Mudassir Lodi
Homework for Lecture 19 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: hw19 with an attachment hw19FirstLast.pdf and/or hw19FirstLast.txt
Also please indicate (EITHER way) whether it is OK to post
1. Carefully read the Maple code for procedure SIRSdemo(N,IN, gamma,nu,h,A) in the Maple code , and run it for the following parameter values (with h = 0.01)
For all of them Population: 1000 people. Initial infected individuals: 200 people; No one dead (or removed) at the start, t = 0.
For β = 0.3N/ν, β = 0.9N/ν, β = 3.9N/ν, answer
At time t = 10: How may people died (or were removed) (recall that the total is always 1000), if the values of ν and γ are as follows:
(i)	ν = 1, γ = 3
SIRSdemo(1000, 200, 1, 3, 0.01, 10)
(ii)	ν = 2, γ = 3
SIRSdemo(1000, 200, 2, 3, 0.01, 10)
(iii)	ν = 3, γ = 7
SIRSdemo(1000, 200, 3, 7, 0.01, 10)
2.	Using RandNice([x,y],8) 3 times, (call it F) each time. For each of them find the following:
(i): The set of equilibrium points
EquPts(F, var)

(ii): The set of stable equilibrium points
StEquPts(F, var)
(iii): Using Dis2(F,x,y,pt+[0.1,0.1],0.01,10);
confirm numerically that for pt in the stable set it converges to that point, but for the other equilibrium points, the orbit goes elsewhere (and very possibly all over the place, i.e. to ∞).
3.	Use EquPts(F,var) together with SIRS to find all the equilibrium points of the SIRS model, confirming Equations (29a) (29b) in Edelstein-Keshet, section 6.6.
SIRS:=proc(s,i,beta,gamma,nu,N):[-beta*s*i+gamma*(N-s-i),beta*s*i-nu*i]:end:
      EquPts(F, var)
4.	Write a Maple code Chemostat(N,C,a1,a2) (analogous to SIRS) giving the underlying transformation of the Chemostat model with parameters a1,a2 as given by Equations (19a),(19b) in Edelstein-Keshet section 4.5. Then use EquPts(F,var) to confirm Eq. (25a),(25b) there.
5.	(Optional, 10 brownie points):
(i)Write Dis3(F,x,y,z,pt,h,A), a 3D analog of Dis2(F,x,y,pt,h,A).
(ii) Run RandNice([x,y,z],10), 2 times, and find, for each case, the equilibrium and stable equilibrium points, and confirm stability (and non-stability) using your Dis3. (similar to question 2, but in three dimensions).