Y. H. Hsieh 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 Impact of Travel between Patches for Spatial Spread of Disease (Joint work with (Joint work with P. van den Driessche and Lin Wang, University of Victoria, Canada)

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

Y. H. Hsieh Geographical map of SARS cases as of July

Y. H. Hsieh Spread of avian flu (H5N1) as of February of 2006 (Science 2006)

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

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

Y. H. Hsieh 1. should be screened for possible SARS at the point of departure. 1. International travelers departing from areas with local transmission should be screened for possible SARS at the point of departure. 2. 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. 3. 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.

Y. H. Hsieh Measures taken by individual governments during SARS outbreak 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).

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

Y. H. Hsieh Model equations

Y. H. Hsieh Remarks 1. 1.Purpose: to study the impact of (restricting) travel by the exposed and infective travelers 2. 2.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)

Y. H. Hsieh 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.

Y. H. Hsieh is the basic reproduction number of the ith patch in isolation is the modified reproduction number of the ith patch modified by travel

Y. H. Hsieh Theorem 3.2. For the model,. Furthermore, if then

Y. H. Hsieh 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.

Y. H. Hsieh Model equations

Y. H. Hsieh

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 ( )

Y. H. Hsieh 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

Y. H. Hsieh * Increase in travel result in disease becoming endemic in the previously low prevalence patch 2

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

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

Y. H. Hsieh Fig. 5. Simulation of infective populations (I 1 and I 2 ) decreasing to 0 when m=0.5 and hence R 0 <1.

Y. H. Hsieh Fig. 6. Simulation of infective populations (I 1 and I 2 ) decreasing to endemic equilibrium when m=0.2 and hence R 0 >1.

Y. H. Hsieh Fig. 7. Banning travel of symptomatic traveler from patch 1 to 2 ( from top) and all other same as Fig. 2, resulting increase in R 0 could adversely driving R 0 above 1 for a range of parameters.

Y. H. Hsieh Fig. 8. Banning travel of symptomatic traveler from patch 2 to 1 ( ) and all other same as Fig. 2, resulting in R 0 decreases to less than 1.

Y. H. Hsieh Fig. 9. Simulation of infective populations (I 1 and I 2 ) approaching a larger endemic equilibrium compared to Fig. 6, when (banning symptomatic travelers from patch 1 to 2) and hence R 0 >1.

Y. H. Hsieh Fig. 10. Simulation of infective populations I 1 to a large endemic equilibrium and I 2 approaching disease-free equilibrium as compared to Fig. 6, when ( banning all travelers from patch 1 to 2) and hence R 0 >1.

Y. H. Hsieh Fig. 11 Simulation of infective populations (I 1 and I 2 ) approaching disease-free equilibrium compared to Fig. 6, when and hence R 0 <1.

Y. H. Hsieh Fig.12. Banning travel of all traveler from patch 2 to 1 ( ) and all other same as Fig. 2, resulting in R 0 decreases to less than 1 as travel from patch 1 to 2 increases.

Y. H. Hsieh 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.

Y. H. Hsieh 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.

Y. H. Hsieh 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: (i) (i) issue travel warning for travelers to avoid all but essential travel to affected areas (decrease ), (i) (i) recommend border screening; and yet not recommending restriction on travel out of affected areas ( ).

Y. H. Hsieh Acknowledgement YHH is supported by grant (NSC M ) 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-FRP ). PvdD is partially supported by NSERC of Canada and MITACS. LW is supported by PIMS and MITACS PDF fellowships.