The final size of a SARS epidemic model without quarantine

In this article, we present the continuing work on a SARS model without quarantine by Hsu and Hsieh [Sze-Bi Hsu, Ying-Hen Hsieh, Modeling intervention measures and severity-dependent public response during severe acute respiratory syndrome outbreak, SIAM J. Appl. Math. 66 (2006) 627-647]. An "acting basic reproductive number" ψ is used to predict the final size of the susceptible population. We find the relation among the final susceptible population size $$ , the initial susceptible population $$ , and ψ. If $$ , the disease will prevail and the final size of the susceptible, $$ , becomes zero; therefore, everyone in the population will be infected eventually. If $$ , the disease dies out, and then $$ which means part of the population will never be infected. Also, when $$ , $$ is increasing with respect to the initial susceptible population $$ , and decreasing with respect to the acting basic reproductive number ψ.