Modeling and Control of Contagious Phenomena

The annual economic losses due to influenza in the USA are estimated at over $80 bn. The HIV/AIDS prevalence among 15-49 years old women in Botswana and Zimbabwe is above 25%. These two figures alone show the huge size of the humanitarian and economic consequences of communicable diseases, even not counting numerous other epidemic diseases in human or animal populations. Evidently, the development of new medical tools is of immense value, but the ability to predict the evolution of epidemic diseases and to compose efficient policies (educational and medical prevention, monitoring, treatment, etc.) is of crucial importance as well, and even modest improvements may have large impacts. The present project aims to further develop the mathematical modeling and simulation tools and to facilitate the dynamic policy optimization in this area. The most important specific feature of the project is that its starting point of investigation are relevant dynamic models of heterogeneous populations. Heterogeneity (with respect to genetic factors, habits, behavioral hazard, age etc.) may play a substantial role in the evolution of communicable diseases. On the other hand, models of infectious diseases in heterogeneous populations that explicitly take into account the heterogeneity are not only hard for numerical processing but, more importantly, require distributed data that are often either unavailable or unreliable. For this reason one of the two main goals of the study is to develop new aggregation techniques capable of implicitly taking into account the heterogeneity within models that need not be supplied with detailed distributed data and, on the other hand, have a much simpler analytic structure. This will allow a more realistic numerical simulation, hence prediction, of the evolution of infectious diseases and a more profound qualitative investigation. These techniques will also be applied to other social processes driven by contagious phenomena, such as illicit drug "epidemics". The second main goal of the project is to further develop and implement methods from optimal control theory for designing optimal prevention and treatment policies. Different "performance" criteria of humanities- related/demographic or economic nature will be involved. The novelty is again the explicit or implicit consideration of the population`s heterogeneity. Optimal control policies obtained by using explicit modeling of heterogeneous populations are hard to apply in reality due to lacking data or to technical reasons, while those obtained by using the aggregated models developed within the first goal mentioned above require less data and are easier to implement. The efficiency and the qualitative properties of the latter control policies will be compared with those of the former ones. The proposed investigation will have the additional effect of promoting the application of the involved powerful mathematical tools, in particular those from optimal control theory, in the areas of epidemiology and health economics.

It is well recognized in the mathematical modelling of epidemic diseases that taking into account the heterogeneity of the population with respect to disease-relevant traits (such as age, immunity, behavioural hazard, etc.) is important for the quality of prediction and counter-disease measures. Various mathematical methods are capable to provide relevant descriptions of the evolution of a disease in a heterogeneous population and respective computational tools are available or their development is unproblematic. One substantial difficulty, however, which usually prevents the applicability of these methods is, that heterogeneous models require information that is also heterogeneous. For example, it may be necessary to know how many individuals have a certain level of immunity, and this information should be available for each immunity level, or how many individuals have a given frequency of a certain type of risky contacts, which should be known for every frequency and for each type. Such information is either not available or rather vague (incomplete and inexact).For this reason, it is important to develop methods that instead of the detailed heterogeneous information incorporate and use only partial, or aggregated, or uncertain information. Since reliable statistical information may also be unavailable, a substantial part of the project is devoted to the so-called set-membership estimations, where the prediction of the state of the disease (a vector with components such as number of susceptible individuals, number of infected, number of recovered, etc.) at any time is represented by a set of such vectors, rather than by a single one. This set has to contain all possible hypothetical disease states that are consistent with the available information (i.e., cannot be excluded as possible outcomes) and, on the other hand, to be as small as possible. The computation of set-membership estimations of the aggregated states of epidemic diseases in heterogeneous populations requires solving specific non-standard optimization problems for distributed systems. The project develops the theory, numerical methods and software for such problems. This enables efficient computation of set-membership estimations of the disease progress. At the same time, slight modifications of these theoretical and computational tools can be used for designing optimal prevention and treatment policies. Optimal vaccination and optimal chemoprophylaxis are investigated in more details, where the optimality is understood as minimal aggregated number of infected individuals in the long run, or as minimal discounted economic losses due to the disease.Another source of uncertainty in epidemic models is the presence of stochastic disturbances, which may result from a random import of disease, or white noise in the strength of infection. The projects contribution to the stochastic epidemiologic modelling is twofold. First, the recently developed theory of warning signs of critical transitions is elaborated for a class of heterogeneous epidemic models, which allows a more accurate forecast of potential disease outburst. Second, a problem of optimal treatment under white noise is solved in a feedback or open-loop form, depending on the measurements information pattern. An important task in the mathematical epidemiology is to characterize and determine the long run behaviour of the disease, in particular whether the disease will die out or will reach an endemic state. This issue is closely related to the so-called basic reproduction number (characterizing the reproduction capacity of the disease) and the stability of the steady states of the disease dynamics (disease-free or endemic). These issues are also investigated within the project for several simple heterogeneous population models, and also under stochastic uncertainty, which allows to predict outburst of an epidemic disease and facilitates the prevention control design.