Modeling the SARS epidemic in Hong Kong Dr. Liu Hongjie, Prof. Wong Tze Wai Department of Community & Family Medicine The Chinese University of Hong Kong Dr. James Derrick Department of Anaesthesia & Intensive Care Prince of Wales Hospital May 13, 2003

Modeling the SARS epidemic in Hong Kong We aim to construct a model of the SARS epidemic in community (i.e., we have excluded the outbreaks among health care workers or the “common source” outbreak in Amoy Garden. The model only applies to a person-to- person mode of transmission of SARS.

Objectives of the Media Release: 1.Explain the natural course of an epidemic – the relationship between the population that is infectious (including patients and infected individuals who will become patients), susceptible population and immune (recovered) population; 2.Show how the natural progression of an epidemic is affected by the effectiveness and timeliness of public health measures, by introducing our mathematical model; 3.Using the assumptions and limitations of our model, discuss the current situation in terms of the trend of the epidemic and the likelihood of its resurgence;

Dynamics of disease and of infectiousness at the individual level Times (days) Clinical onset Incubation period Time of infection Resolution Relapse Symptomatic period Susceptible immune carrier dead recovered Dynamics of disease Onset of infectiousness Latent period End of infectiousness infectious period Susceptible Dynamics of infectiousness

Dynamics of infectiousness at the population level Susceptible S t Infectious I t Recovered / immune R t SIR

Basic Reproductive Number (R 0 ) The average number of individuals directly infected by an infectious case during his/her entire infectious period In a population if R 0 > 1 : epidemic if R 0 = 1 : endemic stage if R 0 < 1 : sucessful control of infection If population is completely susceptible measles : R 0 = smallpox : R 0 = 3 – 5 SARS: ???

Basic (R 0 ) reproductive number R 0 = Number of contact per day Transmission probability per contact Duration of infectiousness xx     D    D = Average number of contacts made by an infective individual during the infectious period:   D e.g. 2 persons per day X 5 days = 10 persons Number of new infections produced by one infective during his infectious period: No. of contacts during D (   D ) X transmission probability per contact (  ) e.g. 10 persons X 0.2 = 2 infected cases

Basic (R 0 ) reproductive number R 0 = Number of contact per day Transmission probability per contact Duration of infectiousness xx     D    D = Preventive measures targeting reducing any parts of the components will halt SARS epidemic

SIR Model Susceptible S t Infectious I t Recovered / immune R t SIR S t : Proportion of population (n) that is susceptible at time t I t : Proportion of n that is currently infected and infectious at time t R t : Proportion of n that is recovered / immune SIR mode is used to predict the three proportions at different scenarios.

Estimate of the 3 proportions changing over time t Susceptible S t Infectious I t Recovered / immune R t SIR Time derivatives of 3 proportion dX/dt, where X could be S, I or R At any time t during the epidemic, the 3 equations will be: dS/dt = -     S  I dI/dt =  S  I – I/D dR/dt = I/D

A close look at dS/dt = -     S  I Susceptible S t Infectious I t Recovered / immune R t SIR In population, there are 6 different types of possible contacts Susceptible meets susceptible (S - S, no transmission) Susceptible meets infectious (S - I, transmission) Susceptible meets resistant (immune) (S - R, no transmission) Infectious meets infectious (I - I, no transmission) Infectious meets resistant (I - R, no transmission) Resistant meets resistant (R - R, no transmission)

Assumptions of this m odel 1. the average household size is 3 (census data in 2001); 2. the interval between onset of disease and admission to hospitals is 5 days (based on the paper by Peiris et al. Coronavirus as a possible cause of SARS. Lancet online, 8 April, 2003); 3. Once SARS patients are hospitalized, they are not able to disseminate the infection back to the community; 4. Patients are infectious one day before the onset of their illness till hospitalized.

Guideline for estimate R 0 In Hong Kong, the mean life expectancy is about 80 years. The average age at the SARS infection is about 40, thus R 0  80 / 40  2 The value of R 0 is used to estimate the parameters in modeling the natural history of the SARS epidemic in Hong Kong. (Ref. Anderson and May. Infectious Diseases of Humans: Dynamics and Control, 1991) R 0  Mean life expectancy / Average age at infection

Estimate of parameters----- Natural history Duration of infectivity (day): 6 days 1 day before onset of symptoms 5 day-delay in seeking treatment (Peiris’s paper)  : No. of contacted person: 14 Household (HH): 2 (Average household size is 3 according HK censes in 2001) Social contacts (SC): 12 No. of contacted persons / day: 14/6 = 2.33  : Risk of transmission per contact HH: 0.25 SC: 0.1 Weighted average  : Two infectious cases enter the susceptible population

Proportions of S, I and R Natural history of SARS epidemic Susceptible Recovered/Immune Infectious

Proportions of infectious population (Control started on different dates) R o = 1.44 R o = 2.01 Natural Day 10 Day 40 Day 30 Day 20

Proportions of infectious population at different R o R o = 2.01 R o = 1.44 R o = 1.39 R o = 1.3 R o = 1.17 Control started on day 10

Predicted number of SARS infectious cases Control Stage 1 starting from on day 20:   D = 5, R o = 1.44 Control Stage 2 starting from on day 30:   D = 3, R o = 0.84

Proportions of S, I and R on log scale (Control at two stages) Susceptible Recovered/Immune Infectious

Computer Assisted SARS Modeling

Main Messages to bring across 1.If the epidemic is allowed “to run its natural course”, in other words, to die down by itself, up to several million people will fall victim to SARS. Sufficient herd immunity that will protect the community from further epidemics will only be achieved at the expense of this magnitude of community infection; 2.An epidemic will die down only when the basic reproductive number, R o (number of people infected by a patient) is less than one. This can be achieved only in two ways:- when herd immunity is high enough (natural course of events), or when effective public health measures limit the spread of the epidemic;

Main Messages to bring across: 3. At present, with all the public health measures in place, it appears our public health measures are capable to reduce the number (Ro) to <1; 4. To effectively control the epidemic, efforts must be sustained keep Ro to <1. Otherwise, the epidemic can start again at any time, because the proportion of immune individuals in our population (herd immunity) is far too low to offer any “natural protection”.