New approximations, and policy implications, from a delayed dynamic model of a fast pandemic

Chandrika Prakash Vyasarayani; Anindya Chatterjee

which matches the (loss of) stability condition of Young et al. [9] as well as our equation (17) . In other words, the condition for a solution growing from zero is exactly the same condition for the existence of another equilibrium for strictly positive P ; and this condition, though obtained here from the long-wave approximation, matches the theoretical result exactly because it is near this stability boundary that the long-wave approximation is asymptotic. Finally, when the coefficient of P in equation (65) is indeed positive, then the two characteristic roots are real: one is positive and one is negative. The positive root leads to the growing solution, as above. The negative root absorbs the apparently free initial condition, contributes an exponentially decaying term that dies soon, and has no influence on the growing solution provided the initial conditions are sufficiently early in the outbreak. A numerical example is shown in figure 6 for two cases: βτ = 1.15 and βτ = 1.25. The parameter p = 0.98 implies that the probability of detecting infected individuals is not perfect. Further, γ = 0.1 models a small fraction of the infectious population recovering without being quarantined. In the figure, the long-wave solution perfectly matches the final saturation state obtained from numerical integration of the DDE (equation (18) ), due to the equivalence of equations (64) and (16) .