Algorithmic Methods for Computing Threshold Conditions in Epidemic Modelling Christopher W. Brown 1⋆ , M’hammed El Kahoui 2 , Dominik Novotni 3 , and Andreas Weber 4 1 Department of Computer Science, United States Naval Academy, U.S.A.; E-mail: wcbrown@usna.edu 2 Department of Mathematics, Faculty of Sciences Semlalia, Cadi Ayyad University, P.O Box 2390 Marrakech, Morocco; E-mail: elkahoui@ucam.ac.ma 3 Bonn-Aachen International Center for Information Technology, Bonn, Germany; E-mail: novotni@cs.uni-bonn.de 4 Institut f¨ ur Informatik II, Universit¨at Bonn, R¨omerstr. 164, 53117 Bonn, Germany; E-mail: weber@cs.uni-bonn.de Abstract. The calculation of threshold conditions for models of infectious diseases is of central importance for developing vaccination policies. Frequently coupled systems of ordi- nary differential equations are used as models and the computation of threshold conditions can be reduced to the question of stability of the disease free equilibrium. We show how the computations of threshold conditions for such models can be done fully algorithmically by using techniques of quantifier elimination for real closed fields and related simplification methods for quantifier-free formulas. 1 Introduction Modelling epidemics of childhood infections are important for public health programs, especially for planning vaccinations. Commonly the epidemics are modelled by systems of ordinary differ- ential equations. Examples are the SEIR model [8,18], which is frequently used to model measles epidemics, models considering reinfections such as the SIRS model [14,10,20], the SEIRS model [15] (see Example 1), the MSEIRS4 model [20], or models taking into account the effects of treatments [19,2] such as the SEIT model [19](see Example 2). The division of the population into different groups in their relation to various stages of infectiousness, immunity, susceptibility, behavior, etc. is a standard technique to incorporate disease specific characteristics into the model while staying in the realm of ordinary differential equations [1,6,11,19]. These models do incorporate a priori medical knowledge on the specific disease but are very often strong simplifications. The justification of the simple models is then obtained a posteriori: The simple models—often involving only 3 or 4 variables — are good if they can reproduce the complex dynamics of the epidemic outbreaks. As non-linear dynamical systems can be complex, such a justification has been found in several cases [8,16]. These epidemiological models involve many parameters, such as the rate of loss of immunity or parameters describing how infections are transmitted, that differ for specific diseases, locations or populations. Estimating these model parameters falls in the realm of numeric techniques [13,17]. However, there is also a very important calculation involving these parameters that is best formu- lated as a symbolic problem, namely the computation of threshold conditions. Threshold conditions are conditions that the parameters must satisfy in order for the model to exhibit certain kinds of qualitative behaviours. When infections are present in a population, a disease outbreak may progress in qualitatively different ways: the disease may die out, or it may reach an endemic stage, in which the disease is always present in the population. The latter will be the case if the number of secondary infections from each infected individual exceeds one. This concept, formalized by the basic reproduction ratio R 0 , is the key concept in the literature behind modelling threshold conditions (see e.g. [6,11,19]). Since the value of R 0 for a given model is a function of the parameters, R 0 provides a threshold condition for the parameters. As some parameters can be changed by vaccinations or behavior, knowledge of threshold conditions can be very important for public health policy. ⋆ Supported in part by NSF grant number CCR-0306440.