Come calcolare il modello SIR di Kermack-McKendrick: equazioni differenziali ordinarie compartmentale per modellare diffusione epidemia in popolazione, formulato da William Ogilvy Kermack (UK 1898-1970) Royal Edinburgh Hospital + Anderson Gray McKendrick (UK 1876-1943) Royal Edinburgh Hospital 1927 paper "A Contribution to the Mathematical Theory of Epidemics" Royal Society A 115:700-721. Formule: dS/dt = -βSI/N (suscettibili), dI/dt = βSI/N - γI (infetti), dR/dt = γI (recuperati), N = S + I + R popolazione totale costante. β tasso trasmissione (contatti effettivi × probabilità infezione), γ tasso recovery (1/durata contagiosità). R₀ = β/γ basic reproduction number numero medio di nuove infezioni causate da un infetto in popolazione completamente suscettibile. Soglia: epidemia se R₀ > 1, die-out se R₀ < 1. Herd immunity threshold = 1 - 1/R₀: per R₀=3 immunity gregge 67%; R₀=18 measles 94%; R₀=1.3 flu seasonal 23%. Periodo contagiosità infectious_period = 1/γ giorni. Esempi: COVID-19 R₀≈3 67% herd immunity, measles R₀=12-18 92-95%, flu seasonal R₀≈1.3 23%, smallpox R₀=5-7 80-86%, polio R₀=5-7 80-86%, pertussis R₀=12-17 92-94%, mumps R₀=4-7 75-86%.

Strumento essenziale per: sanità pubblica Italia ISS Istituto Superiore Sanità Roma fondato 1934 Roma viale Regina Elena, Ministero Salute Roma viale Lungotevere Ripa, AIFA Agenzia Italiana del Farmaco Roma EMA pharma; AIFA contratti vaccino COVID-19 Pfizer-BioNTech Moderna AstraZeneca Janssen Johnson; ISTAT statistiche sanità Roma 1926 fondato; vaccination policy Italia Calendario Nazionale Vaccinazioni MIUR Ministero Istruzione coperture obbligatorie 12 vaccini bambini Decreto Lorenzin 119/2017; CDC Centers Disease Control Atlanta GA USA fondato 1946 100000 employees; ECDC European Centre for Disease Prevention Control Stockholm; WHO World Health Organization Geneva fondato 1948; contact tracing Immuni Italian app COVID-19 maggio 2020 lancio; sviluppo modelli epidemic forecasting Università Italia: Sapienza Roma DIET, Bologna alma mater 1088 prima Europa Statistica DEMS, Pisa SNS Scuola Normale Superiore Mathematics, Politecnico Milano DEIB DAER, Federico II Napoli, Padova DEI Statistica.

Formula modello SIR

Sistema equazioni differenziali: dS/dt = -βSI/N (suscettibili), dI/dt = βSI/N - γI (infetti), dR/dt = γI (recuperati), N = S + I + R popolazione costante (no nascita-morte demografica). β tasso trasmissione effettivo, γ tasso recovery. Sistema 3D non-lineare ODE prima ordine, conservazione N totale.

Basic reproduction number R₀: R₀ = β/γ numero medio infezioni secondarie causate da un caso index in popolazione fully susceptible. Soglia epidemica: R₀ > 1 epidemia outbreak, R₀ < 1 epidemia die-out. R₀ depends biological transmissibility + behavioral contacts + environmental conditions. R₀ effective Reff = R₀ × S/N decreases population becomes immune.

Herd immunity threshold: H = 1 - 1/R₀ percentuale popolazione necessaria immune per stop epidemia. Per R₀=2 H=50%, R₀=3 H=67%, R₀=5 H=80%, R₀=10 H=90%, R₀=18 measles H=94%. Achieved by natural infection OR vaccination, no distinction model. Real-world considerations: vaccine efficacy <100%, waning immunity, age-structured contacts.

Final size epidemic: in classical SIR after epidemic complete (I → 0), fraction never infected ε = S(∞)/N satisfies transcendental equation: ε = exp(-R₀(1 - ε)). Solutions numerical. For R₀=3: ε ≈ 0.06 (94% infected). For R₀=2: ε ≈ 0.20 (80% infected). For R₀=1.5: ε ≈ 0.42 (58% infected). Attack rate complement = 1 - ε.

Peak infected timing: dI/dt = 0 when βSI/N = γI, equivalently S/N = γ/β = 1/R₀. Peak occurs when susceptible fraction = 1/R₀, i.e., when herd immunity reached during natural epidemic. Time to peak depends initial conditions and parameters, typically 30-90 days for emerging pandemic.

