Limits...
A double epidemic model for the SARS propagation.

Ng TW, Turinici G, Danchin A - BMC Infect. Dis. (2003)

Bottom Line: It is important both for predicting the future of the present outbreak and for implementing effective prophylactic measures, to identify the causes of this behavior.In this report, we show first that the standard Susceptible-Infected-Removed (SIR) model cannot account for the patterns observed in various regions where the disease spread.Finally, we could, within the framework of the model, fix limits to the future development of the epidemic, allowing us to identify landmarks that may be useful to set up a monitoring system to follow the evolution of the epidemic.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Mathematics, The University of Hong Kong, Hong Kong, China. ntw@maths.hku.hk

ABSTRACT

Background: An epidemic of a Severe Acute Respiratory Syndrome (SARS) caused by a new coronavirus has spread from the Guangdong province to the rest of China and to the world, with a puzzling contagion behavior. It is important both for predicting the future of the present outbreak and for implementing effective prophylactic measures, to identify the causes of this behavior.

Results: In this report, we show first that the standard Susceptible-Infected-Removed (SIR) model cannot account for the patterns observed in various regions where the disease spread. We develop a model involving two superimposed epidemics to study the recent spread of the SARS in Hong Kong and in the region. We explore the situation where these epidemics may be caused either by a virus and one or several mutants that changed its tropism, or by two unrelated viruses. This has important consequences for the future: the innocuous epidemic might still be there and generate, from time to time, variants that would have properties similar to those of SARS.

Conclusion: We find that, in order to reconcile the existing data and the spread of the disease, it is convenient to suggest that a first milder outbreak protected against the SARS. Regions that had not seen the first epidemic, or that were affected simultaneously with the SARS suffered much more, with a very high percentage of persons affected. We also find regions where the data appear to be inconsistent, suggesting that they are incomplete or do not reflect an appropriate identification of SARS patients. Finally, we could, within the framework of the model, fix limits to the future development of the epidemic, allowing us to identify landmarks that may be useful to set up a monitoring system to follow the evolution of the epidemic. The model also suggests that there might exist a SARS precursor in a large reservoir, prompting for implementation of precautionary measures when the weather cools down.

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Daily new number of confirmed SARS cases from Hong Kong: hospital, community and the Amoy Gardens.
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Figure 2: Daily new number of confirmed SARS cases from Hong Kong: hospital, community and the Amoy Gardens.

Mentions: Figure 2 shows the daily new number of confirmed SARS cases in Hong Kong from 17 March, 2003 to 10 May, 2003 (source: Hong Kong SAR Government press release, ). The daily number of SARS cases are grouped into three categories: i) the Amoy Gardens, ii) the community, iii) the hospital staff. The graph can be considered as an approximation of . The graph has several peaks and it therefore looks very different from the general shape of the graph of the function I. In the SIR model once the I curve decreases, it will decrease to the zero value Therefore, the standard SIR model could not be used to model the outbreak of SARS in Hong Kong. If we take away those cases from the hospital staff and the Amoy Gardens Estate, that is if we only consider the cases from the community, then the graph looks closer to the general shape of a graph for a function I. However, it is still difficult for a general SIR model to produce a I curve whose values are within a narrow range (say 15 to 25 cases) for a relatively long time interval. This motivated us to replace the SIR model by the SEIR model. In this model there is a fixed period between exposure and becoming infectious, called latent period (this is in fact the case for SARS and it is another reason why we consider the SEIR model). Thus, rather than an exposed susceptible becoming immediately infectious, it enters the Exposed class, labeled E, remaining there a fixed period of time. More detailed description of this model will be given in the next section. It is worth mentioning here that the SEIR model can produce much more interesting and complicated I, R and curves. It is also possible to produce a I curve whose values are within a narrow range for a relatively long time interval and result thus in a relatively large R∞. Note that the SIR model is the limiting case of the SEIR model when the time interval from the infection to onset is zero; for given parameters r and a the total size of the epidemic is the same in the two models; however the duration is longer for the SEIR model which will display lower admission per day curves (or the I curves); if we see this result from the reciprocal point of view, the same level of admissions per day will result in a higher total epidemic size for the SEIR model than for the SIR model because of the Exposed class which is "hidden" in the sense that it has no effects on the other classes until the individuals move to the Infective class and thus contributing to the propagation of the epidemic. A numerical illustration of this phenomena is given in Figure 3 that plots the evolution of the S,E, I and R classes for the two models with the same parameters r and a.


A double epidemic model for the SARS propagation.

