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A SIDARTHE Model of COVID-19 Epidemic in Italy
Giulia Giordano, Franco Blanchini, Raffaele Bruno, Patrizio Colaneri, Alessandro Di Filippo, Angela Di Matteo, Marta Colaneri, the COVID19 IRCCS San Matteo Pavia Task Force
TL;DR
Inapparent transmission complicates epidemic prediction and control planning. The paper develops the SIDARTHE model and finds that very strong social distancing markedly reduces infections and ICU admissions, making it preferable to partial measures.
Problem
The paper asks how to predict epidemic trends and assess control strategies when transmission may occur through people with no or mild symptoms.
Method
The paper formulates an eight-equation SIDARTHE model distinguishing detected from undetected infections and mild or moderate from major or extreme symptoms.
Results
Very strong social distancing anticipates infection and ICU-admission peaks while markedly decreasing total infections and ICU admissions, whereas partial measures produce smaller decreases.
Takeaways & Limitations
The simulations suggest an urgent need for, and effectiveness of, strong social-distancing measures during Italy’s COVID-19 epidemic.
Takeaways & Limitations
Long-term predictions can be misleading when model coefficients change, requiring updates after imposed restrictions or other countermeasures.
Abstract
from arXiv · showhide
In late December 2019, a novel strand of Coronavirus (SARS-CoV-2) causing a severe, potentially fatal respiratory syndrome (COVID-19) was identified in Wuhan, Hubei Province, China and is causing outbreaks in multiple world countries, soon becoming a pandemic. Italy has now become the most hit country outside of Asia: on March 16, 2020, the Italian Civil Protection documented a total of 27980 confirmed cases and 2158 deaths of people tested positive for SARS-CoV-2. In the context of an emerging infectious disease outbreak, it is of paramount importance to predict the trend of the epidemic in order to plan an effective control strategy and to determine its impact. This paper proposes a new epidemic model that discriminates between infected individuals depending on whether they have been diagnosed and on the severity of their symptoms. The distinction between diagnosed and non-diagnosed is important because non-diagnosed individuals are more likely to spread the infection than diagnosed ones, since the latter are typically isolated, and can explain misperceptions of the case fatality rate and of the seriousness of the epidemic phenomenon. Being able to predict the amount of patients that will develop life-threatening symptoms is important since the disease frequently requires hospitalisation (and even Intensive Care Unit admission) and challenges the healthcare system capacity. We show how the basic reproduction number can be redefined in the new framework, thus capturing the potential for epidemic containment. Simulation results are compared with real data on the COVID-19 epidemic in Italy, to show the validity of the model and compare different possible predicted scenarios depending on the adopted countermeasures.
1 Introduction
The paper introduces the COVID-19 epidemic in Italy as a rapidly spreading outbreak whose transmissions often involve individuals with no or mild symptoms. It proposes the SIDARTHE model to distinguish infection detection, symptoms, and illness severity, then calibrates it with Italian data to examine countermeasure scenarios.
- Background: SARS-CoV-2 was identified in Wuhan in late December 2019 and subsequently spread across multiple world countries.WHO documented 93090 confirmed cases across 77 countries on March 4, 2020, while Italy became the most hit country outside Asia.
- Motivation: Human-to-human transmission frequently occurred through individuals showing no or mild symptoms, while SARS-CoV-2’s estimated basic reproduction number ranged from 2.0 to 3.5.The paper identifies these epidemiological traits as distinctive compared with SARS-CoV and MERS-CoV.
- Motivation: Predicting epidemic trends is necessary for planning effective control strategies and determining how those strategies affect the epidemic’s course.The introduction situates predictive mathematical models, including the widely used SIR model, within this need.
- Contribution: SIDARTHE discriminates detected and undetected infections, asymptomatic and symptomatic cases, and non-life-threatening versus potentially life-threatening illness.Potentially life-threatening cases are defined as major and extreme severity and require Intensive Care Unit admission.
- Experiments: The model parameters were estimated from Italian national epidemic data covering February 20, 2020 to March 12, 2020, and used to discuss longer-term countermeasure scenarios.The scenarios showcase the impact of different measures intended to contain contagion.
2 Results
The Results section develops and analyses the eight-stage SIDARTHE model, including its equilibria, stability, reproduction number, and case-fatality measures. Italian data are used to estimate parameters and compare epidemic forecasts under differing social-distancing measures.