Storia e contesto

William Ogilvy Kermack (UK 1898-1970) Royal Edinburgh Hospital chemist mathematician 1924 lab explosion blinded both eyes age 26, continued research with assistants reading aloud. Anderson Gray McKendrick (UK 1876-1943) Royal Army Medical Corps physician served India 1900-1922 medical research statistical methods, retired Edinburgh Royal Edinburgh Hospital 1922-1933 Director Statistical Research. Collaboration 1927 paper "A Contribution to the Mathematical Theory of Epidemics" Proceedings Royal Society A 115:700-721 introduced compartmental SIR model deterministic ODE foundational mathematical epidemiology. Sequel papers 1932 "Contributions to the Mathematical Theory of Epidemics II" Society A 138:55, 1933 III Society A 141:94 extended model age-structure waning immunity. Daniel Bernoulli (Switzerland 1700-1782) Basel 1760 prima preventive analysis smallpox vaccination Memoires Mathematiques Physiques Académie Royale Sciences Paris pre-cinetic model. William Hamer (UK 1862-1936) Royal Society 1906 mass action transmission concept. Sir Ronald Ross (UK 1857-1932) Liverpool School Tropical Medicine Nobel Prize Medicine 1902 mosquito-malaria transmission Indian medical service, 1911 mass action mathematical theory Theory Happenings. R0 concept George MacDonald (UK 1903-1967) Liverpool 1957 Epidemic Equilibrium Static populations dengue malaria.

Roy Anderson Robert May (UK 1942-2020-AustraliaUK Lord May Oxford-Princeton Imperial College) 1991 'Infectious Diseases of Humans Dynamics Control' Oxford University Press standard textbook. COVID-19 pandemic 2020-2024: SARS-CoV-2 R₀ original Wuhan 2.5-3.5, Alpha variant 4-5, Delta 5-8, Omicron 8-12 reinfection. Vaccine development Pfizer-BioNTech mRNA Mainz Germany Moderna Cambridge MA AstraZeneca Oxford-Cambridge UK Sweden Janssen Johnson Belgium 2020-2021 emergency authorization FDA EMA. Italia ricerca epidemiology: ISS Istituto Superiore Sanità Roma fondato 1934 Roma viale Regina Elena 8000 employees national health institute; Ministero Salute Roma viale Lungotevere Ripa Ministro Schillaci 2022+; AIFA Agenzia Italiana del Farmaco Roma 2003 fondata 1500 employees vaccine pharmaceutical regulator; CTS Comitato Tecnico Scientifico COVID-19 Roma Palazzo Chigi advisory body Pier Cassese Walter Ricciardi Franco Locatelli; INMI Istituto Nazionale Malattie Infettive Lazzaro Spallanzani Roma; INMI Istituto Nazionale Malattie Infettive Lazzaro Spallanzani Roma 1936 fondato Hospital INFN; ISTAT Istituto Nazionale Statistica Roma 1926 fondato statistical office Italia.

Italia COVID-19 storia: paziente 1 Codogno Lombardia 21 febbraio 2020 deceased Brescia 23 febbraio Mattia 38enne sportivo 38 anni; lockdown nazionale 9 marzo 2020 Decreto Conte; vaccino COVID-19 V-Day 27 dicembre 2020 INMI Spallanzani Roma prima Italia Claudia Alivernini infermiera; immunity 80%+ raggiunta gennaio 2022 con Omicron variant peak. Italia Decreto Lorenzin 119/2017 obbligatorio 12 vaccini bambini: poliomielite, difterite, tetano, epatite B, pertosse, Hib, morbillo, parotite, rosolia, varicella, meningococcici, pneumococcici. Coperture vaccinali Italia 95%+ obiettivo OMS herd immunity. USA epidemiology: CDC Centers Disease Control Atlanta GA fondato 1946 100000 employees federale; NIH National Institutes Health Bethesda MD 27 institutes Anthony Fauci NIAID 1984-2022 director COVID-19 advisor White House; FDA Food Drug Administration Silver Spring MD 1906 fondata 18000 employees. ECDC European Centre for Disease Prevention Control Stockholm 2005 fondata 300 employees European Union epidemiologic surveillance. WHO World Health Organization Geneva 1948 fondato 8000 employees. Software simulazione: SIR Python scipy.integrate.odeint, R EpiModel package Centers, NetLogo agent-based simulation, Berkeley Madonna ODE.

