A simple model for Toronto

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A simple model for Torontos SARS outbreak

• Carlos Castillo-Chavez
• Center for Nonlinear Studies
• Los Alamos National Laboratory
• Cornell University

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Outline

• SARS epidemiology and related issues
• SARS transmission model (SEIJR)
• Parameter estimation and data fitting
• The basic reproductive number
• Simulation Results
• Conclusions
• Extensions and related models

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SARS epidemiology and related issues

• SARS (Severe Acute Respiratory Syndrome) is a new
  respiratory disease which was first identified in
  Chinas southern province of Guangdong in
  November 2002.
• The most striking feature of SARS is its ability
  to spread on a global scale. One man with SARS
  made 7 flights from Hong Kong to Munich to
  Barcelona to Frankfurt to London, back to Munich
  and Frankfurt before finally returning to Hong
  Kong.
• An individual exposed to SARS may become
  infectious after an incubation period of 2-7 days
  with 3-5 days being most common.
• Most infected individuals either recover after
  7-10 days or suffer 7 - 10 mortality. SARS
  appears to be most serious in people over age 40
  especially those who have other medical problems.

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SARS epidemiology and related issues, Cont

• Researchers in the Erasmus medical center in
  Rotterdam recently demonstrated that a corona
  virus is the causative agent of SARS.
• The mode of transmission is not very clear. SARS
  appears to be transmitted mainly by
  person-to-person contact. However, it could also
  be transmitted by contaminated objects, air, or
  by other unknown ways. Recently, it has been
  suggested that SARS corona virus may have come
  from an animal reservoir.
• Presently there is no vaccine available. Rapid
  diagnosis and isolation of infectious individuals
  is critical for many reasons.

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SARS transmission model

• U.S. data is limited and sparsely distributed
  while the quality of Chinas data is hard to
  evaluate.
• There appears to be enough data for Toronto, Hong
  Kong and Singapore (WHO, Canada Ministry of
  Health) to build a preliminary model (single
  outbreak).
• Single outbreak model ignores demographic
  processes other than the impact of SARS on
  survival.

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Compartmental model
The model considers two distinct susceptible
classes S1, the most susceptible, and S2.
is the transmission rate to S1 from
E, I and J. E is the class composed of
asymptomatic, possibly infectious individuals.
The class I denotes infected, symptomatic,
infectious, and undiagnosed individuals. J are
diagnosed individuals.
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Initially S1 and S2
where is a fraction of the population at
major risk of SARS infection and N is the total
population. We can write the following system of
ODE-s
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Parameter estimation and data fitting

• The transmission rate was obtained from the
  initial rate of exponential growth r (computed
  from time series data).
• The values of p and q are fixed arbitrarily while
  l and are varied and optimized (least
  squares criterion) for Toronto, Hong Kong and
  Singapore.
• Some restrictions apply. For example
• and
• Members of the diagnosed class J recover at
  the same rate as members of the undiagnosed class
  I.
• All other parameters are taken from the current
  literature.

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Basic reproductive number R0
The number of secondary infectious individuals
generated by a single typical infectious
individual when introduced into a population of
only susceptible individuals.

R0 1.2 (Hong Kong and Toronto) R0 1.1
(Singapore)
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Basic Reproductive Number
R0 1.2 (Hong Kong and Toronto) R0 1.1
(Singapore)
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Initial Rates of Growth

• Initial rates of growth r are computed from data
  provided by the WHO and the Canada Ministry of
  Health. These rates are computed exclusively from
  the cumulative number of cases reported between
  March 31 and April 14.
• The values obtained are 0.0405 (world data),
  0.0496 (Hong Kong) , 0.054 (Toronto) and 0.037
  (Singapore).

