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謝英恆 國立中興大學應用數學系 Ying-Hen Hsieh Department of Applied Mathematics

Impact of Travel between Patches for Spatial Spread of Disease (Joint work with P. van den Driessche and Lin Wang, University of Victoria, Canada). 謝英恆 國立中興大學應用數學系 Ying-Hen Hsieh Department of Applied Mathematics National Chung Hsing University Taichung, Taiwan.

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謝英恆 國立中興大學應用數學系 Ying-Hen Hsieh Department of Applied Mathematics

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  1. Impact of Travel between Patches for Spatial Spread of Disease (Joint work with P. van den Driessche and Lin Wang, University of Victoria, Canada) 謝英恆 國立中興大學應用數學系 Ying-Hen Hsieh Department of Applied Mathematics National Chung Hsing University Taichung, Taiwan

  2. Early Spatial Spread of SARS(From “Learning from SARS: Preparing for the next disease outbreak” 2003 IOM SARS workshop summary)

  3. Geographical map of SARS cases as of July 3 2003

  4. Spread of avian flu (H5N1) as of February of 2006 (Science 2006)

  5. Geographical map of H5N1 human infections in Southeast Asia as of May 2005 (K. Ungchusak briefing at 2005 WHA Assembly)

  6. WHO did not recommend the restriction of travel to any areas WHO recommended measures to limit the international spread of SARS World Health Organization (WHO) measures related to international travel during 2003 SARS outbreak

  7. International travelers departing from areas with local transmission should be screened for possible SARS at the point of departure. Travelers with one or more symptoms of SARS and who have a history of exposure or who have fever or who appear acutely ill may be advised to postpone their trip until they have recovered. Contact of a probable case travel to another country should be placed in voluntary isolation and kept under active surveillance by the health authorities in the country of arrival.

  8. Border control: installing infrared thermal scanning devise to screen travelers in order to detect symptomatic cases in and out (to stop travel of infective persons in and out of a patch). Banning travelers from affected areas (to stop travel of exposed persons into a patch). Measures taken by individual governments during SARS outbreak

  9. Flowchart for Multi-patch SEIRP Model are the respective travel rates of incubating and infective persons from patch j to patch i

  10. Model equations

  11. Remarks • Purpose: to study the impact of (restricting) travel by the exposed and infective travelers • We do not consider asymptomatic compartment • 3. Other related modeling work: • SEIRP model: Hyman and LaForce (2001) • Multi-patch SEIR model: Arino and van den Driessche (2003, 2006) • 2-patch SIR model: Wang and Zhao (2006)

  12. Theorem 3.1. If , then the DFE ( and all others are 0) is locally asymptotically stable; and if , the DFE is unstable. Moreover, if the disease transmission is standard incidence, then the DFE is globally asymptotically stable provided that . is the basic reproduction number for the multi-patch system which is dependent on travel.

  13. is the basic reproduction number of the ith patch in isolation is the modified reproduction number of the ith patch modified by travel

  14. Theorem 3.2. For the model, . Furthermore, if then

  15. Model with two patches • To illustrate, we assume one patch has high disease prevalence ( ), while the • other with low prevalence ( ). • The results hold if the low prevalence patch was disease-free initially.

  16. Model equations

  17. Numerical simulations • For average incubation (1.48 days) and infectivity (2.6 days) periods, we use values from Ferguson et al. (Science 2005) • All other values are theoretical values to illustrate the impact of travel • When in isolation, patch 1 is high prevalence ( ), patch 2 is low prevalence ( )

  18. is travel rate of infectives from patch j to patch i ; in particular, implies the travel of infectives from patch j to patch i is banned is the travel rate of the incubating individuals from patch j to patch i; in particular, implies the travel of the incubating individuals from patch j to patch i is banned

  19. * Increase in travel result in disease becoming endemic in the previously low prevalence patch 2

  20. However, increased travel from patch 1 to patch 2 could decrease to less than one, thus eradicating disease in the two-patch system

  21. Assuming all travel rates are the same, increase travel decreases the chance of disease becoming endemic.

  22. Fig. 5. Simulation of infective populations (I1 and I2) decreasing to 0 when m=0.5 and hence R0<1.

  23. Fig. 6. Simulation of infective populations (I1 and I2) decreasing to endemic equilibrium when m=0.2 and hence R0>1.

  24. Fig. 7. Banning travel of symptomatic traveler from patch 1 to 2 ( from top) and all other same as Fig. 2, resulting increase in R0 could adversely driving R0 above 1 for a range of parameters.

  25. Fig. 8. Banning travel of symptomatic traveler from patch 2 to 1 ( ) and all other same as Fig. 2, resulting in R0 decreases to less than 1.

  26. Fig. 9. Simulation of infective populations (I1 and I2) approaching a larger endemic equilibrium compared to Fig. 6, when (banning symptomatic travelers from patch 1 to 2) and hence R0>1.

  27. Fig. 10. Simulation of infective populations I1 to a large endemic equilibrium and I2 approaching disease-free equilibrium as compared to Fig. 6, when ( banning all travelers from patch 1 to 2) and hence R0>1.

  28. Fig. 11Simulation of infective populations (I1 and I2) approaching disease-free equilibrium compared to Fig. 6, when and hence R0<1.

  29. Fig.12.Banning travel of all traveler from patch 2 to 1 ( ) and all other same as Fig. 2, resulting in R0 decreases to less than 1 as travel from patch 1 to 2 increases.

  30. Conclusions • Banning or restricting travel from low prevalence patch to high prevalence patch ( or small) always contributes to disease control. • Banning or restricting travel of symptomatic travelers only from high prevalence patch to low prevalence patch ( or small) could affect the containment of the outbreak adversely under certain range of parameter values.

  31. Conclusions (continued) • Banning or restricting travel from the high prevalence region to the low prevalence region ( or small) could result in: • Low prevalence patch becoming disease-free, but the disease becomes even more prevalent in the high prevalence patch. • The resulting number of infectives in high prevalence patch alone exceeds the combined number of infectives in both regions if border control had not been in place.

  32. Conclusions (continued) • Border control could be useful to stop spatial spread of disease, if properly implemented. • Our results suggest that, during the 2003 SARS outbreak, World Health Organization had been correct to: • issue travel warning for travelers to avoid all but essential travel to affected areas (decrease ), • recommend border screening; and yet not recommending restriction on travel out of affected areas ( ).

  33. YHH is supported by grant (NSC 94-2115-M005-006) from the National Science Council of Taiwan. YHH is grateful to the Canadian government for their generous financial support to fund YHH’s visit to University of Victoria under a Faculty Research Award (623-2-FRP2005-04). PvdD is partially supported by NSERC of Canada and MITACS. LW is supported by PIMS and MITACS PDF fellowships. Acknowledgement

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