Tabella malattie R₀ herd immunity

• COVID-19 SARS-CoV-2 original Wuhan: R₀=2.5-3.5, herd 60-71%
• COVID-19 Alpha variant 2020: R₀=4-5, herd 75-80%
• COVID-19 Delta variant 2021: R₀=5-8, herd 80-87%
• COVID-19 Omicron variant 2022: R₀=8-12, herd 87-92%
• Measles morbillo: R₀=12-18, herd 92-94%
• Pertussis whooping cough: R₀=12-17, herd 92-94%
• Mumps parotite: R₀=4-7, herd 75-86%
• Rubella rosolia: R₀=6-7, herd 83-86%
• Smallpox vaiolo: R₀=5-7, herd 80-86%
• Polio poliomielite: R₀=5-7, herd 80-86%
• Diphtheria difterite: R₀=6-7, herd 83-86%
• SARS 2003: R₀=2-5, herd 50-80%
• MERS-CoV 2012: R₀=0.4-0.9, no herd needed (sub-threshold)
• Influenza pandemic 1918 Spanish: R₀=2-3, herd 50-67%
• Influenza seasonal: R₀=1.3, herd 23%
• HIV/AIDS: R₀=2-5 untreated, herd 50-80% (chronic)
• Ebola 2014-2016 outbreak: R₀=1.5-2.5, herd 33-60%
• Marburg virus: R₀=1.6, herd 38%
• Italia: 60 mln popolazione
• Italia COVID-19 V-Day: 27 dicembre 2020 INMI Spallanzani Roma
• Italia paziente 1: 21 febbraio 2020 Codogno Lombardia Mattia 38enne
• Italia lockdown nazionale: 9 marzo 2020 Decreto Conte
• Italia decreto Lorenzin obbligo: 119/2017 12 vaccini
• USA: 333 mln popolazione
• USA COVID-19 cases: 100+ mln cases 2020-2024
• USA COVID-19 deaths: 1.2+ mln
• WHO: 1948 Geneva 194 stati membri
• CDC: 1946 Atlanta GA 100000 employees
• ECDC: 2005 Stockholm 300 employees EU
• ISS Roma: 1934 fondato 8000 employees

Esempi numerici dettagliati

Esempio 1 - COVID-19 Italia R₀=3. Italia popolazione N=60 mln, β=0.3 transmission rate giornaliero, γ=0.1 recovery rate (10 giorni infectious period). R₀ = 0.3/0.1 = 3. Herd immunity 1 - 1/3 = 67%. Initial I0=1 paziente Codogno Lombardia 21 febbraio 2020. Senza interventi peak ~30 giorni 5-10 mln infetti contemporanei, attack rate finale ~94% (ε=0.06). Italia lockdown nazionale 9 marzo 2020 Decreto Conte ridusse R₀ a 0.6-0.8 die-out. V-Day 27 dicembre 2020 vaccino Pfizer INMI Spallanzani Roma prima dose Claudia Alivernini infermiera.

Esempio 2 - Measles morbillo R₀=15. Popolazione N=60 mln, β=1.5 (alta transmissibility), γ=0.1 (10 giorni infectious). R₀ = 15. Herd immunity 1 - 1/15 = 93.3%. Senza vaccinazione attack rate near 100%, peak infetti contemporanei 30%+ population. Italia obbligo vaccinazione MPR morbillo-parotite-rosolia Decreto Lorenzin 119/2017 obbligatorio. Italia outbreak measles 2017 4885 casi 4 morti 87.7% non-vaccinated. Coperture 95%+ obiettivo OMS WHO.

Esempio 3 - Influenza seasonal R₀=1.3. Popolazione N=60 mln Italia, β=0.13 transmission, γ=0.1 recovery. R₀ = 1.3. Herd immunity 1 - 1/1.3 = 23%. Piccolo R₀ epidemic limited size, attack rate 35-40%. Italia influenza seasonal cases 5-8 mln annual ISS surveillance system InfluNet, vaccino quadrivalente raccomandato over 65 anni Italia 60% coverage. AIFA Agenzia Italiana Farmaco Roma authorize annual flu vaccines.

Esempio 4 - R₀ < 1 sub-threshold die-out. Patogeno limitato β=0.05 (low transmission, e.g., MERS-CoV), γ=0.1. R₀ = 0.5 < 1. Epidemia muore subito, no outbreak. Reff = 0.5 × 1 = 0.5 each infected causes 0.5 secondary cases on average geometric decline. Initial I0=100 quickly drops to 0. MERS-CoV Saudi Arabia 2012-2024 R₀=0.4-0.9 sub-threshold sustained transmission impossible without superspreaders.

Esempio 5 - Smallpox vaiolo R₀=5 historical. Popolazione N=60 mln, β=0.5, γ=0.1. R₀ = 5. Herd immunity 1 - 1/5 = 80%. WHO global smallpox eradication campaign 1959-1980 vaccination 80%+ coverage worldwide. Last case Somalia 1977 Ali Maow Maalin. Smallpox declared eradicated WHO 1980 8 May World Health Assembly. Italia campagna vaccinazione obbligatoria vaiolo 1962-1981 OMS-Italia. Edward Jenner 1796 first vaccine cowpox-smallpox, vaccinazione term origin Vacca cow Latin.

