Infectious Diseases Bruce Hannon, U of IL, Urbana, IL 61801, b-hannon@uiuc.edu, June 99.
20.1 Basic Model
In this model we consider the spread of an infectious disease within a population. We assume that there is some initial number of individuals already infected with the disease. These individuals can pass on the disease to a group of susceptibles S. We do not model explicitly the agents that cause the disease, such as viruses or bacteria. Doing that would be rather impractical if we would want to apply our model to real world diseases. Tracing the billions of agents that can cause the outbreak with a particular disease is virtually impossible. Therefore, we do not explicitly model the dynamics of individuals in a population of disease-causing agents but deal with their effects in an aggregate way. The law of mass action discussed in Part II of this book has proven to be a powerful analogous way of capturing the spread of a disease in a population. The two "reactants" in our case are the susceptible individuals S and the infective ones I. We define a contact rate BETA at which these two groups of individuals make contact and propagate the disease. This contact rate BETA is analogous to the reaction rates in chemical reactions. We also model an influx of susceptibles into the stock S. Additionally, we assume that once an individual had the disease that individual ultimately becomes immune to the disease. We therefore remove those infectives from the stock I and let them disappear in a "cloud" they will not further affect the spread of the disease and, therefore, we need not keep track of them. This is a very simple epidemic model but can you anticipate the resulting dynamics? Make an educated guess before you run the model. How do the dynamics change with a change in the rate of NONIMMUNE IMMIGRANTS. This rate is here set to 7 per time period, and the initial stocks for S and I are 1000 and 20, respectively.
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What are the effects of a vaccine on the spread of the disease? Assume that only 20% of the population receives the vaccine and that it is only 90% effective among the susceptibles. Infectives are not immunized. Change the model to include an incubation period and reassess the ability of your vaccine to limit the spread of the disease.
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