The Phase Plane Analysis Paper

With several recent outbreaks of new types of viruses, such as, Zika, Ebola, and Hepatitis, I wanted to research the basic types of mathematical models to see the rate at which a virus is transmitted. To understand the complexity of viruses, I first needed to understand what an epidemic was and the basic types of growth models. Upon my research, I found that there was a lot to cover with these growth models, so I constrained myself to the models known as the Exponential and the Logistic models. Lastly, Dr. Antoniou suggested that after analyzing these models, which are differential equations, we should move to a whole new topic which was not covered in our Differential Equations class; the Phase Plane Analysis.

First, an epidemic is the rapid spread of infectious disease to a number of people, in a given population. We can model the rate at which a disease is spread, with our first model,

Note that there are indefinitely many solutions to this differential equation; so, we need an initial condition, which is our initial number of infected, to determine the unique solution. Using this model, we can predict how many people will have the virus or disease at any time t. There are a couple conditions that need to be met in order for a population to undergo exponential growth. The first and second, being unlimited space and resources to grow. The last is no threat from predators. If these conditions are met and we have exponential growth, then our population will grow larger and larger until it approaches an infinitely large size. Since we have a fixed growth rate k, our quantity of infected will have a doubling time or period which remains constant. The “doubling time” is the period it takes for a quantity to double in size or value. We can prove this by taking our function, and solving for time one and time two. We want , where our function is . Solving for , we get . Where is the difference between time 1 and time 2. Hence, our doubling time is