# Math Insight

### Model of an infectious disease without immunity

Elementary dynamical systems
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Total points: 3
1. In the SIS model of an infectious disease, individuals are either infected or susceptible to infection, so that infected individuals become susceptible once again when they recover. If the state variable $I(t)$ is the fraction of infected individuals at time $t$, the evolution of the model can be written as \begin{align*} \diff{I}{t} = \alpha I (1-I) - \mu I, \end{align*} where $\alpha$ and $\mu$ (“alpha” and “mu”) are positive parameters.
1. Why don't we need a state variable to track the fraction of susceptible individuals? Because each individual is either
or
. In this model, we don't allow individuals to be in any other state (like recovered or removed).

Because of this fact, we can write the fraction of susceptible individuals in terms of the state variable $I$. If the fraction of infected individual is $I$, then the fraction of susceptible individuals is
.

2. Explain the meaning of the two terms on the right hand side of the differential equation, including the meaning of the two parameters $\alpha$ and $\mu$.

The term $\alpha I (1-I)$ is the rate at which
individuals
. Since a
person needs to encounter an
person to get the infection, the rate is proportional both to the fraction
of infected individuals and the fraction
of susceptible individuals. (Write an answer in terms of the state variable in the last two blanks.) The parameter $\alpha$ captures how likely it is for a
person to encounter an
person and then subsequently
.

The term $-\mu I$ is the rate at which
individuals
and become
again. This rate is just proportional to the number of
individuals. The parameter $\mu$ captures how quickly
individuals
.

2. Analysis of equilibria.
1. What are biologically plausible values of the state variable $I$?
$\le I \le$
2. Find the equilibria. $I_e =$
3. You should have found two equilibria, one of which was zero. For each equilibrium, give a condition on the parameters so that the equilibrium is biologically plausible.

The zero equilibrium is biologically plausible for
values of the parameters $\alpha$ and $\mu$.

The other equilibrium is biologically plausible if the following condition on the parameters is satisfied:
. (This condition should be an inequality involving $\alpha$ and $\mu$).

4. For each equilibrium, give the conditions on the parameters for which the equilibrium is stable and is unstable.

The zero equilibrium is stable if
. The zero equilibrium is unstable if
.

The other equilibrium is stable if
. The other equilibrium is unstable if
.

There is one case where the stability theorem does not tell us if the equilibria are stable or unstable. This happens when $\alpha =$
.

5. Based on your analysis, for what values of the parameters will the infectious disease die out with time?
For what values of the parameters will the disease persist in the population?
(Assume the initial conditions were for a positive $I(0)$.)

3. The above analysis should have identified two cases for parameter values that give different results. For each case, draw the phase line with equilibria, restricted to biologically plausible values of $I$. Then, on a plot of $I$ versus $t$, sketch solutions $I(t)$ for representative initial conditions $I(0)$ allow with equilibria solutions, using a solid line for stable equilibria and a dashed line for unstable equilibria.

For the first case, let $\alpha=0.5$ and $\mu=2$.

Feedback from applet
equilibria:
number of equilibria:
stability of equilibria:

On this plot of $I$ versus $t$, in addition to the equilibria, show solutions for the initial conditions $I(0)=0.1, 0.5,$ and $0.9$.

Feedback from applet
Equilibria:
Final values of curves:
Initial conditions of curves:
Stability of equilibria:

For the second case, let $\alpha=2$ and $\mu=0.5$.

Feedback from applet
equilibria:
number of equilibria:
stability of equilibria:

On this plot of $I$ versus $t$, in addition to the equilibria, show solutions for the initial conditions $I(0)=0.1, 0.5,$ and $0.9$.

Feedback from applet
Equilibria:
Final values of curves:
Initial conditions of curves:
Stability of equilibria:

4. Fix $\alpha=0.5$. Then, draw a bifurcation diagram of $I$ versus the parameter $\mu$. On the diagram, indicate stable equilibria with a solid line and unstable equilibria with a dashed line. Include just biologically plausible values of $I$.
Feedback from applet
location of branches:
number of branches:
stability of branches:

A bifurcation point is a value of $\mu$ where the number or stability of the equilibria changes. The above bifurcation diagram should have a bifurcation point, which we'll label $\mu^*$. $\mu^*=$
.