

Fig 4. Effect of removal rate γ on dynamics with delay Δ = 3.

Discrete- and continuous-time results are in blue and orange, respectively. Other parameters as in Fig 1 with I₀ = 1.