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Computing Threshold Conditions
for Models of Infectious Diseases
with Reinfections

Andreas Weber
Wilhelm-Schickard-Institut für Informatik
Universität Tübingen, 72076 Tübingen, Germany

Paul Milligan
MRC Labs, Fajara, Banjul, The Gambia


The spread of many infectious diseases can be modeled by dynamical systems. The system of ordinary differential equations corresponding to these dynamical systems are very often parametric, and equilibria very often exist only for certain values of these parameters.

These in turn indicate the conditions under which the infectious disease will die out. Thus the computation of the corresponding threshold conditions is of great importance. We compute such threshold conditions for various models of infectious diseases with reinfections focusing on ones which might describe epidemics caused by Respiratory Syncytial Virus (RSV).

For the simpler models we could use recently improved computer algebra systems that perform quantifier elimination on real closed fields [5,6,7,8] to calculate these threshold conditions fully automatically. Our main tool was the REDLOG system [3]. The results of the quantifier elimination was too large to be very useful; however, simplification procedures available in REDLOG [4,3] succeeded to reduce the results to a useful size.

For more complicated models the results of the quantifier elimination were not useful in their unsimplified form, and the simplification procedures of REDLOG failed within the available time. After performing some simplifications on the input ``by hand'' or interactively with a traditional computer algebra system such as Maple [1,2] the quantifier elimination package and its simplification tools succeeded in some interesting cases.

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IMACS ACA'98 Electronic Proceedings