Introduction
In the year 431 before Christ, Thukydides noted down about a tragedy that devastated Athens citizens. The symptoms were dramatic. Young, healthy adults suddenly came down with an unexplainable disease. This outbreak was the start of an era where epidemics were recorded. Up to the 20ths century contagious diseases ran rampant and often incurable, which dramatically reduced the population in case of an outbreak. From the scores of epidemics occurred in the history, some of them have a special impact up to now: The outbreak came nearly two millennia after the outbreak in Athens, which brought death and bane over the world. Medical historians discovered that the plague occurred in the year 1331 in China, and deci- mated the population by 50 percent. Over existing trade routes the plague reached Krim in the year 1346, and from this hub Europe, Northern Africa and the Middle East. The name of this disease became the embodiment of horror: Black Death. At that time the disease and the infection was mysterious, but today we know that a bacterium named Yersinia pestis spread using flea living on black rats, infected individuals, and killed between 1347 and 1351 a third of the Europeans back then. Above all the plague bacterium can be transmitted from an infected individual to a healthy individual, which is known as airborne infection. The outbreak changed the behavior of the population in various ways. Some kept themselves away from remaining population to prevent from population contact, while others started to live an extensive life. The next outbreak of the plague appeared in 1896, and spread to nearly every part of the globe. By 1945, the death toll reached approximately 12 million.
Between 1918 and 1920 the Spanish Flu pandemic killed about 20-50 million people, especially young adults and teens with well working immune systems. No infection, no war, and no famine have ever had claimed that much victims in a little while. Surprisingly, the outbreak of the Spanish Flu had no evident impact, because in the heads the scare of the war was present, and nobody wanted to write about this epidemic.
An outbreak of the Asian Flu in 1957 resulted in an estimate of one million deaths. The Hong Kong Flu killed a population of about 700.000 individuals. AIDS, caused by the human immunodeficiency virus (HIV), was first recognized in the 1980s, and it has killed over 20 million people until now. This disease is now a pandemic, with an estimate of more than 40 million infected individuals at present. Apparently there are several factors, which perpetuate the spread of AIDS and other infectious diseases, including incautiousness (both sexually and drug abuse), misconceptions of the transmission and the immense belief in the development of modern medicine. It is worth pointing out in this context that about 90 percent of the death from infectious diseases worldwide is caused by only a few of diseases.
Most contagious diseases can be modeled using mathematical approaches to an- alyze and understand the epidemiological behavior or for predicting the process. Therefore, different approaches have been developed in the past. The classic S-I-R epidemic model, where class S denotes the number of susceptibles, class I denotes the number of invectives and class R denotes the number of recovered individuals. The sum of the given initial value problem is S(t)+I(t)+R(t)=N, with N being the number of observed population. However, the SIR model is not adequate to model natural birth and death, immigration and emigration, passive immunity and spatial arrangement adequately. To model infection diffusion through space, partial differ- ential equations (PDE) are needed. With PDE models it is possible to simulate the spreading of a disease over a population in space and time. However, the integration of geographical conditions, demographic realities, and keeping track over each individual is impossible.
For this purpose cellular automaton (CA) models can be used. A CA model is a dynamical system in which time and space is discrete and is specified by a regular discrete lattice of cells and boundary conditions, a finite set of cells and states, a defined neighborhood relation, and a state transition function that is responsible for computing the dynamics of the cells over the time. CA models for highly dynamic disease spread simulation are widely known and shape-space interactions were introduced for enabling to simulate complex interacting systems. Dynamic bipartite graphs for modeling physical contact patterns were introduced, which resulted in more precisely modeling of individuals’ movements. The graph can be built on actual census and available demographic data. When analyzing those graphs, the existing hubs can be found easily. It could be figured out that by using strategies like targeted vaccination combined with early detection without resorting to mass vaccination of a population an outbreak could be contained. The simulation application EpiSims which has been developed at Los Alamos allows simulating different scenarios by modeling the interaction of the different in- dividuals participating in the simulation. The knowledge about the paths enables to perform arrangements like quarantine or targeted vaccination to prevent the disease from further spreading. The model EpiSims was a reproduction of the city Portland (Oregon), but not a facsimile, because to model the habits of about 1,6 million indi- viduals would be nearly impossible and furthermore a massive intrusion into privacy. EpiSims allows to set parameter values for the within-host disease model on demo- graphics of each person, but also simulating the introduction of counter-measures such as quarantine, vaccination or antibiotic use can be done. The human mobility information is derived from the TRANSIMS model, which estimates the movement of people based on census data and activity maps taken from defined samples of the population. Using this specific information “social network” can be modeled for understanding how epidemiology depends on those characteristics, and furthermore the calculation of the overall economic pecuniary impact is possible.
Sample of a Virus Disease Spread Simulation
In the first simulation scenario a map of Austria was used. The model was simplified due to a homogenous population density over the whole country.
For the second simulation scenario the state Tyrol was chosen. Tyrol has 660.000 inhabitants, where about 115.000 inhabitants are living in the capital Innsbruck. The total area is 10.628 square kilometers. The area of settlement is about 1.600 square kilometers.
