Simulation of Influenza Epidemics with a Hybrid Model - Combining Cellular Automata and Agent Based Features

Transcription

1 Simulation of Influenza Epidemics with a Hybrid Model - Combining Cellular Automata and Agent Based Features Štefan Emrich 1), Felix Breitenecker 1), Günther Zauner 2), Nikolas Popper 2), 1) Institute for Analysis and Scientific Computing, Vienna University of Technology, 2) Wiedner Hauptstrasse 8-10, 1040 Vienna, Austria die Drahtwarenhandlung Simulation Services, Neustiftgasse 57-59, 1070 Vienna, Austria semrich@aurora.anum.tuwien.ac.at, fbreiten@osiris.tuwien.ac.at, guenther.zauner@drahtwarenhandlung.at, niki.popper@drahtwarenhandlung.at Abstract. To understand and predict epidemic patterns ODEs and PDEs have been used since the beginning of the last century. But these approaches have a quite relevant shortcoming. Trying to model a multiply heterogeneous population (e.g. with individual characteristics, varying population densities) increases complexity beyond limits. To bring individual effects into epidemic models a new approach is necessary. Agent-based (AB) models as well as Cellular Automata (CA) represent tools which allow incorporating such influences. In this paper we shall present a hybrid model that combines the flexibility of an AB-framework with the computational efficiency of CAs. We will also look at the potential benefit of such a structure by taking a look at first (academic) results. Keywords. Epidemics, Hybrid modeling, Cellular Automata, Agent-based Systems 1. A historical perspective of epidemic modeling Humanities urge to understand or in the best case even predict the course of epidemic patterns is probably as old as diseases themselves. It is clear, that without proper knowledge of the disease as well as without modeling techniques and tools this urge cannot be met. Thus, from the first known mathematical attempt undertaken by Hippocrates, it took several hundred years before reasonable models have been developed. At the beginning of the last century ODE- & PDE-approaches have been developed in order to describe the behavior of epidemic outbreaks. One of those early (and very well known) models being the description of a SIR-type epidemic made by Kermack and McKendrick. One major drawback of these approaches is, that they only reflect a perfectly homogeneous population. If one tries to include different subpopulations or spatial distributions the complexity of the models is quickly growing beyond reason. If it would be possible to create models that do respect individual characteristics and spatial distributions more realistic outcomes or at least a better understanding of epidemics could be expected. During the second half of the last century two interesting modeling approaches have been developed, that are capable of representing individual and spatial features. Namely Cellular Automata (CA), developed in the 1940ies and 50ies and Agent-Based (AB) Systems also referred to as multi agent systems (MAS) that have been developed during the 1990ies. Although these systems are fairly different they show also quite some overlapping, as we will see. 2. Brief Comparison of CA and MAS 2.1. Cellular Automata Both, Cellular automata and multi-agent systems are classified as bottom-up approach. Also they describe (complex) systems only by

2 definition of local interaction rules. Within CA theses interaction rules are very strict and usually quite simple. Generally classical cellular automata can be described by four points (adopted from [6]): Cell geometry CA consist of equal (geometrical) cells which are arranged in a (regular) lattice. Neighborhood definition A cell neighborhood is defined for the automaton. The neighborhood determines which cells a r e i n f l u e n c i n g e a c h o t h e r. T h i s neighborhood is valid for all cells during the whole runtime. Cell states A finite number of (discrete) states is defined. Every cell can assume one of those states. The state of the cell is subject to change. The transition between states is determined by the transition rules. The states of the cells are updated synchronously (after discrete time steps) for all cells. Transition rules These rules describe how the cells change its states. The rules only depend on the state of the neighboring cells and on the state of the cell itself. One set of transition rules is valid for the whole CA over the whole runtime AB-Systems and MAS For agent-based systems a comparably strict and exact definition does not exist. Throughout literature one can find many varying definitions, often depending on the authors needs, in general AB modeling is lacking a common standard, although this is not a big problem. One can define multi agent systems by starting with the agent itself, as being a computer system situated in an environment. It further has the capabilities to flexibly and autonomously act in this environment in order to reach (predefined) objectives/goals. This requires specifying the following three terms more detailed: Situated in our case means that the agent is interacting with its environment, it is capable to receive input from the surrounding (e.g. via sensors) and can also manipulate its environment to some extent. The definition of autonomy needs to be handled with care (we are talking about a pre-programmed computer system). It is satisfactory if the agent can reach decisions without (human) interaction. Flexibility is required in multiple ways. Firstly one demands that the system is operating and acting in reasonable time. Secondly the agents are not to be solely reactive but goal-oriented or in the best case anticipating. And thirdly agents may have the capability to communicate or interact with other agents and/or real humans. Multi agent systems consist as the name indicates of a number of such agents. Further we want MAS to have following characteristics: agents have a limited point of view (incomplete information and/or problem solving capabilities), the absence of global system control, decentralized data and asynchronous computation of the agents. This definition is adopted from [3]. Based on these definitions of CA and ABS we will now sketch a short comparison of the methods Comparison of the Methods Looking at CA their smallest unit is the cell; it is by definition fixed and may hold discrete states. Agents in MAS on the other hand are much more flexible and may hold several characteristic features (e.g. height, color, diameter, health status, speed, pressure, e.g.). The processing of time within CA happens in a very discrete way. All cell states are updated simultaneously. Between these updates nothing changes. For AB systems one may either use discrete or virtually continuous time, and changes may occur at any given time. The spatial structure of cellular automata is defined by its lattice and usually symmetrical (see [2] for an exception). Agent-based systems on the other hand do not require any (spatial) structure. Agents may or may not be spatially bound, and any desired structure can be introduced. The cells of CA do have a fixed range of influence the neighborhood. Agents within MAS on the other hand may have a very variable sphere of influence which does not necessarily need to be connected to spatial neighborhoods. Concluding we can generally say, that AB systems are more flexible than cellular automata. One could even say that agent-based systems are an extension of CA (see [4]). On the other hand this extension and freedom when using MAS comes at a price a trade off becomes necessary. The hardware demands of AB systems are