Ng TW, Turinici G, Danchin A - BMC Infect. Dis. (2003)

Daily new number of confirmed SARS cases from Hong Kong: hospital, community and the Amoy Gardens.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC222908&req=5

Figure 2: Daily new number of confirmed SARS cases from Hong Kong: hospital, community and the Amoy Gardens.
Mentions: Figure 2 shows the daily new number of confirmed SARS cases in Hong Kong from 17 March, 2003 to 10 May, 2003 (source: Hong Kong SAR Government press release, ). The daily number of SARS cases are grouped into three categories: i) the Amoy Gardens, ii) the community, iii) the hospital staff. The graph can be considered as an approximation of . The graph has several peaks and it therefore looks very different from the general shape of the graph of the function I. In the SIR model once the I curve decreases, it will decrease to the zero value Therefore, the standard SIR model could not be used to model the outbreak of SARS in Hong Kong. If we take away those cases from the hospital staff and the Amoy Gardens Estate, that is if we only consider the cases from the community, then the graph looks closer to the general shape of a graph for a function I. However, it is still difficult for a general SIR model to produce a I curve whose values are within a narrow range (say 15 to 25 cases) for a relatively long time interval. This motivated us to replace the SIR model by the SEIR model. In this model there is a fixed period between exposure and becoming infectious, called latent period (this is in fact the case for SARS and it is another reason why we consider the SEIR model). Thus, rather than an exposed susceptible becoming immediately infectious, it enters the Exposed class, labeled E, remaining there a fixed period of time. More detailed description of this model will be given in the next section. It is worth mentioning here that the SEIR model can produce much more interesting and complicated I, R and curves. It is also possible to produce a I curve whose values are within a narrow range for a relatively long time interval and result thus in a relatively large R∞. Note that the SIR model is the limiting case of the SEIR model when the time interval from the infection to onset is zero; for given parameters r and a the total size of the epidemic is the same in the two models; however the duration is longer for the SEIR model which will display lower admission per day curves (or the I curves); if we see this result from the reciprocal point of view, the same level of admissions per day will result in a higher total epidemic size for the SEIR model than for the SIR model because of the Exposed class which is "hidden" in the sense that it has no effects on the other classes until the individuals move to the Infective class and thus contributing to the propagation of the epidemic. A numerical illustration of this phenomena is given in Figure 3 that plots the evolution of the S,E, I and R classes for the two models with the same parameters r and a.

Bottom Line: It is important both for predicting the future of the present outbreak and for implementing effective prophylactic measures, to identify the causes of this behavior.In this report, we show first that the standard Susceptible-Infected-Removed (SIR) model cannot account for the patterns observed in various regions where the disease spread.Finally, we could, within the framework of the model, fix limits to the future development of the epidemic, allowing us to identify landmarks that may be useful to set up a monitoring system to follow the evolution of the epidemic.

View Article: PubMed Central - HTML - PubMed

Affiliation: Department of Mathematics, The University of Hong Kong, Hong Kong, China. ntw@maths.hku.hk

ABSTRACT

Background: An epidemic of a Severe Acute Respiratory Syndrome (SARS) caused by a new coronavirus has spread from the Guangdong province to the rest of China and to the world, with a puzzling contagion behavior. It is important both for predicting the future of the present outbreak and for implementing effective prophylactic measures, to identify the causes of this behavior.

Results: In this report, we show first that the standard Susceptible-Infected-Removed (SIR) model cannot account for the patterns observed in various regions where the disease spread. We develop a model involving two superimposed epidemics to study the recent spread of the SARS in Hong Kong and in the region. We explore the situation where these epidemics may be caused either by a virus and one or several mutants that changed its tropism, or by two unrelated viruses. This has important consequences for the future: the innocuous epidemic might still be there and generate, from time to time, variants that would have properties similar to those of SARS.

Conclusion: We find that, in order to reconcile the existing data and the spread of the disease, it is convenient to suggest that a first milder outbreak protected against the SARS. Regions that had not seen the first epidemic, or that were affected simultaneously with the SARS suffered much more, with a very high percentage of persons affected. We also find regions where the data appear to be inconsistent, suggesting that they are incomplete or do not reflect an appropriate identification of SARS patients. Finally, we could, within the framework of the model, fix limits to the future development of the epidemic, allowing us to identify landmarks that may be useful to set up a monitoring system to follow the evolution of the epidemic. The model also suggests that there might exist a SARS precursor in a large reservoir, prompting for implementation of precautionary measures when the weather cools down.

Show MeSH
Related in: MedlinePlus