Parameters
The simulated infectious disease used for the simulation is similar to the avian flu, except for the imperative difference that this virtual virus can be transmitted between human beings directly with a relatively high likelihood. Therefore, this virtual form of the H5N1 avian flu virus can be considered a dangerous mutation, which could have the power to effect an epidemic/pandemic situation.
| Description | Value |
| Latent period in days | 3 |
| Infectious period in days | 10 |
| Recovered or removed after days | 15 |
| incubation period in days | 3 |
| Symptomatic period in days | 4 |
| Natural birth rate in percent | 0.002 |
| Natural death rate in percent | 0.001 |
| Virus morbidity in percent | 0.63 |
| Spontaneous infection rate in percent | 0.00001 |
| Vectored infection rate in percent | 0.4 |
| Contact infection rate in percent | 0.6 |
| Movement probability in percent | 0.4 |
| Immigration rate in percent | 0.0000001 |
| Re-Susceptible (temporary immunity) after days | 100 |
| Temporary immunity after birth in days | 20 |
State transition function δ
The algorithm iterates through each cell of the CA. Each cell represents a small area of the used geographical map and performs the operations of the n individuals placed in the cell (=location). The above described method performCellAction() computes the next discrete time step by considering following steps:Handle the natural death cases
Handle the natural birth cases
Compute death caused by the diseaseCompute the immigrants
Compute vectored infections
Compute contact infections
Compute spontaneous infections
Handle recovered individuals
Handle re-susceptible
Perform movement operations of the individuals
Adapt parameters according specification
Create output for actual time step
The steps are performed until the specified number of time steps for the simulation is reached. During the simulation process snapshots of the actual distri- butions are created and furthermore, the data for subsequent statistical analysis is generated and stored. With this information it is possible to track each individual and to reconstruct the occured interactions. This enables the usage of statistical approaches for better understanding the disease spread mechanisms and to identify the best possible way to stop the spreading.
Simulation Results
Austria
Three scenarios were simulated. The infection seed point was set to the capital of Austria, Vienna. In scenario A, neither medical treatment was provided nor was quarantine declared. In scenario B, two different medications were used for the treatment, but quarantine was not considered. The medication was aimed at increasing the healing chances by 45-55 percent. In scenario C, individuals were submitted to both, medical treatment and quarantine. Furthermore, the social behavior changes of the individuals during the simulation was considered. These behaviors werde modeled because when a disease is circulating, individuals are very cautious contacting others to minimize their own risk of infection.
Figures depicts the development of the susceptibles over the time. As expected from declaring a quarantine status in scenario C, the infection spread stops.
Figure shows the characteristics of the infection over the time in percent and in Figure the fatal cases are illustrated. Assuming that there is no medica- tion, and no quarantine declared, the highest death toll is observed. The difference between scenario B and C is based on the fact that in scenario B the medication is given from the first day on, whereas in scenario C the medication and the quarantine start 50 days after the outbreak.
Tyrol
Eight different scenarios were simulated The seed point of the infection was set to the capital Innsbruck. In the first scenario (scenario A), the disease spread in the state Tyrol where medical treatment was performed. Two different drugs are available for infected individuals. Drug one reduces the death rate by 55 percent, whereas drug two reduces the death rate by 45 percent. The social behavior of the individuals changes during the simulation time, which would also occur in a real situation. When a fatal disease is circulating, individuals are very cautious contacting others to minimize their infection risk. The second scenario (scenario B) is similar to scenario A with the difference that no medical treatment is performed. Scenario C and D is equal to A and B with the difference that there is no adaptation of the social behavior. Scenario E and F is equal to scenario A and B with the difference that after 50 time steps a strictly controlled quarantine is introduced. In the last two scenarios (An, Bn), the same simulation parameters were applied as in A and B with the difference that no geographical and population density was used. Therefore, each cell covers the mean number of individuals from the state Tyrol model. By comparing these scenarios with A and B, it is possible to find out the relevance of geographical (natural barriers) and population density information. The blue color (Ind) is used for the population, red color (S) depicts the susceptible individuals, yellow (I) is used to visualize the infected individuals and the green color (R) was taken to depict the removed or temporarily immune individuals. The virus’s transmissibility (R0 value) is such that each infectious case gives rise to 3.4 secondary infectious cases. The following figures depict the classes susceptible (S), infected (I), and removed (R).
Summary
The framework enables the simulation of different communicable diseases by spec- ifying the disease parameters and demographic characteristics. Furthermore, the population can be divided into subgroups, which enables to simulated different im- pacts of the disease on each individual. The described scenarios demonstrated that CA and agent based models can be used for simulating and visualizing the spread and the adherent impact of infectious diseases. Furthermore, the simulation envi- ronment allows the access to any individual parameter at any point in time of the simulation, which enables detailed statistical analysis. Although the connections and the affiliated behavior between the individuals can be modeled using different
neighborhood relations, the behavior of any individual in these models depends on functions using random numbers. For upgrading the behavior algorithms, social behavior approaches should be used and the computation of the economic impact should be computed for better creation of public health strategy plans for managing fatal diseases. The natural manner – let us call it “Groundhog Day” – that most individuals do have a way of living caused by their daily workflow, can not be mod- eled correctly, using such functions. To solve this problem, virtual worlds could be helping. The well-known Second Life, where millions of people do live an additional life and do also have a behavior, which is very similar compared to their own, should be observed for simulations. One has to keep in mind, that building a population model from census and demographic data statistically equal to one in the real world would be very complex but also a deep impact to privacy, and therefore, afflicted with many of problems.
Recapitulatory one can state, that the proposed CA framework is able to support public health offices by providing them with information for creating plans to manage such situations and to prevent serious long-term economic repercussions.