3 considerably higher than those of CA. Thus the time consumed for simulation of large models increases when using MAS, whereas CA are computationally extremely efficient and can if programmed in an adequate manner also take advantage of parallel computing which is apparently not possible for MAS. On the other hand it is possible to introduce interesting behavior to CA models by loosening the tight rules (see [5]). This comparison shows that a combination of MAS and CA may be of quite some interest, if one could take advantage of the flexibility of AB modeling and also achieve computational efficiency comparable to those of CA. 3. Defining the Model Structure To set up the structure let us consider the most important things that our model must incorporate. Since we want to make a simulation with a (close-to) realistic population we have to respect its heterogeneous structure. For influenza transmission the sex does not play a role, thus we only need to look at the different age pools. Depending on the strain of the circulating influenza virus, different age groups bare different infection risks. Generally speaking very old and very young people are at highest risk. To represent the social behavior of the population we shall break down the daily routine into 3 parts: working time or time spent in schools/child care facilities respectively, leisure time and time at home. Although this division is a fairly rough one, it will show to be satisfying for our problem. From this division we can deduce that every individual does need a work place and household to which it returns every day. In addition we shall also assign an explicit neighborhood which we shall interpret as the regular contacts of an individual (friends, relatives, etc.) Together with the infection probabilities it seems to make sense to divide our population into the subgroups of infants (0-3 years), kindergarten children (3-6), elementary school (6 10), high school (10-18), working population (18 65) and senior citizens (65 +). As said before, we do not consider the sex of the individuals. The demographic structure of the population can be obtained fairly easy via demographic institutions (for Austria this is Statistik Austria ). The (average) size of work places, schools and similar are usually also available from such sources. Based on this population frame we can start to build our model. As said before we are about to combine MAS with CA in such an order, that we benefit from their distinctive strengths (MAS: flexibility, CA: efficient computation) and as far as possible omit their weaknesses. With respect to our complex population structure it does seem sensible to store the individual properties within an agent-based framework. The implementation of the model is realized in MATLAB, primarily because of its efficient matrix-operations. Thus we shall store the individual s characteristics in a huge matrix, our look-up table. This table does hold following ten entries for every agent within the system: - unique agent ID - current health status - age of agent - household ID - neighborhood ID - (age specific) workplace ID - time stamp - infection probability - control variable (if symptomatic) - help variable This matrix is initialized at the beginning of the model according to the demographic structure. A very convenient effect of this look up table is the possibility to easily add extra characteristics to refine the model if necessary or desired. Within the model this AB framework is used to transfer the population, according to the daily structure, from workplaces/schools, to their leisure time neighborhood and into their households. For a reasonable runtime the schools/work places/child care facilities as well as neighborhoods are modeled by lattice-gas cellular automata (LGCA) of type FHP-I (see [6] for further information). But instead of uniform particles the LGCA are filled with the unique agent IDs. On collision they are put at risk of infection with their distinct probability. The CAs are initialized one after another, starting with the work places/schools continued by the neighborhood, by the main routine. The CA is filled with all associated agents IDs and then computed for the given runtime. Changes of health status are written directly into the agent look-up matrix. Households are modeled stochastically, on one hand because of their small (population) sizes and on the other hand because contact between agents can be taken for granted. This structure simplifies the programming process, since it is necessary to program only one CA. The size of the CA does of course depend of the number of agents in it and on a given density.

