Nonlinear dynamics problem from Strogatz textbook

[Star01111]

Homework Statement

I am working on problem 3.7.6 on Strogatz textbook Nonlinear_Dynamics_and_Chaos. It basically says there is a simple model for the evolution of an epidemic. Suppose that the population can be divided into three classes:
x ( t ) = number of healthy people; y ( t ) = number of sick people; z ( t ) = number of
dead people. Assume that the total population remains constant in size, except
for deaths due to the epidemic.
Then the model is
$$\dot x = -kxy$$
$$\dot y = kxy-ly$$
$$\dot z = -ly$$
where k and l are positive constants.

I am fine with a) to f) but having some troubles with g) to j) or k)

g) Show that the maximum of $\dot u (t)$occurs at the same time as the maximum of both $\dot z (t)$ and $\dot y (t)$
This time is called the peak of the epidemic, denoted$t_{peak}$ peak . At this time, there are more sick people and a higher daily death rate than at any other time.

h) Show that if $b<1$, then $/dot u (t)$ is increasing at $t=0$ and reaches its maximum at some time $t_{peak}>0$ Thus things get worse before they get better. Show that $\dot u(t)$ eventually decreases to 0.
i) On the other hand, show that $t_{peak}=0$ if $b>1$. (Hence no epidemic occurs if $b>1$.
j) The condition b = 1 is the threshold condition for an epidemic to occur. Can you give a biological interpretation of this condition?
k) Kermack and McKendrick showed that their model gave a good fit to data from the Bombay plague of 1906. How would you improve the model to make it more appropriate for AIDS or COVID-19? Which assumptions need revising?

Relevant Equations

Some equations that might be useful
$$z\dot = l[N-z-x_{0} exp(-kz(t)/l)]$$
which can be nondimensionalized to $$\frac {du}{d\tau} = a-bu-e^{-u}$$
where $$\tau = kx_0 t$$, $$u=\frac {kz}{l}$$
(I hope my nondimensionalization is correct)

For g) how should I argue this claim? To me it seems straight forward because u is linearly related to z. Then so do their derivatives. And by the description of the model, $\dot z$ is linear to y. So it's quite obvious but not sure what I should pay more attention to when I write my proof.

My main problem starts at h) My attempt is that I take the derivative of $\dot u(t)$ w.r.t. t after rewriting $\tau$ in terms of t. By plugging in the $z$value at t=0, I can argue the derivative is positive then increasing at $t=0$. But how should I show its maximum at some time $t_{peak}>0$ and $\dot u (t)$ eventually decreases to 0? I don't have an explicit equation for $\dot u (t)$ which depends on t. I am getting lost on how to tackle this analytically. Maybe I should solve this completely by drawing the graph and then argue from there?

k) I think the key factor that is missing to implement the model for AIDS and COVID is medical intervention and physically isolation. Both can be represented by a correction term at the end of the model that is first introduced. Am I in the right direction?

Comments: I would say problem a) to f) in 3.7.6 are pretty standard (if someone interested can look it ), but the rest of problem are a bit vague and not described that clearly on how to start. So I would like some help from you guys on this. Thank you so much!

 

[pasmith]

Homework Statement: I am working on problem 3.7.6 on Strogatz textbook Nonlinear_Dynamics_and_Chaos. It basically says there is a simple model for the evolution of an epidemic. Suppose that the population can be divided into three classes:
x ( t ) = number of healthy people; y ( t ) = number of sick people; z ( t ) = number of
dead people. Assume that the total population remains constant in size, except
for deaths due to the epidemic.
Then the model is
$$\dot x = -kxy$$
$$\dot y = kxy-ly$$
$$\dot z = -ly$$
where k and l are positive constants.

I think you want $\dot z = ly$; otherwise the number of dead people would decrease in proportion to the number of infected people.

I am fine with a) to f) but having some troubles with g) to j) or k)

Not everyone has access to (your specific edition of) your textbook. Posting partial questions means that we are less able to help you; for example. how does $u$ relate to $x$, $y$ or $z$? What is $b$?

 

[Star01111]

Sorry about the typo. You're right. It should be $$\dot z = ly$$. And as I said in the relevant equations, $u$ is only linearly dependent on $z$, and it should have no dimension. $a$,$b$ are constants without dimensions which depends on constants $k$,$l$ and $x_0$.($x_0$=$x(0)$ is just the initial condition of $x$). The constraints to $a$ and $b$ are $a>= 0$ and $b>0$. And I don't think the exact meanings of a and b matter here, just treat them as non negative or positive constants.

I think you want $\dot z = ly$; otherwise the number of dead people would decrease in proportion to the number of infected people.



Not everyone has access to (your specific edition of) your textbook. Posting partial questions means that we are less able to help you; for example. how does $u$ relate to $x$, $y$ or $z$? What is $b$?

 

[Star01111]

I think it might be helpful to post a script of the textbook question from 3.7.6. (i) and (ii) are assumptions of the model and there are subquestions from a) to k). The parts I need some help are g) to k). I would appreciate any help!