Mudassir Lodi
Homework for Lecture 18 of Dr. Z.’s Dynamical Models in Biology class
Email the answers (either as .pdf file and/or .txt file) to ShaloshBEkhad@gmail.com by 8:00pm Monday, Nov. 8,, 2021. Subject: hw18 with an attachment hw17FirstLast.pdf and/or hw17FirstLast.txt
Also please indicate (EITHER way) whether it is OK to post
1.
Carefully read the answer to the first question in https://sites.math.rutgers.edu/~zeilberg/Bio21/att18S.pdf .
understand it and then write a Maple program that inputs numbers a,b,c,d,e, call it, C(a,b,c,d,e) and outputs the answer to the following question a chickens lay b eggs in c days, how many eggs do d chickens lay in e days?
Check that	 3) = 6 (as in the original problem).
((b/a) / c ) * e * d
2.
Carefully read the answer to the second question in https://sites.math.rutgers.edu/~zeilberg/Bio21/att18S.pdf .
understand it and then write a Maple program that inputs numbers a,b,k, call it, W(a,b,k) and outputs the answer to the following question • A and B can fill a cistern in a hours,
•	A and C can fill the same cistern in b hours,
•	B can fill k-times as fast as C
Find how long C would take to fill the cistern, working alone.
Check that W(4,5,2) = 20.
A + B = 1/a
A + C = 1/b
B = Ck
A + Ck = 1/a
A + C = 1/b
3. Recall that in order to find all the equilibrium points of a continuous-time (first-order) system
	x0(t) = F(x(t))	,
where F is a transformation from Rk to Rk you do the following
•	(i) Use algebra to solve F(x) = 0, getting a (usually) finite set of points in Rk. These are all the equilibrium points (that live in Rk) of the dynamical system (but so far you don’t know whether they are stable or not).
•	(ii) Find the Jacobian matrix, J(x) in general, featuring x1,...,xk.
•	(iii) For each of the points that you found in the algebraic step (i), plug it in the general Jacobian matrix of (ii), getting a certain numerical k×k matrix. Ask Maple to find its eigenvalues. If all of them have negative real part then the examined equilibrium point is stable, otherwise not. (In the borderline case of purely imaginary eigenvalue (or 0) it is sometimes called semi-stable.)
Use the above to find all the equilibrium points of the 2-dimensional continuous dynamical system
	x0(t) = x(t)(1 − x(t) − y(t))	,
	y0(t) = x(t)(3 − 2x(t) − y(t))	.
Then find out which ones of them are stable and which ones are not.

4.	Using procedure Dis2(F,x,y,pt,h,A), with h = 0.01, and A = 10 https://sites.math.rutgers.edu/~zeilberg/Bio21/M18.pdf .
confirm numerically the answers of problem 3.. Take pt to be close (but not the same) as the above-mentioned equilibrium points, (for example, if the given equilibrium point is [a,b] take pt = [a + 0.1,b + 0.1]), and see whether it tends to [a,b] or rather escapes away.
5.	Read section 6.6. of Leah Edelstein-Keshet’s book
https://sites.math.rutgers.edu/~zeilberg/Bio21/keshet/keshet6.pdf
and convince yourself that the SIRS dynamical system given in Eqs (28) (once we use R = N−S−I) is represented by procedure
SIRS(s,i,beta,gamma,nu,N) ,
i.e. it gives the underlying transformation where we used s and i rather than S and I (in Maple
√  I = −1).
Using procedure
Dis2(SIRS(s,i,beta,gamma,nu,N),x,y,[N-30,30],0.01,10) in https://sites.math.rutgers.edu/~zeilberg/Bio21/M18.pdf .
with β = 0.01, ν = 1 (so βν = 100)
with N = 50, N = 80, N = 120, confirm the prediction that the epidemic will be eradicated
(eventually the number of infected individuals will go to 0) if N < βν = 100, but will persist if  = 120, in the long-run, how many individuals will be infected?
Comment   is the famous R0-factor, aka infectuous contact number.