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Research Areas for PASI '06: Math Models in Population Dynamics
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The general topic of the Institute is Mathematical Modeling of Population Dynamics. This is interpreted to include human populations and those of mosquitoes or other disease-spreading agents in the case of epidemics. In other cases it includes fish or shrimp populations and tracking their migrations. The modeling of these phenomena requires several components:
- Data and Statistical Methods: data collection, parameter estimation, spatial analysis, stochastic processes
- Mathematical Models: ordinary differential equations models (SIR-type), space-time models (PDE), discrete models
- Computational and Visualization Methods: numerical ODEs, reaction-diffusion PDEs, fluid flow
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Data and Statistical Methods
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The available information regarding the rate of infection of a disease is typically incomplete due to underreporting, unavailable demographic data, and variable collection methods. Similar issues are present in collected data for the population of shrimp or fish in various rivers or lakes. In spite of this, the mathematical models designed to predict population behaviors are more relevant when they make use of available data. A typical model will involve several parameters whose ranges of values must be estimated from the data in order for the results to be relevant to a specific epidemic. One of the subtopics of the PASI is the statistical methods that can be used to transform raw data into information that can be included in a mathematical model. One of the most important aspects of this subtopic is the estimation of various statistical parameters and their variability. The results inform the mathematical models of the statistical significance of the parameters involved and the likelihood of those parameters to be in a given range. Specifically, the Institute will feature:
- Data collection, quality control, and preliminary analysis: The collection, aggregation, and evaluation of empirical data are all critical components of any statistical analysis. At this stage, data are inspected for quality control and the appropriateness of underlying modeling assumptions. The presentations and discussions will include the type of data currently being collected, the methods of collection, and the challenges faced in the process.
- Modeling and parameter estimation: Depending on the type of data available in a particular case, generalized linear models can be used to estimate the relative rates of disease for individuals from various populations. The significance and correlation of demographic factors may be estimated through modeling, allowing for the selection of the most critical variables.
- Spatial analysis: The geographic nature of observed cases is a critical component of the modeling process. Spatial methods may be employed to determine whether the observed distribution of cases is significantly clustered. While such clustering will be expected for contagious diseases such as viruses, clustering in non-contagious causes may be caused by topographical and environmental factors.
- Stochastic processes: Additionally, there will be presentations on stochastic processes as a mechanism for introducing uncertainty into model equations.
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Mathematical Models
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Mathematical models of population dynamics can be designed based on a variety of assumptions, leading to completely different model equations which require distinct mathematical methods of analysis. Three types of models will be addressed.
- SIR-type models: In SIR-type models (Susceptible, Infectious and Removed), the levels of populations of various groups are tracked as functions of time. The rate of change of each population group is influenced by the others, resulting in a system of ordinary differential equations. These are almost always nonlinear equations since the terms that account for the interactions between two groups (e.g., infectious mosquitoes and susceptible humans) often involve the product of the two populations multiplied by a parameter that represents the inverse of the time scale in which the interaction occurs. These parameters (e.g., infection rate and recovery rate) must be estimated for particular applications based on available data. The mathematical techniques used to analyze the resulting system of ordinary differential equations come from dynamical systems and involve classifying equilibrium states, determining basins of attraction of stable equilibria, linearization, Lyapunov functions, etc. Uncertainty can also be introduced into the model by adding a stochastic component to the parameters. Additional topics related to these models include accounting for the age structure of the population and the effect of travel, which affect the nature of the interactions between different groups.
- Space-time models: The second type of mathematical model may be viewed as an extension of the SIR-type model that includes the spatial distribution of the population. In this case each population group is a function of space and time, leading to a system of partial differential equations. The mobility of the individuals can be modeled by a diffusion process when the populations tend to move about but staying relatively close to a single location (e.g., within a neighborhood). To account for long-distance travel, diffusion models are not appropriate and one has to include terms that represent long-range interactions. Populations of shrimp or fish that are affected by water currents require the interaction between the fluid flow and the population. Fluid flow is modeled with the Navier-Stokes equations or some simplified variation of them and the resulting fluid motion can be used in the population model to simulate their displacement. The presentations will address simple fluid flow models and how to combine them with the population dynamics.