Limiti e considerazioni

1. Modello deterministico ignora stocasticità: real epidemics demographic stochasticity important small numbers, extinction probability, superspreader events.
2. Mass action assumption: assume homogeneous mixing population, real contacts heterogeneous network structure age-structured spatial.
3. Population costante N: ignora birth-death demographics, valid short-term outbreaks, longer term need SIR with vital dynamics.
4. Permanent immunity: SIR assume R category permanent, real waning immunity SIRS model needed (susceptible-infected-recovered-susceptible).
5. R₀ context-dependent: not intrinsic property pathogen, depends behaviour environment, varies time location.
6. Single pathogen single strain: real diseases multiple strains, cross-immunity, mutation evolution dynamics SIR insufficient.
7. Spatial homogeneity: assume mixing perfect, real geographic spatial heterogeneity diffusion equations needed.
8. Age-structure: contacts age-dependent (children school adults workplace), age-structured SEIR models PR Hethcote 2000 standard.
9. Asymptomatic infections: SIR assume infected detected, asymptomatic transmission COVID-19 35-40% required modification.
10. Numerical integration: Euler unstable for stiff ODE, use Runge-Kutta RK4 or implicit methods.

Errori comuni

1. Confondere R₀ con Reff effective: R₀ initial fully susceptible, Reff = R₀ × S/N decreases over time.
2. Trascurare unit time consistency β γ giorni vs settimane vs mesi: same units required.
3. Pretendere herd immunity 100% needed: 1 - 1/R₀ sufficient (67% per R₀=3).
4. Confondere R₀ R0 (recovered) notation: R₀ basic reproduction number vs R(t) recovered category.
5. Ignorare vaccine efficacy <100%: real H = (1 - 1/R₀) / V_efficacy, correzione necessaria.
6. Pretendere SIR predittivo precisione: tool didattico, real public health complex behavioral non-linear.
7. Trascurare incubation period: SIR ignora periodo incubation, SEIR exposed category needed accuracy.
8. JavaScript Math.exp Math.log naturale: controllare base function, R₀ logistic transcendental equation numerical.
9. Pretendere SIR deterministic exact: intrinsic stochasticity small populations, demographic noise extinction.
10. Confondere attack rate cumulative cases vs current infected: attack final size != peak.

Applicazioni numerate

1. Italia ISS sanità: Istituto Superiore Sanità Roma fondato 1934 Roma viale Regina Elena 8000 employees national health institute; Ministero Salute Roma viale Lungotevere Ripa.
2. Italia AIFA vaccini: Agenzia Italiana del Farmaco Roma 2003 fondata 1500 employees vaccine pharmaceutical regulator; AIFA contratti vaccino COVID-19 Pfizer-BioNTech Moderna AstraZeneca Janssen.
3. Italia COVID-19 storia: paziente 1 Codogno Lombardia 21 febbraio 2020 Mattia 38enne; lockdown nazionale 9 marzo 2020 Decreto Conte; V-Day 27 dicembre 2020 INMI Spallanzani Roma.
4. Italia obbligo vaccinazione: Decreto Lorenzin 119/2017 obbligatorio 12 vaccini bambini polio difterite tetano epatite B pertosse Hib MPR varicella meningococcici pneumococcici.
5. USA CDC NIH FDA: CDC Centers Disease Control Atlanta GA fondato 1946; NIH Bethesda MD 27 institutes; FDA Silver Spring MD 1906 fondata 18000 employees.
6. Europe ECDC: European Centre for Disease Prevention Control Stockholm 2005 fondata 300 employees European Union epidemiologic surveillance.
7. WHO: World Health Organization Geneva 1948 fondato 194 stati membri 8000 employees global epidemic surveillance smallpox eradication 1980.
8. Italia università epidemiologia: Sapienza Roma DIET Probabilità applicata; Bologna alma mater 1088 prima Europa Statistica DEMS; Pisa SNS Scuola Normale Superiore Mathematics; Padova DEI Statistica.
9. INMI Spallanzani Roma: Istituto Nazionale Malattie Infettive Lazzaro Spallanzani Roma 1936 fondato Hospital INMI infectious diseases reference.
10. ISTAT statistiche: Istituto Nazionale Statistica Roma 1926 fondato statistical office Italia mortality cases incidence rates.

Per supporto

Per domande tecniche: info@calcoi.com. Strumento didattico epidemiologia matematica. Per applicazioni Kermack WO McKendrick AG 1927 'A Contribution to the Mathematical Theory of Epidemics' Royal Society A 115:700-721, Anderson RM May RM 1991 'Infectious Diseases of Humans Dynamics Control' Oxford University Press standard textbook, Hethcote H 2000 'The Mathematics of Infectious Diseases' SIAM Review 42:599, Bernoulli D 1760 smallpox vaccination preventive analysis Académie Royale Sciences Paris.

Aggiornato a luglio 2026