Mathematical Analysis of The Role of Quarantine and Isolation in Epidemiology
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The quarantine of people suspected of being exposed to a disease, and the isolation of those with clinical symptoms of the disease, constitute what is probably the oldest infection control mechanism since the beginning of recorded human history. The thesis is based on using mathematical modelling and analysis to gain qualitative insight into the transmission dynamics of a disease that is controllable using quarantine and isolation. A basic model, which takes the form of an autonomous deterministic system of non-linear differential equations with standard incidence, is formulated first of all. Rigorous analysis of the basic model shows that its disease-free equilibrium is globally-asymptotically stable whenever a certain epidemiological threshold (denoted by Rc) is less than unity. The epidemiological implication of this result is that the disease will be eliminated from the community if the use of quarantine and isolation could result in making Rc < 1. The model has a unique endemic equilibrium whenever Rc > 1. Using a Lyapunov function of Goh-Volterra type, it is shown that the unique endemic equilibrium is globally-asymptotically stable for a special case. The basic model is extended to incorporate various epidemiological and biological aspects relating to the transmission dynamics and control of a communicable disease, such as the use of time delay to model the latency period, effect of periodicity (seasonality), the use of an imperfect vaccine and the use of multiple latent and infectious stages (coupled with gamma-distributed average waiting times in these stages). One of the main mathematical findings of this thesis is that adding time delay, periodicity and multiple latent and infectious stages to the basic quarantine/isolation model does not alter the essential qualitative features of the basic model (pertaining to the persistence or elimination of the disease). On the other hand, the use of an imperfect vaccine induces the phenomenon of backward bifurcation (a dynamical feature not present in the basic model), the consequence of which is that disease elimination becomes more difficult using quarantine and isolation (since, in this case, the epidemiological requirement Rc < 1 is, although necessary, no longer sufficient for disease elimination). Numerous numerical simulations are carried out, using parameter values relevant to the 2003 SARS outbreaks in the Greater Toronto Area of Canada, to illustrate some of the theoretical findings as well as to evaluate the population-level impact of quarantine/isolation and an imperfect vaccine. In particular, threshold conditions for which the aforementioned control measures could have a positive or negative population-level impact are determined.