© 2014 by Princeton University Press 4.3 Spread of SARS Angela B. Shiflet and George W. Shiflet Wofford College © 2014 by Princeton University Press

Spread of Disease The first case of SARS (several acute respiratory syndrome) occurred in November, 2002 in southern China. SARS is spread by close personal contact and perhaps by airborne transmission. It was contained by July, 2003, but resulted in 812 deaths. Many models of the spread of disease (including SARS model) are extensions of SIR Model (three populations are considered: Susceptible, Infected, Recovered). In SIR Model, assume the rate of change of the number of recovered is proportional to the number of infected.

Spread of Disease Also, assuming no births, deaths, immigration, or emigration in SIR Model. recovery rate a = 1/(number of days infected) = 1/d. Thus, dR/dt = a*I = (1/d)*I The rate of change of the number of susceptible is proportional to the product of the number of infected and the number of susceptible (due to the interactions between S and I): dS/dt = -r*S*I where r is called the transmission constant and usually is a small constant.

Spread of Disease Another explanation is that the rate of change of S is minus the product of the mean number of contacts per day an infected has (k), the probability such a contact is with a susceptible (S/N where N is the total population size), the probability that the disease is spread during such a contact (b), and the number of infected: dS/dt = -k*(S/N)*b*I = -(k*b/N)*S*I = -r*S*I The rate of change of the infected: dI/dt = - dS/dt – dR/dt I gains from what S has lost; and what I loses, R acquires.

Spread of Disease Diagram of model? Differential equations? dS / dt = -rSI dI / dt = rSI - aI dR / dt = aI

Spread of Disease

Spread of SARS and SEIR Model Efforts including quarantine the exposed individuals to separate them from the susceptible population, and isolate those who had SARS. SEIR Model (susceptible, exposed, infected, recovered) has an intermediate exposed population of individuals who have the disease but are not yet infectious. The assumptions are

Spread of SARS and SEIR Model

Spread of SARS and SEIR Model

As in the SIR Model, the product Iu*S gives the total number of possible interactions. Thus, (k/N0)*b*Iu*S = k*b*Iu*S/N0 is the number of new cases of SARS each day. Of these new cases, a fraction of (q) go into category exposed_quarantined (EQ), while the fraction (1 – q) go into exposed (E). For those transferring from susceptible (S) to susceptible_quanrantined (SQ), although they have been exposed, the disease was not transmitted. The total Number of non-transmission contact is (k/N0)*(1-b)*IU*S = k*(1-b)*IU*S/N0. The rate of change of those going from Susceptible (S) to SQ is q*k*(1-b)*IU*S/N0.

Reproduction Number Due to exponential growth, it’s important that R < 1. If R < 1, there is no epidemic. For R > 1, there is an epidemic. Three paths from IU to recovered_immune at rate of v, to SARS_death at a rate of m, to ID at a rate of w. The total rate of change leaving IU is (v + m + w)/day. Thus, 1/(v + m + w) is the average duration of infectiousness. R0 = k*b*(1-q)/(v + m + w).