- Model formulation: The model partitions the population into eight stages, distinguishing undetected and detected infections, symptom severity, recovery, and death.The stages are S, I, D, A, R, T, H, and E, governed by eight ordinary differential equations.
- Model formulation: Transmission, symptom-progression, life-threatening-symptom, and recovery rates encode diagnosis, disease severity, treatment, and immunity effects.Undetected infected individuals generally transmit more than diagnosed or symptomatic individuals, while therapies and immunity can reduce severe progression and increase recovery rates.
- Equilibria and stability: The positive bilinear system has disease-free long-term states containing only susceptible, healed, and deceased populations, so the epidemic eventually ends.H(t) and E(t) are cumulative variables depending on the other state variables and their initial conditions.
- Equilibria and stability: Stability is determined by the susceptible threshold ¯S∗, while the limiting susceptible population cannot exceed this threshold and satisfies ¯S ¯R0 < 1.Larger R0 values imply stronger population involvement according to the model’s epidemic criterion.
- Italian epidemic scenarios: R0 = 2.38 under the fitted baseline parameters, indicating significant growth of infected individuals.The parameter setting includes κ = ξ = σ = 0.017.
- Italian epidemic scenarios: Without further countermeasures, 73% of the population contracts the virus and about 5.2% dies over 300 days; stronger distancing reduces R0 below 1 and reverses growth.Milder measures yield R0 = 1.13 and delay and reduce the peak, whereas stronger measures yield R0 = 0.787 and anticipate the peak.
3 Discussion
The discussion emphasizes that distinguishing diagnosed from undiagnosed infections and symptom severity reveals distortions in epidemic statistics and supports scenario-based evaluation of control measures. Simulations indicate that strong social distancing is urgently needed, while predictions remain highly sensitive to uncertain parameters and healthcare capacity.
- Model contribution: The model distinguishes detected from undetected infections and separates mild or moderate cases from major or extreme symptom classes.This structure captures differences in diagnosis status and symptom severity within the epidemic.
- Model contribution: Diagnosed-versus-undiagnosed distinctions expose bias in estimated infection counts, transmission rates, and case fatality rates.The discussion contrasts the diagnosed-case CFR with the actual CFR among all infected individuals.
- Limitations: Long-run predictions are extremely sensitive to uncertain parameter values and must account for changes caused by government measures.The model is less sensitive to initial conditions, while parameters can vary with population density, habits, environment, and age distribution.
- Control measures: Social distancing reduces infection coefficients, but peak timing is nonmonotonic, and mild containment may increase the fraction with life-threatening symptoms.Partial restrictions initially postpone the peak, whereas strong restrictions anticipate it.
- Control measures: Strong social-distancing measures are urgently needed and are predicted to be effective in controlling the Italian epidemic.The model evaluates control scenarios by predicting epidemic and ICU-admission peak timing and magnitude.
- Control measures: Partial restrictions delay epidemic and ICU peaks but produce only moderate reductions in total infections and ICU admissions.Stronger restrictions are contrasted with partial implementation in the scenario analysis.
4 Methods
The methods characterize disease-free equilibria and derive a reproduction parameter governing their stability, then estimate model parameters from Italian epidemic data while excluding distorted death data from fitting.
- Equilibria: Equilibria have the form (S̄, 0, 0, 0, 0, T̄, H̄), with S̄ + T̄ + H̄ = 1.They arise when either S = 0 or all infected compartments I, D, A, and R equal zero.
- Stability analysis: The linearised system has three null eigenvalues and four eigenvalues determined by the polynomial D(s) = (s + r1)(s + r2)(s + r3)(s + r4)(s + r5).The rate combinations are r1 = ϵ + ζ + λ, r2 = η + ρ, r3 = θ + µ + κ, r4 = ν + ξ, and r5 = σ + τ.
- Stability analysis: R0 = G(0) is the H∞ norm of the transfer function, and equilibrium stability occurs when S̄ R0 < 1.For the positive system, the H∞ norm equals the static gain G(0) = N(0)/D(0).
- Parameter estimation: Parameters were inferred from Italian official data spanning February 20 through March 12, 2020, with observations converted into fractions of the approximately 60-million population.The fitted data included infected individuals by symptom category and diagnosed individuals who recovered.
- Parameter estimation: The model fit excluded death data because early death/infected ratios were highly overestimated by statistical distortion.The distortion was attributed largely to Italy’s older population and extensive intergenerational contacts.