Start of the Outbreaks

• Estimates of the start of the outbreaks in
  Toronto, Hong Kong and Singapore are obtained
  from the formula

That is, we assume an initial exponential growth
(r is the model-free rate of growth from the
time series x(t) of the cumulative number of SARS
cases).
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Estimated start of the outbreaks
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Semilog plot of the cumulative number of SARS
cases
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The case of Toronto

• The total population in Ontario is approximately
  12 million. However, the population at major risk
  of SARS infection lives in the southern part
  which is approximately 40 of the total
  population ( in our model).
• Hospital closures (Scarborough Grace hospital on
  March 25th and York Central hospital on March
  28th), the jump in the reported number of
  Canadian SARS cases on March 31st and the rapid
  rise in recognized cases in the following week,
  indicate that doctors were rapidly diagnosing
  pre-existing cases of SARS (in either E or I on
  March 26th). At this point the diagnosis and
  isolation rates were changed

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Simulation results

• (circles) data
• (lines) cumulative number of SARS cases C
• Prior to March 26 1/6, l0.76
• The fit to data is given by 1/3, l0.05
  (rapid diagnosis and effective isolation of
  diagnosed cases, dashed line)
• The second curve is given by 1/6, l0.05
  (slow diagnosis and effective isolation of
  diagnosed cases, dotted line)
• The third curve is given by 1/3, l0.3
  (rapid diagnosis with improved but imperfect
  isolation, dash-dot line)

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Simulation results, Cont
l 0.38 (Hong Kong) and l0.4 (Singapore).
Singapore has 0.68. All other parameters
are from Table 1. The data is fitted starting
March 31st because of the jump in reporting on
March 30th.
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Conclusions

• By examining two cases with relatively clean
  exponential growth curves we are able to
  calibrate the SEIJR model. We use the model to
  study the non-exponential dynamics of the Toronto
  Outbreak after intervention. Two features of the
  Toronto data, the steep increase in the number of
  recognized cases after March 31st and the rapid
  slowing in the growth of new recognized cases,
  robustly constrain the SEIJR model by requiring
  that and days-1.
• The fitting of data shows that initial rates of
  SARS growth are quite similar in most regions
  leading to estimates of R0 between 1.1 and 1.2.

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Conclusions, cont

• Both good isolation and rapid diagnosis are
  required for control.
• In our model "good control" means (a) at least a
  factor of 10 reduction in l (effectiveness of
  isolation) and (b) simultaneously a maximum
  diagnostic period of 3 days. The model is
  sensitive to these parameters, so they should be
  treated as absolutely minimal requirements
  better is better.
• Although our work doesn't give us a technical
  basis for the following statement, it is
  reasonable to speculate that these (necessary)
  drastic isolation measure could bog down medical
  services and create huge expense for those who
  have to implement them (particularly if there is
  a delay in implementation). It will help the
  situation enormously if it becomes possible to
  rapidly diagnose SARS cases with a laboratory
  test then many of the people with suspected SARS
  would only need to be isolated on a precautionary
  basis until the test results were in. The folks
  developing the test should have the maximum
  usable resources thrown at them.

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EXTENSIONS/THOUGHTS

• Transportation Systems
• Transient Populations

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Subway Transportation Model
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Proportionate mixing (Mixing Axioms)
(1) Pij gt0 (2) (3)
ci Ni Pij cj Nj Pji   Then is the only
separable solution satisfying (1) , (2), and
(3).  
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Definitions
the mixing probability between non-subway
users from neighborhood i given that they
mixed. the mixing probability of
non-subway and subway users from neighborhood i,
given that they mixed. the mixing
probability of subway and non-subway users from
neighborhood i, given that they mixed.
the mixing probability between subway users from
neighborhood i, given that they mixed.
the mixing probability between subway users from
neighborhoods i and j, given that they mixed.
the mixing probability between non-subway
users from neighborhoods i and j, given that
they mixed. the mixing probability between
non-subway user from neighborhood i and subway
users from neighborhood j, given that they mixed.
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Formulae of Mixing Probabilities (depends on
activity level and allocated time)
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Plot R0 (q1, q2) vs q1 and q2
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Curve R0 (q1, q2) 1
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Cumulative deaths One day delay (q1 q20.8)
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Forthcoming BookMathematical and Modeling
Approaches in Homeland Security

• SIAM's, Frontiers in Applied Mathematics
• Edited by
• Tom Banks and Carlos Castillo-Chavez ,
• On sale on June 2003.