1️⃣
Simulations & Modeling - Part 1
Euler and the geometric growth of populations
The model
Analytical solution
Introducing births and deaths
Life expectancy
Carrying capacity models
Lotka & Volterra
Interactions between agents
Model with three species
Combine Lotka-Volterra model with carrying capacity and competition
Scenario 1
Scenario 2
Scenario 3
Scenario 4
Bass model
LVBC diachronic model
SIR compartmental model
Why using epidemiological models?
Introduce demography in the SIR model
Find the equilibium points
SEIR model
Heterogeneities
Pros
Cons
Modelling Risk Structure
Super-Shedders
Super-Spreaders
Modelling Age Structure
Model Calibration
Parametrization vs Calibration
RMSE
Pros and Cons
Maximum Likelihood Estimator
Markov Chain Monte Carlo method
Stochastic Process
Simulations & Modeling - Part 1 1
, Epidemiological Parameters
Method to estimate R0
Epidemics growth rate λ- homogeneous mixing
Bayesian Inference
Next Generation Matrix approach
Euler and the geometric growth of
populations
The model
Population dynamics model. Science that tries to explain how different type of
population varies over time.
A population Pn grows at a constant rate r over time. At each perio, the
population is 1 + r times greater than the previous one
⁍
This is called the geometric growth of population where basically you are
saying that the population time n + 1is equal to 1 + r , where r is the growth
rate and it's a positive number, times the population P0 in n.
If we repeat the steps, starting from P0 , we get the equation of the geometrical
or exponential growth of the population.
Pn = (1 + r)n ⋅ P0
Estimating population in a certain time point as a function of the growth rate and
the population in the previous time. This is the differential equation:
P ′ (t) = r ⋅ P (t)
Assumption: Offsprings of fish are proportional to fish already present in the
lake.
Fish generated between
t and t + hare proportional to fish at time tand to the interval h.
Simulations & Modeling - Part 1 2
, P (t + h) = P (t) + P (t) ⋅ r ⋅ h
P (t + h) + P (t)
= r ⋅ P (t)
h
P′ (t) = r ⋅ P(t)
Malthusian population growth or Malthusian model
Population increases at each period are called generations.
Analytical solution
P ′ (t) = r ⋅ P (t)
P ′ (t)
=r
P (t)
P ’(t)/P (t)can be seen as the derivativeof the logfunction, such that:
∂ log P (t)
= log ′ P (t) = r
∂t
By integrating both sides:
log P (t) = r ⋅ t + c
P (t) = ert ⋅ ec
Where ec is the initial population, in fact when t = 0, P (t) = ec
Introducing births and deaths
Birth → Alive individuals → Death:
P (t + 1) = P (t) + B − D
In the previous model r = B − D.
Difference equation
P (t + h) = P (t) + b ∗ P (t) ∗ h − m ∗ P (t) ∗ h
where:
bis the birth rate
mis the mortality rate
Simulations & Modeling - Part 1 3
Simulations & Modeling - Part 1
Euler and the geometric growth of populations
The model
Analytical solution
Introducing births and deaths
Life expectancy
Carrying capacity models
Lotka & Volterra
Interactions between agents
Model with three species
Combine Lotka-Volterra model with carrying capacity and competition
Scenario 1
Scenario 2
Scenario 3
Scenario 4
Bass model
LVBC diachronic model
SIR compartmental model
Why using epidemiological models?
Introduce demography in the SIR model
Find the equilibium points
SEIR model
Heterogeneities
Pros
Cons
Modelling Risk Structure
Super-Shedders
Super-Spreaders
Modelling Age Structure
Model Calibration
Parametrization vs Calibration
RMSE
Pros and Cons
Maximum Likelihood Estimator
Markov Chain Monte Carlo method
Stochastic Process
Simulations & Modeling - Part 1 1
, Epidemiological Parameters
Method to estimate R0
Epidemics growth rate λ- homogeneous mixing
Bayesian Inference
Next Generation Matrix approach
Euler and the geometric growth of
populations
The model
Population dynamics model. Science that tries to explain how different type of
population varies over time.
A population Pn grows at a constant rate r over time. At each perio, the
population is 1 + r times greater than the previous one
⁍
This is called the geometric growth of population where basically you are
saying that the population time n + 1is equal to 1 + r , where r is the growth
rate and it's a positive number, times the population P0 in n.
If we repeat the steps, starting from P0 , we get the equation of the geometrical
or exponential growth of the population.
Pn = (1 + r)n ⋅ P0
Estimating population in a certain time point as a function of the growth rate and
the population in the previous time. This is the differential equation:
P ′ (t) = r ⋅ P (t)
Assumption: Offsprings of fish are proportional to fish already present in the
lake.
Fish generated between
t and t + hare proportional to fish at time tand to the interval h.
Simulations & Modeling - Part 1 2
, P (t + h) = P (t) + P (t) ⋅ r ⋅ h
P (t + h) + P (t)
= r ⋅ P (t)
h
P′ (t) = r ⋅ P(t)
Malthusian population growth or Malthusian model
Population increases at each period are called generations.
Analytical solution
P ′ (t) = r ⋅ P (t)
P ′ (t)
=r
P (t)
P ’(t)/P (t)can be seen as the derivativeof the logfunction, such that:
∂ log P (t)
= log ′ P (t) = r
∂t
By integrating both sides:
log P (t) = r ⋅ t + c
P (t) = ert ⋅ ec
Where ec is the initial population, in fact when t = 0, P (t) = ec
Introducing births and deaths
Birth → Alive individuals → Death:
P (t + 1) = P (t) + B − D
In the previous model r = B − D.
Difference equation
P (t + h) = P (t) + b ∗ P (t) ∗ h − m ∗ P (t) ∗ h
where:
bis the birth rate
mis the mortality rate
Simulations & Modeling - Part 1 3
