SIR Model for Portugal - North Region

Last Updated: 2020-08-17 13:00:49 UTC

1 Disclaimer

This article is an independent simulation and does not reflect any portuguese governmental position about the subject.

This article is a work in progress.

2 Concepts

• Incubation Period: is the period between infection and the visible appearance of symptoms.
• Latency period: is the period between infection and the ability to infect someone.

3 Model Hypothesis

3.1 Compartments

The population is assumed to be divided in compartments, for example:

• Susceptible individuals
• Infected, symptomatic and infectious
• Recovered, immune from further infection

3.2 Discrete-time

The observations, for simplicity, will be analyzed as a discrete time-series. In this case, we will use days.

3.3 Population

The total number of individuals is modelled using currently available data for birth and death rates.

3.4 Other assumptions

• The population is considered large, so random effects can be ignored.
• The population is considered homogeneous at any point in time. That means that the individuals of any compartment are distributed randomly.
• The disease is transmitted by proximity or contact between an infected and a susceptible individual (individual contact model - ICM).
• The susceptible individual becomes infected instantaneously.
• The infected individuals may or may not be infectious until they get symptoms.
• The infected eventually get recovered and become immune or die.
• The population pyramid is taking into account for disease severity.

4 Extended Model

A more complex model is proposed by Tim Churches¹, adding several compartments and permit a better exploration of the potential effects of various combinations and timings of interventions on COVID-19 spread.

Attention to the description of each compartment:

Compartment	Functional definition
S	Susceptible individuals
E	Exposed and infected, not yet symptomatic but potentially infectious
I	Infected, symptomatic and infectious
Q	Infectious, but (self-)isolated
H	Requiring hospitalisation (would normally be hospitalised if capacity available)
R	Recovered, immune from further infection
F	Case fatality (death due to COVID-19, not other causes)

4.1 Transition diagram

4.2 Parameters

There are many parameters. We will start with the defaults and adjust accordingly to new evidence.

The full list of parameters is available in the source code.

4.3 Time-variant parameters

Some parameters also accept a vector of values which allows the model to change itself accordingly, for example, to government policies like self-isolation, or social distancing. This feature is compelling for computational models such as this, and one very hard to incorporate into purely mathematical models.

Most of the parameters for which time-variation is useful supports it. If they do not, an error message is raised.

5 Evidence

5.1 COVID-related

Last checked: 2020-04-11

• Recovery (h, d) = takes at least 15 days
• Exposition to symptoms (b) = median of 5
• Incubation period: 5.1 (2.2 - 11.5)²
• Symptoms to death: ~15 (11 - 17) (So, Infection to death ~20 days)²
• Days to double: 8-9 (Case report estimation is 31-48% at 2020-04-11, for R 1.3 and CFR 2.0³)
• Mortality rate: 0.85 - 1.31%³

6 Baseline simulation

6.1 Parameters used

• Population: 3,574,394 (estimation of 2019 from Census)⁴
• Birth rate: 7.7 ‰⁴
• Baseline death rate (any cause): 9.86 ‰⁵
• Baseline hospital death rate (any cause): 50.5 ‰⁵
• Rates of exposition (table below)
• Report rate (table below)
• Ages and severity (table below)

Interaction rates between compartments
Date	Event	E-S*	I-S*	Q-S*
08 March	Suspension of visits to health facilities and prisons	16	10	5.0
10 March	Suspension of flights from the most affected areas of Italy	15	10	5.0
11 March	WHO declares pandemic	15	10	5.0
13 March	Portugal alert status statement	10	7	3.5
15 March	Restrictions on access to commercial spaces	7	5	2.5
16 March	Suspension of classes	5	3	1.5
18 March	Declaration of State of Emergency	5	2	1.0
02 April	Extension of the State of Emergency	3	2	1.0
04 May	End of the State of Emergency	7	2	1.0
17 May	Extension of the alert status	7	2	1.0

* Empirical values.

Age pyramid and disease fatality rate
Ages	Population	Prevalence	Fatality rate*
>=0	8.03%	1.63%	0.00%
>=10	10.34%	3.00%	0.20%
>=20	11.16%	11.59%	0.20%
>=30	12.35%	14.04%	0.20%
>=40	15.51%	16.76%	0.40%
>=50	15.36%	16.95%	1.30%
>=60	12.76%	11.55%	3.60%
>=70	8.68%	8.76%	8.00%
>=80	5.79%	15.72%	14.80%

* Data from China CDC⁶.

6.2 Distribution of duration in key compartments

6.3 Baseline Prevalence

6.3.1 Overall - Interactive Plot

6.3.2 Comparison with real data - Interactive Plot

6.3.3 Basic reproduction number \(R_{0}\)

Estimated R is 1.22 (±0.01)

7 Running intervention experiments

Work in progress…

7.1 End of State of Emergency

First Scenario (pessimistic):

• E-S, I-S, Q-S are set back to March, 15

Second Scenario (optimistic):

• E-S, is set back to March, 15. I-S and Q-S still under control.

8 Acknowledgements

These simulations are heavily based on Tim Churches’ code, available here.

1. Churches T. Tim churches health data science blog: Modelling the effects of public health interventions on covid-19 transmission using r - part 2. Published online 2020. https://timchurches.github.io/blog/posts/2020-03-18-modelling-the-effects-of-public-health-interventions-on-covid-19-transmission-part-2/

2. MIDAS-Network. COVID-19 parameter estimates. https://github.com/midas-network/COVID-19/tree/master/parameter_estimates/2019_novel_coronavirus

3. CMMID. Inferring covid-19 cases from deaths of confirmed cases. https://cmmid.github.io/visualisations/inferring-covid19-cases-from-deaths

4. Instituto nacional de estatística. https://www.ine.pt

5. Transparência sns. https://transparencia.sns.gov.pt

6. Novel Coronavirus Pneumonia Emergency Response Epidemiology Team". "The epidemiological characteristics of an outbreak of 2019 novel coronavirus diseases (covid-19) — china, 2020". "China CDC Weekly". "2":"113". "http://weekly.chinacdc.cn//article/id/e53946e2-c6c4-41e9-9a9b-fea8db1a8f51"