LAB 11: MODELING OF EPIDEMIC PROPORTIONS AND INITIALVALUE ODE PROBLEMS
Mathematics:
Mathematical
modeling of infection diseases among human's population is based on solving the
initialvalue problem for a system of ordinary differential equations.
A mathematical model divides a total population into groups of people
interacting to each other with respect to a particular infection disease and
describes the number of people in each group as a function of time. This
project exploits a lethal disease when the total number of population is
divided into four groups: notyetinfected, infected, medicated and dead
people. The infection is assumed to result from contacts with an infected
individual.
Denote
the time as t, the number of susceptible (notyetinfected)
people as S = S(t), the number of infected people as I =
I(t), the number of medicated people as M = M(t), and the
number of dead people as D = D(t). The total number of population
is N = S + I + M + D. If the natural birth and death processes
are neglected on the time intervals when the infection disease blows up,
spreads out and eventially disappears, then N is constant in time
t.
Suppose
a is the rate at which the disease spreads per day and also that
the number of people who gets infected is proportional to the number of
interactions between susceptible (S) and infected (I)
people. Then, the rate, at which the number of susceptible people S(t) decreases,
is:
_{}=  a S(t) I(t)
Suppose
r is the death rate of infected people per day and also that the
number of dead people is proportional to the number of infected people (I).
Then, the rate at which the number of dead people D(t) increases,
is:
_{}= r I(t)
Suppose
b is the constant number of shots per day that save people's
lives and that people who received the medicine can no longer be infected and
are not contageous. Then, the rate, at which the number of medicated people M(t)
increases, is constant as:
_{}= b
Since
the medicine is given only to sick people, it is natural to assume that b
< I(t) for any t > 0. The rate, at which the number
of infected people I(t) changes, can be found from the condition
that the total number of people N remains unchanged for any time t
> 0:
_{} = a S(t) I(t) – r
I(t) – b
By
summing all four differential equations above, we confirm that:
_{} = _{}+ _{} + _{} + _{} = 0.
Objectives:
·
Understand
how to solve the initialvalue problems for systems of differential equations
·
Exploit
modeling of the epidemic proportions with the Euler method (discrete
differences)
· Exploit modeling of the epidemic proportions with MATLAB ODE solvers (RungeKutta and Adams)
In
numerical methods of solving the initialvalue problem for systems of
differential equations, the solution is to be found at discrete times: t_{0},t_{1},t_{2},…t_{n},
starting with the initial time t_{0} = 0 and
ending at the final time t_{n} = T. If the discrete times
are taken with a constant step size: h = t_{1} – t_{0} =
t_{2} – t_{1 }= … = t_{n} – t_{n1},
the derivative of any function f'(t_{k}) at any time t
= t_{k} can be approximated by the first forward divided
difference:
f'(t_{k}) = _{}
This
is the Euler method for solving the system of differential equations.
Replacing the first derivatives by the first forward differences at the times t
= t_{k}_{ }for k = 1,2,…,n; the system of
differential equations becomes the discrete dynamical system, which is suitable
for numerical iterations (loops):
S_{k+1} = S_{k }– h a S_{k} I_{k}
_{ }D_{k+1} = D_{k }+ h r I_{k}
M_{k+1} = M_{k }+ h b_{}
I_{k+1} = I_{k }+
h a S_{k} I_{k }– h r I_{k} – h b
Again,
the total number of people is preserved: N_{k+1 }= N_{k}.
A typical pattern of dynamical behaviour of the number of susceptible (S)
and infected (I) people is shown as a
function of time t:


Steps
in writing the MATLAB script:
Exploiting
the MATLAB script:
The
number of survived people at any time t is
R(t) = S(t) + M(t) + I(t) =
N – D(t)
The
number of survived people after 15 days is R = 6673
for b = 0 and R = 8582 for b = 10.
MATLAB
ODE solvers include:
·
ode23:
based on the variablestep RungeKutta method of second and third order
·
ode45:
based on the variablestep RungeKutta method of fourth and fifth order
·
ode113:
based on the Adams method of the variable order from one to thirteen
·
ode15s:
based on the implicit methods of the variable order for stiff differential
equations
Steps
in writing the MATLAB script:
Exploiting
the MATLAB script:
QUIZ:

Euler method 
ode23 
ode45 
ode113 
ode15s 
b = 0 
6673 




b = 10 
8582 



