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Computational Methods in Epidemiology
Paula Gonzalez1, Leticia Velazquez1,2, Miguel Argaez1,2, Carlos Castillo-Chávez 3 , Eli Fenichel 4
1Computational Science Program, University of Texas at El Paso.
2 Department of Mathematical Sciences, University of Texas at El Paso.
3Department of Mathematics and Statistics, Arizona State University.
4School of Life Sciences, Arizona State University.
4.
Introduction
There is an increasing interest in modeling
risk associated with emerging infectious
diseases (EIDs). Disease risks, (like risk
associated with invasive species), are
endogenous, however most model treat risk as
exogenous and intrinsic. The SPIDER group,
in collaboration with researchers from the
University of Texas at El Paso (UTEP) and
Arizona State University (ASU), will study
models both to forecast and estimate the risk
associated with emerging infectious diseases
in humans livestock and plants. In this poster
we present a simple example of how we can
model infectious diseases..
Newton’s Method
Problem Formulation
We consider a simple Susceptible - Infected
( ) model, i.e. the total population is divided
into susceptible (S) and infected (I)
individuals.
Assume that is the effective contact rate.
We do not consider natural mortality and birth
rate.
We assume that the infected do not recover,
but eventually die at a rate
.
Also we consider that the infected population
disperses in one direction according to Fick's
motion law:
Numerical Results
We solve the problem with the followings
initials conditions
Let
and parameters
with
Where
We obtain the followings blocks in the Jacobian matrix
Our goal is to find u such that
.
SPIDER project
.
SPIDER: “Working group for Synthesizing and
Predicting Infectious Disease while account for
Endogenous Risk”.
Goal
Step 2: For h = 0, 1, … until
for a dispersion constant .
We assume that susceptible does not disperse.
The previous model is an example of how we can
model infectious diseases. We solve a system of
differential equations using implicit time
discretization. The resulting nonlinear system are
solved using Newton’s method. Our future work
includes the addition of physical constraints to the
model, i.e, nonnegative on certain variables likes
rates of population S and I. Further, we will
colaborate in the
SPIDER working group
described next.
Newton's Algorithm:
Step 1: Given an initial point
We present a simple example of how we can
model infectious diseases.
We study a
dispersion epidemic model where the total
population is divided into Susceptible (S) and
Infected (I). We consider that infected
individuals do not recover and eventually die.
Also we consider that the infected population
disperses in one direction according to Fick’s
motion law. We solve the system of
differential equations using implicit time
discretization. The resulting nonlinear system
is solved using Newton’s method.
The
algorithm is implemented using MATLAB.
We plotted the population concentration
versus space and time scales in order to study
the behavior of the individuals.
Conclusion and future work
F (uh )  
Step 3: Calculate the Newton step
.
from:
The model is given by the system of
differential equations
For n = 20 and the previous initial conditions
and parameters. We obtain
Step 4: Update
The Jacobian Matrix
matrix:
.
is a
The SPIDER working group will bring together
disease
ecologist,
economist
and
mathematicians. To facilitate the development of
predictive models to assist in the development of
risk management strategies.
The SPIDER group proposed to develop a
unified framework for developing predictive
models and policy that combines global trade,
learning, inspection programs, and incentive for
experts.
block
Figure 1
We discretize the previous equations with
respect to the time for obtaining a sequence of
ordinary differential equations
where
are
matrices whose components are
diagonal
The matrix C is a tridiagonal matrix. As an
example , for
we have
Acknowledgments
Figure 1 shows six time plots of infected and
susceptible versus space x . As time increases the
location of the largest concentration of infected move
from left to right. The left side of the infected will
decrease as time increases because the concentration of
susceptible decreases.
References
1.
2.
3.
4.
The authors thank the support from the
Computational Science Program and the National
Institute for Mathematical and Biological
Synthesis (NIMBioS) Grant, "Synthesizing and
Predicting Infectious Disease while account for
Endogenous Risk (SPIDER)". Special thanks to
Dr. Ming-Ying Leung and the Bioinformatics
program from UTEP for the travel support.
We use half cells at the boundaries finite
differences with
.
So there are
unknowns for both
and
, this allow s to pose the
problem as a square nonlinear system of
equations.
Horan, R. D., C. A. Wolf, E. P. Fenichel, and K. H. J. Mathews. 2008. Joint
management of wildlife and livestock disease. Environ. Resource Econ.
41:47-70.
Horan, R. D., and E. P. Fenichel. 2007. Economics and ecology of
managing emerging infectious animal diseases. Am. J. Agr. Econ 89:12321238.
Project: Working group for Synthesizing and Predicting Infectious Disease
while account for Endogenous Risk. SPIDER group.
White, Robert. Computational Mathematics.
Contact Information
Figure 2
Figure 2 shows the infected and susceptible population
versus time. As time increase, susceptible individuals
decreases and infected one increases.
Paula A. Gonzalez-Parra
Computational Sciences Program
[email protected]