Complex dynamics induced by delayed adaptive behavior during epidemics

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

The basic reproduction number, the number of secondary cases produced by the introduction of one infected person into an entirely susceptible population, R 0 , is a fundamental tool of epidemiology because it delineates a threshold between an epidemic and eradication [35, 36] . R t , the reproduction number at time t, is the eigen-value of this system or I t+1 It [37, 38] . When R t is greater than 1, the epidemic will continue to grow. When it is less than 1, it will shrink. Reducing R t to less than 1 is therefore a central goal of epidemic intervention. The effect of endogenous behavior change on R t in this model is illustrated in Fig. 3 . Figure 1 : Return map for system with ∆ = 0 whereÎ 2 is a stable equilibrium and between 0 and N .Î 0 is unstable andÎ 1 > N . This is valid whenĉb 0 > γ N and γ > 0. R t > 1 if I 0 <Î 2 unless I t becomes greater thanÎ 2 at which point R t < 1. The system thereby regulates itself to approach theÎ 2 equilibrium when ∆ = 0. The copyright holder for this preprint (which was not peer-reviewed) is the . https://doi.org/10.1101/2020.04.14.028407 doi: bioRxiv preprint Figure 2 : Return map for system whenÎ 0 is stable,Î 2 is negative and unstable, andÎ 1 > N and unstable whenĉb 0 < γ N . Here, R t < 1 for all legitimate values of I t (i.e. when 0 < I t < N ).