Math – actuarial science, Python programming language

HW 3 – due by 15:00 on Thursday, March 25
Upload to Canvas a single pdf containing your work before the deadline.
Numerical computation should be done using the Python programming language. I recommend
you use Spyder, the Scientic Python Development Environment, which is a free integrated devel
opment environment (IDE) that is included with the Anaconda Python distribution.
Figures should be created using computational software. Your code should be included as an ap-
pendix.
1. Consider the dynamics of a directly transmitted viral microparasite to be modelled by the
system
dX
dt
= bN ? XY ? bX;
dY
dt
= XY ? (b + r)Y;
dZ
dt
= rY ? bZ;
where b,  and r are positive constants and X, Y and Z are the number of susceptibles, infectives
and immune populations respectively. Here the population is kept constant by births and deaths
(with a contribution from each class) balancing. Show that there is a threshold population size,
Nc, such that if N < Nc = (b + r)= the parasite cannot maintain itself in the population and
both the infectives and the immune class eventually die out. The quantity N=(b+r) is the basic
reproductive rate of the infection.
2. Using Python, numerically solve the logistic ODE using the 4th-order Runge-Kutta method.
Include a plot showing both the analytical and numerical solutions.
dN
dt
= rN(1 ?
N
K
); r;K > 0; N(0) = N0:
Note that information about the RK4 method can be found at https://en.wikipedia.org/wiki/
RungeKutta_methods.
Given y0(t) = f(t; y(t)); y(t0) = y0, the RK4 method is given as
yn+1 = yn +
1
6
h(k1 + 2k2 + 2k3 + k4);
1
where h is the step size and where
k1 = f(tn; yn);
k2 = f(tn +
h
2
; yn + h
k1
2
);
k3 = f(tn +
h
2
; yn + h
k2
2
);
k4 = f(tn + h; yn + hk3):
3. Using Python, numerically solve the ODE system below (pendulum) using Euler method with
step size h = 0:1, midpoint method (RK2 method) with step size h = 0:1, and RK4 method with
step size h = 0:1. Include a plot showing all three numerical solutions, and explain the dierences
and similarities.
d
dt
= y;
dy
dt
= ?sin ;
with initial conditions (0) = 0:5 and y(0) = 0.
2