RESUMEN

Global dynamics of a compartmental model which describes virus propagation in vivo is studied using the direct Lyapunov method, where the incidence rate of the infection and the removal rate of the virus are assumed to be nonlinear. In the case where the functional quotient between the force of infection and the removal rate of the virus is a nonincreasing function of the virus concentration, the existence of a threshold parameter, i.e., the basic reproduction number or Basic reproductive ratio, is established and the global stability of the equilibria is discussed. Moreover, in the absence of the above-mentioned monotonicity property, estimations for the sizes of the domains of attraction are given. Biological significance of the results and possible extensions of the model are also discussed.

RESUMEN

The 2003 severe acute respiratory syndrome (SARS) epidemic came and left swiftly, resulting in more than 8,000 probable cases worldwide and 774 casualties. It is generally believed that quarantine of those individuals suspected of being infected was instrumental in quick containment of the outbreaks. In this work we propose a differential equation model that includes quarantine and other intervention measures implemented by the health authority, including those to prevent nosocomial infections and decrease frequency of contacts among the general public. We also consider the possible behavior change by the general populace to avoid infection, in response to the severity of the outbreak in general and to these intervention measures in particular. Complete analysis is given for the model without quarantine. For the general model with quarantine, a basic reproduction number is derived and full description of its dynamics is provided. We will show that introducing quarantine measures in the model could produce bistability in the system, thus changing the basic dynamics of the model. We give numerical examples of parameter values with which bistable steady states, where one is disease-free and the other endemic, could exist. However, realistic parameter values indicate that, assuming limited imported cases, the occurrence of the stable endemic steady state or bistability is unlikely. The modeling results indicate that for an infectious disease with infectivity and patterns of transmission typical of SARS\@, the outbreak can always be eradicated by implementing border control of imported cases and limited quarantine, along with the public's social response to avoid infections. Moreover, the results also suggest that quarantine measures will be effective in reducing infections only if the quarantined/isolated SARS patients and their potential contacts can successfully reduce their contact rate and/or transmission probabilities. Hence the effectiveness of quarantine for infectious diseases like SARS\@, for which no infection is being prevented during the quarantine period, can only be indirect and therefore must be combined with other intervention measures in order to quickly contain the outbreaks.