Machine Learning for Epidemiological Models (Enrico Meloni)

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MACHINE LEARNING FOR
EPIDEMIOLOGICAL MODELS
E. Meloni, A. Zugarini, A. Panizza, A. Betti, M. Gori
September 24, 2020
SAILa...

TABLE OF CONTENTS
1. SIR Models
2. The data
3. The learning problem
4. Regularization
5. Fitting the data
1

SIR Models

WHY LEARN EPIDEMIC MODELS?
During 2020, the spread of COVID-19 infection aﬀected the
whole world.
Some countries resorted ...

WHY LEARN EPIDEMIC MODELS?
Deterministic epidemic models, such as SIR, usually model the
evolution through some constant p...

THE MODEL
A classical deterministic approach to model an epidemic is the
susceptible, infectious, removed (SIR) model.
A p...

THE MODEL
The dynamics of the epidemics then can be described by the
following system of ODEs1:



x (t) = −βx(t)y(...

SIDARTHE
S I
D
R
H
E
T
A
α, β, γ, δ
ε
η
ξ
ζ
µ
τ
ρ
λ
κ
ν
σθ
A more sophisticated epidemiological model is SIDARTHE2,
where ...

SIDARTHE
Compartments:
• S: Susceptible
• I: Asymptomatic undetected infected individuals
• D: Aymptomatic detected infect...

DYNAMIC PARAMETERS
The epidemiological parameters are usually considered constant in
time.
This prevents the model from de...

The data

OFFICIAL DATA
We pre-processed the data3 to obtain 5 time-series which map to
5 SIDARTHE compartments:
• D: detected asymp...

TRAIN/VALIDATION/TEST SPLIT
All time-series were split into three sets:
• Train Set (size T): 0 ≤ t ≤ T
• Validation Set (...

The learning problem

GENERIC STATEMENT
We call u(t) the concatenation of all parameters at time t.
u(t) = (α(t), β(t)...)
Problem
Learning u(t)...

PRECISE FORMULATION
Given the supervisions D, R, T, Hd and E
Given ˆDu, ˆRu, ˆTu, ˆHu
d and ˆEu the solutions of SIDARTHE
...

GRADIENT DESCENT THROUGH ODES
The learning is done through Gradient Descent.
The SIDARTHE equations are computed through n...

Regularization

SMOOTHNESS
In case of overﬁtting, the learned parameters could result in a
discontinuous and wrinkled function of time.
To...

MOMENTUM
Since the solution is computed through integration, the value of
parameters at initial time steps have a bigger i...

TARGET NORMALIZATION
D, R, T, Hd and E have very diﬀerent scales of values.
E.g. max(Hd ) ≈ 200 000 and max(T) ≈ 4 000
Thi...

Fitting the data

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