Please place your name(s) in the upper-right corner.
Let’s model the dynamics of a disease where the population size
(or density) is constant (this means that \(S
+ I = 1\)) and there are two groups that make up the population:
susceptible, \(S\) and infected, \(I\). We sometimes make the assumption of a
constant population size because that’s sometimes the biology, but it
also means that it’s easier to do the math; e.g., substituting one
variable for another. This will be a model for a transmissible disease
from which one does not recover, like the herpes simplex virus in
humans. If the transmission rate is \(\beta\), then what are the equilibria
(hint, there are > 1) when looking at the rate of change of infecteds
in the equation: \[\frac{\mathrm{d}I}{\mathrm{d}t} = \beta
SI\] Please show your work.
It’s often a concern for disease
biologists to determine if a disease will spread through a susceptible
population. For what numeric values of transmission rate will
allow a disease to spread in a near-fully susceptible population (i.e.,
\(S \rightarrow 1\) and \(I \rightarrow 0\))? What values of
transmission rate will not allow spread in a fully susceptible
population?
Reflect on the implication of this
question for 30 seconds. What are your thoughts?
If we wanted to model the dynamics of a different disease—one where susceptibles become infected and recover only to become susceptible again, like influenza or COVID in humans—then you could imagine extending the equation from 1 by adding a term for loss of infecteds; e.g.: \[\frac{\mathrm{d}I}{\mathrm{d}t} = \beta SI - \nu I\] Find the equilibiria of the equation above.
Please write out a system of differential equations for a disease
with 4 groups: susceptible, exposed, infected, and recovered. The
susceptibles interact with infecteds and the disease is transmitted at
rate \(\beta\). The susceptibles then
become exposed, the exposed become infected (really, infectious, here)
at rate \(\mu\), and the infecteds
recover at rate \(\nu\).
Make a “hypothetical” flow/box-and-arrow diagram for a zombie or
vampire model. Create this in any way that you desire.
Write one thing that you learned or focused on from the chapter that you expected to be on the quiz.