- Discrete models: In addition to the time-continuous models described above, other modeling frameworks have been used in population biology and epidemiology. These include time-discrete dynamical systems, stochastic models, and interaction-based models. Time discrete models are of the form Pt = F(Pt-1), where Pt represents population density at time t. This density is typically given in terms of an integer, since in most cases the population consists of a collection of discrete individuals or insects. Since available experimental data often come in the form of time series of observations, this type of models is natural to consider.
Another type of discrete model is the so-called interaction-based or agent-based model. This model type is a derivative of the cellular automaton invented in the 1950s. The population is represented by a collection of variables, each variable, or agent, representing a member of the population. Each agent is equipped with a collection of rules that specify motion, if a spatial dimension is present, and the interaction with other population members. This type of model is particularly useful for epidemiological models. One of the most sophisticated instantiations of this type is EpiSims, a model of the spread of an infection caused by an airborne pathogen in an urban area, using Portland, Oregon, as an example. EpiSims is built on top of TranSims, a cellular automaton based model of the population dynamics in urban areas. The advantage of interaction-based models is that global dynamics is generated by local interactions of individuals, which mimics reality closely in many situations. A disadvantage is that this type of model does not lend itself easily to mathematical analysis; however, statistical studies using multiple simulation runs are used to assess the model performance.
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Computational and Visualization Methods
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The mathematical models of population dynamics described above lead to systems of differential equations in the continuous models or difference equations in the discrete models. There is much analysis that can be performed to determine equilibrium states and asymptotic solutions; however, most commonly the equations can only be solved numerically. For this reason it is very important to include in the scientific program numerical methods that can be used to nd approximate solutions to the model equations and to display them for visualization. The computational and visualization methods subtopic of the scientific program will include computational techniques for systems of ordinary differential equations, systems of partial differential equations from fluid dynamics and reaction-diffusion type. In the case of ODEs, the emphasis will be on the use of available software such as MATLAB that is known to be efficient and reliable. In the case of fluid flow problems and reaction-diffusion equations, numerical methods based on finite-differences and fundamental solutions will be addressed.
- Numerical methods for ODEs: Standard explicit methods such as Runge-Kutta are typically robust and efficient for the solution of systems of ODEs when there is a single time scale in the problem. When more than one time scale is present and two time scales are vastly different, the system is stiff and it is generally advantageous to use an implicit or semi-implicit method. These methods require the solution of a linear system of equations which can be done by inverting a matrix in small systems or by iterative methods or matrix decomposition methods for large systems. The main ideas of these methods will be addressed and the software MATLAB will be used for demonstration of the time step limitations in explicit methods and the advantages and challenges of implicit methods. In some cases it is necessary to find the long-term solution of the system (up to a steady state) for which high accuracy may be desirable.
- Numerical methods for reaction-diffusion PDEs: For evolution equations of the type Qt = c^2(Qxx+Qyy+Qzz)+F(Q, t), one can use finite differences to discretize the space derivatives and reduce the equation to a system of ODEs for the value of the variable Q(x, y, z, t) at each of the grid nodes. The methods for ODEs are then appropriate. The talks will address the concepts of local truncation error, stability and efficiency of the methods with the goal of giving an overview of the methodology and the factors that enter into the decision of which methods to use in specific applications.
- Numerical methods for fluid flow: In the simulation of streams that carry a dilute set of particles (that may represent small sea life), numerical methods based on finite-differences, such as projection methods, are available. These methods are currently an active area of research. Lagrangian methods such as vortex methods or the method of regularized Stokeslets are also excellent for the simulation of fast and creeping flows, respectively. Lagrangian methods are based on fundamental solutions of the PDEs and result in large systems of coupled ODEs which can be solved using commercial software. The idea behind these methods, the connection to PDE theory, and their numerical implementation will be explained and relevant examples will be shown.
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