Dec-10-2020, 02:53 PM
Hey there, please help me with my assignment
""""
Exercise 1: three types of people: susceptible (S), infected (I) and recovered ®, the differential equations are given by:
1) d*S/d*t= -ß*S*I
2) d*I/d*t=ß*S*I-y*I
3) d*R/d*t= y*I
->Implement the differential equation system in Python. Simulate it
with the initial conditions S(0) = 997, I(0) = 3, R(0) = 0 for the parameter
values ß = 0.0004 and y = 0.04.
Systematically vary the infection rate ß and in
particular observe how the progression of the epidemic changes when the infection
rate is reduced, as for example by social distancing. Discuss the term “flattening the
curve”.
In the following modified version, population turnover is included. It is assumed that
new uninfected individuals are born with a constant rate a. Moreover, a constant
relative death rate δ is assumed for all groups. To reflect that infections can cause
death, a higher death rate constant ϵ > δ for infected people is assumed. Introducing
µ = y + ϵ, the system of differential equations reads
1) d*S/d*t= a-ß*S*I-δ * S
2) d*I/d*t=ß*S*I-µ*I
3) d*R/d*t= y*I- δ*R
"""
"""
Exercise 2: Show that this system exhibits two steady states, one reflecting a
disease-free population, and one an endemic state.
For your control, the states are
(S(0), I(0), R(0))=[(a/δ), (0, 0)]
(S(1), I(1), R(1))=[(µ/ß),(a*ß-δ*µ/ß*µ),(y/δ) * (a*ß-δ*µ/ß*µ)]
"""
"""
Exercise 3: Investigate the system by performing dynamic simulations.
A good starting point are the following parameters: a = 96.5, µ = 16, δ = 0.028, y = 0.7µ, ß= 3.05 δ*µ/a
and the initial conditions (S(0), I(0), R(0)) = (1070, 15, 0). Explore the
influence of the different parameters and interpret the results.
"""
"""
Exercise 4: Be creative! The SIR model is only the simplest version of an epidemiological models. Explore other versions, think how to implement them, collect
information in the internet, or make a completely new model of your own. Possible
ideas include:
a) The SEIR model (susceptible, exposed, infected, resistant) takes into account
that individuals that have been exposed to the disease to not immediately
become infectuous.
b) The SIS model assumes that people can be infected more than once.
c) How would you model a spatial component? Think of two locations, between
which people are randomly exchanged with a small rate. Or even think of
several locations and a hub, such as a popular holiday resort.
d) How would you assess the e↵ect of a vaccination?
e) Munz et al (2009) predicts the spread of zombieism, a disease which can be
modelled much like a typical SIR model, but with the caveat that the infected,
replaced with zombies in the model and denoted with a Z, can return from the
recovered population (which here refers to ’removed’ individuals, or zombies
who have been ’removed’ from the population by being killed).
The initial equations they use to simulate this model are as follows:
1)d*S/d*t = π-ß* S* Z- δ* S
2)d*Z/d*t = ß*S*Z +y* R -a*S*Z
3)d*R/d*t = δ*S + a*S*Z - y* R
Where S represents the susceptible population, Z the zombie population, and
R the population who have been ’removed’, or have died. π represents the
birth rate, δ the non-zombie related death rate, ß the rate of zombification,
y the rate of resurrection of removed people into zombies, and a the rate at
which the susceptible population kill zombies.
How could this model be adapted to more accurately adapt the behaviour of
zombies in particular media? In 28 Days Later, the zombies do not ressurrect,
and in Shaun of the Dead, they display an incubation time before returning
from the removed population.
"""
If anyone can solve these tasks, please help me.
""""
Exercise 1: three types of people: susceptible (S), infected (I) and recovered ®, the differential equations are given by:
1) d*S/d*t= -ß*S*I
2) d*I/d*t=ß*S*I-y*I
3) d*R/d*t= y*I
->Implement the differential equation system in Python. Simulate it
with the initial conditions S(0) = 997, I(0) = 3, R(0) = 0 for the parameter
values ß = 0.0004 and y = 0.04.
Systematically vary the infection rate ß and in
particular observe how the progression of the epidemic changes when the infection
rate is reduced, as for example by social distancing. Discuss the term “flattening the
curve”.
In the following modified version, population turnover is included. It is assumed that
new uninfected individuals are born with a constant rate a. Moreover, a constant
relative death rate δ is assumed for all groups. To reflect that infections can cause
death, a higher death rate constant ϵ > δ for infected people is assumed. Introducing
µ = y + ϵ, the system of differential equations reads
1) d*S/d*t= a-ß*S*I-δ * S
2) d*I/d*t=ß*S*I-µ*I
3) d*R/d*t= y*I- δ*R
"""
"""
Exercise 2: Show that this system exhibits two steady states, one reflecting a
disease-free population, and one an endemic state.
For your control, the states are
(S(0), I(0), R(0))=[(a/δ), (0, 0)]
(S(1), I(1), R(1))=[(µ/ß),(a*ß-δ*µ/ß*µ),(y/δ) * (a*ß-δ*µ/ß*µ)]
"""
"""
Exercise 3: Investigate the system by performing dynamic simulations.
A good starting point are the following parameters: a = 96.5, µ = 16, δ = 0.028, y = 0.7µ, ß= 3.05 δ*µ/a
and the initial conditions (S(0), I(0), R(0)) = (1070, 15, 0). Explore the
influence of the different parameters and interpret the results.
"""
"""
Exercise 4: Be creative! The SIR model is only the simplest version of an epidemiological models. Explore other versions, think how to implement them, collect
information in the internet, or make a completely new model of your own. Possible
ideas include:
a) The SEIR model (susceptible, exposed, infected, resistant) takes into account
that individuals that have been exposed to the disease to not immediately
become infectuous.
b) The SIS model assumes that people can be infected more than once.
c) How would you model a spatial component? Think of two locations, between
which people are randomly exchanged with a small rate. Or even think of
several locations and a hub, such as a popular holiday resort.
d) How would you assess the e↵ect of a vaccination?
e) Munz et al (2009) predicts the spread of zombieism, a disease which can be
modelled much like a typical SIR model, but with the caveat that the infected,
replaced with zombies in the model and denoted with a Z, can return from the
recovered population (which here refers to ’removed’ individuals, or zombies
who have been ’removed’ from the population by being killed).
The initial equations they use to simulate this model are as follows:
1)d*S/d*t = π-ß* S* Z- δ* S
2)d*Z/d*t = ß*S*Z +y* R -a*S*Z
3)d*R/d*t = δ*S + a*S*Z - y* R
Where S represents the susceptible population, Z the zombie population, and
R the population who have been ’removed’, or have died. π represents the
birth rate, δ the non-zombie related death rate, ß the rate of zombification,
y the rate of resurrection of removed people into zombies, and a the rate at
which the susceptible population kill zombies.
How could this model be adapted to more accurately adapt the behaviour of
zombies in particular media? In 28 Days Later, the zombies do not ressurrect,
and in Shaun of the Dead, they display an incubation time before returning
from the removed population.
"""
If anyone can solve these tasks, please help me.