BIOMATH STUDENT SEMINARS BY Ji Zhang and Matt Hamilton




Presented by:  Ji Zhang

Measuring Stochasticity of Ross-MacDonald Malaria Model Using MVN Moment Closure Method

ABSTRACT

Malaria is a very serious human disease, causing more than one million deaths every year. With the 
discovery of the transmission cycle of malaria, people started to model the spread of the infection, 
with the Ross-MacDonald model being the earliest mathematical description. However, the basic 
Ross-MacDonald model is deterministic, so it cannot account for the randomness in transmission that 
results from the finite size of a real world population (demographic stochasticity). This randomness 
can lead to extinction of the infection in situations for which the deterministic model predicts 
persistence. Measures of randomness, such as the variance of the behavior of the model at a given time, 
provide information on the importance of stochastic effects on the dynamics and persistence of malaria.

Numerical simulations (Monte-Carlo simulations) are commonly used to study stochastic disease models. 
However, they generally take a very long time to achieve for large population sizes. In this project 
we compare the results of, and the time taken to undertake, multivariate normal (MVN) moment closure 
approximation and Monte-Carlo simulation of the Ross-MacDonald malaria model. We show that the MVN moment 
closure approximation is a good substitute for numerical simulation in measuring stochasticity.
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Presented by:  Matt Hamilton

Saddle points and sudden transitions: the effect of migration in “rock-paper-scissors” competition

Abstract

Ecological systems sometimes exhibit sudden and dramatic transitions between qualitatively different states. 
Saddle points can provide a way to explain sudden changes based on dynamics alone. However, the viability of 
such explanations depends upon how robust the saddle structure is to perturbation and noisiness in nature. May 
and Leonard (1975) showed that for certain parameter values, the classic Lotka-Volterra competition model for 
three species displays a particular type of competitive intransitivity (commonly known as “rock-paper-scissors”
dynamics) which is based on a saddle point structure. We present work based on the same model which investigates 
the effect on this saddle structure of migration between two rock-paper-scissors populations. It is shown 
that migratory contact with another population can stabilize the system and eliminate sudden transitions 
between states.