4 By such it is possible to have other densities for work places than for schools (which also reflects the real life situation). Further this structure guarantees, that at the end of a model-day all agents spent the same time within CA although they only were in suspend-mode most of the time. 4. Simulation Results As for every simulation the correct parameterization of the model is a crucial task. In the case of the present model some additional difficulties have been experienced. A main problem is the lack of infection probabilities. To the author s knowledge databases or literature on disease transmission of influenza do not exist. Given they would exist the problem would still persist to some (reduced) extend, since such values strongly depend on the circulating virus and influenza viruses are highly mutant. In addition exact data for verification & validation of the model was also not available although existent. Thus the results must be looked upon as being academic. First of all the overall behavior of the model does seem reasonable, if comparing Fig.1 and Fig.2. Fig.1: EISS-Index, Germany 2004/05 x-axis: Week of Year Where Fig.1 shows the influenza activity according to the EISS-Index for Germany (Season 2004/05), taken from [6]. Fig.2: Simulation results of Model (infected with and without symptoms) x-axis: Day of Simulation, y-axis: Individuals in Fig.2 illustrates the result of simulation with the present model within a population of 1 million. The higher curve being the number of infected with symptoms, the lower one without. The jags are explained by the fact that during weekends the agents are not passing through the work/school CAs. They spend that time within the spare time CA instead. Both curves are characterized by an elevated plain that is sustained for several weeks, instead of reaching a single peak as observed in most (simple) ODE models and a fairly delayed outbreak of the epidemic. As we know the model population consists of six different sub-populations (age pools) which do bear different infection probabilities. If taking a look at spread of the epidemic within these pools we notice quite a different pattern (see Fig. 3 and Fig.4). Where Fig. 3 is the epidemic pattern for the age pool of elementary school children (6-10) and Fig.4. shows the pattern for adults (18-65) It is obvious, that the infection spreads much faster among the elementary students than among the working population. Also the peaks height in terms of maximum affected individuals per age pool is clearly higher. Various interpretations for this observation are possible. For sure the different infection rates are playing an essential role. But it is very likely that in addition to the transmission probabilities the density of the work place is a major factor. One could even go as far as referring to schools as the breeding ground for infections. Firstly students become infected, then they go home/ into their neighborhoods and affect their surrounding.

5 population will stay at home (this probability is specific for every age pool). Although the results are academic, the effect is dramatic: The peak in Fig.5 is 45% higher than that in Fig.6! Fig.3: Percentage of infected elementary school students Fig.5: Ratio of infected in elementary school (low stay-home probability) Fig.4: Percentage of infected adults Of course this model should not only be used for reproduction of the status quo. Soon the question arises what kind of counter measures can be taken to suppress or weaken the outbreak of epidemics. Such counter measures could be to vaccinate large parts (or even the whole) population. In our model this can be implemented by a simple change of the infection rates, thus it does not seem to represent a very interesting scenario. Especially since pharmaceutical companies tend to present their vaccine as only solution to epidemic outbreaks alternative measures would be more interesting. One very easy possibility would be to increase the percentage of people that stay at home after becoming symptomatic. Although in the real world this would require quite some changes (e.g. raising awareness, change of working policies, etc.). Since the effects become most visible within the population of elementary school children these sub-populations have been picked for comparison. Fig.5 and Fig.6 do show two different model results. The only difference between both initial settings is a change of the probability to stay at home after becoming symptomatic. This probability was increased (in Fig.6) in such a way that an additional 20% of the overall Fig.6: Ratio of infected in elementary school (high stay-home probability) 5. Conclusion We did compare the overall model result with real epidemic patterns. This comparison seems to indicate that the model s structure is mapping reality in a fairly sound manner. Together with the fact that no data for parameterization was available the combination of CA and MAS to model the spread of influenza seems promising and can in this case be seen as success. Since the model structure is still fairly simple it offers quite a potential for further improvement. For example one could implement hot-spots for the transmission of diseases such as hospitals. Enhancements could also be made at the daily routine.

6 What probably is most interesting about this model is the capability to investigate the spread of an epidemic for different age pools and population groups. This could lead to new insights and a better understanding for the prediction of epidemics. Finally this understanding can be used to examine different counter measures regarding their efficiency. With respect to the current financial situation of public health insurance (in Austria) this could lead to significant savings. 6. References [1] A r b e i t s g e m e i n s c h a f t I n f l u e n z a, Abschlussbericht der Influenzasaison 2004/05, Berlin, 2005 [2] Hegselmann, Flache. Do Irregular Grids make a Difference? Relaxing the Spatial Regularity Assumption in Cellular Models of Social Dynamics. JASS Journal of Artificial Societies and Social Simulation, 2001 [3] Jennings, Sycara, Wooldridge. A Roadmap of Agent Research and Development (Autonomous Agents and Multi-Agent Systems). Kluwer Academic Publishers [4] Lustick J. Agentbased Modelling of Collective Identity: Testing Constructivist Theory. JASS, 2000 [5] Roli, Zambonelli. On the Emergence of Macro Spatial Structures in Dissipative Cellular Automata, and its Implications for Agentbased Distributed Computing. [6] Wolf-Gladrow D. A. Lattice-Gas Cellular Automata and Lattice Boltzmann Models. Springer, 2000