Cross-scale dynamics and the evolutionary emergence of infectious diseases

Sebastian J. Schreiber; Ruian Ke; Claude Loverdo; Miran Park; Priyanna Ahsan; James O. Lloyd-Smith

The i-th coordinate of q(t) = G t (0, . . . , 0) is the probability of extinction by generation t when there is initially one infected individual with initial viral load (i, N − i). The probability of emergence is given by 1 − q where q = lim t→∞ q(t) is the asymptotic extinction probability. The limit theorem of branching processes implies that q = (q 0 , . . . , q N ) is the smallest (with respect to the standard ordering of the positive cone), non-negative solution of the equation q = G(0, . . . , 0). These extinction probabilities can be non-zero if and only if the dominant eigenvalue of the Jacobian matrix DG(1, 1, . . . , 1) is greater than one. Equivalently, the reproductive number given by the dominant eigenvalue of the next generation matrix of DG(1, 1, . . . , 1) is greater than one [92] . Note that the linear map s → DG(1, 1, . . . , 1)s corresponds to the mean-field dynamics of the embedded multi-type branching process.