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

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)

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Geographical map of SARS cases as of July 3 2003

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

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

Y. H. Hsieh

World Health Organization (WHO) World Health Organization (WHO) measures related to international travel measures related to international travel

during 2003 SARS outbreakduring 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. International travelers departing from areas with local transmission should be should be screenedscreened for possible SARS at for possible SARS at the the point of departurepoint of departure. .

2.2. Travelers Travelers with one or more symptoms of SARSwith one or more symptoms of SARS and and who have a history of exposure or who have fever or who have a history of exposure or who have fever or who appear acutely ill may be advised to who appear acutely ill may be advised to postpone postpone their triptheir trip until they have recovered. until they have recovered.

3.3. Contact of a probable case travel to another country Contact of a probable case travel to another country should be placed in voluntary isolationshould be placed in voluntary isolation and kept under and kept under active surveillance by the health authorities in the active surveillance by the health authorities in the country of arrival. 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).

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Flowchart for Multi-patch SEIRP Flowchart for Multi-patch SEIRP Model Model

iAiI

infectiveiR

recoverediE

incubating

iPpartia lly

im m une

i ip ( )i i i i iN PI i iR

i iE i iI

i iI

( ) ( )i i i i i iN S I E iS

susceptible

and E Iij ijm m are the respective travel rates of incubating

and infective persons from patch j to patch i

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Model equationsModel equations

1 1

( ) ( )n n

S Sii i i i i i i i i i i ij j ji i

j j

dSA N S I E d S P m S m S

dt

1 1

( )[ ( ) ] ( )n n

E Eii i i i i i i i i i i i ij j ji i

j j

dEN S I E PI d E m E m E

dt

1 1

( )n n

I Iii i i i i i ij j ji i

j j

dIE d I m I m I

dt

1 1

( )n n

R Rii i i i i ij j ji i

j j

dRI d R m R m R

dt

1 1

( ) ( )n n

P Pii i i i i i i i i i ij j ji i

j j

dPR d P N PI m P m P

dt

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RemarksRemarks

1.Purpose: to study the impact of (restricting) travel by the exposed and infective travelers

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)

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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.

0 1R

0 1R

0 1R

0R

* *i iS N

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is the basic reproduction number of the ith patch in isolation

is the modified reproduction number of the ith patch modified by travel ( )

0i

R

( )0iR

( )0i i i i i

i i i

b bR

c a c

( )0

1 1 1( )( )

i i i i in n nE I E

i ji i ji i jij j j

b bR

c m a m c m

where , ( ) , for 1,2,..., n.i i i i i i i i i i ia d b N N c d i

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Theorem 3.2. For the model, .Theorem 3.2. For the model, .

Furthermore, if Furthermore, if

then then

( )0 0

1max i

i nR R

, , and for 1, 2, ..., n,i i i ia a d d i

( ) ( ) ( )0 0 0 0

11 1max max ,min max .i i i

i ni n i nR R R R

Y. H. Hsieh

Model with two patchesModel with two patches

( )0 1iR

• 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.

( )0 1iR

Y. H. Hsieh

Model equationsModel equations1

1 1 1 1 1 1 1 1 1 1 1 12 2 21 1

11 1 1 1 1 1 1 1 1 1 1 1 12 2 21 1

11 1 1 1 1 1 12 2 21 1

11 1 1 1 1 12 2 21 1

11 1 1 1 1 1

( ) ( )

( )[ ( ) ] ( )

( )

( )

( )

S S

E E

I I

R R

dSA N S I E d S P m S m S

dtdE

N S I E PI d E m E m EdtdI

E d I m I m IdtdR

I d R m R m RdtdP

R d Pdt

1 1 1 1 12 2 21 1

22 2 2 2 2 2 2 2 2 2 2 21 1 12 2

22 2 2 2 2 2 2 2 2 2 2 2 21 1 12 2

22 2 2 2 2 2 21 1 12 2

22 2 2 2 2 21 1 12

( )

( ) ( )

( )[ ( ) ] ( )

( )

( )

P P

S S

E E

I I

R R

N PI m P m P

dSA N S I E d S P m S m S

dtdE

N S I E P I d E m E m EdtdI

E d I m I m IdtdR

I d R m R m Rdt

2

22 2 2 2 2 2 2 2 2 2 21 1 12 2( ) ( ) P PdPR d P N P I m P m P

dt

Y. H. Hsieh

0

0

For the two-patch model, assume that the disease transmission is

standard incidence. Then the DFE is globally asymptotically stable if 1 and is

unstable if 1.

R

R

Theorem 4.1.

1 2 1 2

1 2 1 2 1 2 0

If for , 1,2 and , ,

, , , then an increase in or decreases .

E Iij ijm m m i j

a a a c c c b b a m R

Theorem 4.2.

(1)0 0

(2) (1) (2)0 0 0 0

With the conditions of Theorem 4.2, as the travel rate become

large, the basic reproduction number approaches the mean value of and

1, i.e. ( ) as . In this

2

m

R R

R R R R m

Remark 4.1.

limit case, the two patches

merge into one.

1 2 1 2 1 2 1 2

01 2 0

If , 0 for , 1,2 and , , , ,

, then 0, i.e., the basic reproduction number decreases as increases.

E Iij ijm m m i j a a c c

Rb b R m

m

Theorem 4.3.

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 ( )(1)

0 1.58R (2)0 0.26R

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• 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

Iijm

0Eijm

Eijm

0Iijm

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

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

0R

0R

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

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

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

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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.

21 210, 0.6, 0.8, 1.0I Em m

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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.

12 21 210, 0.6I E Im m m

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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.

21 0Im

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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.

21 21 0I Em m

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Fig. 11 Simulation of infective populations (I1 and I2) approaching disease-free equilibrium compared to Fig. 6, when and hence R0<1.12 0Im

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Fig.12.Fig.12. Banning travel of all traveler from patch 2 to 1 ( Banning travel of all traveler from patch 2 to 1 ( ) and all other same as Fig. 2, resulting in ) and all other same as Fig. 2, resulting in RR00 decreases to less than 1 as travel from patch 1 to 2 decreases to less than 1 as travel from patch 1 to 2 increases.increases.

21 21 0I Em m

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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.

12 12 0I Em m

21 0Im

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Conclusions (continued)

Banning or restricting travel from the high prevalence region to the low prevalence region ( or small) could result in:

1. Low prevalence patch becoming disease-free, but the disease becomes even more prevalent in the high prevalence patch.

2. 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.

21 21 0I Em m

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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) issue travel warning for travelers to avoid all but essential travel to affected areas (decrease

),

(i) recommend border screening; and yet not recommending restriction on travel out of affected areas ( ).

12 12 and I Em m

21 21 and I Em m

Y. H. Hsieh

AcknowledgementAcknowledgement

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.