NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR
AUTOMATED ISOCHRONES AND THE LOCATION OF EMERGENCY MEDICAL SERVICES IN CITIES:
A NOTE
L.D. Mayhew
July 1981 WP-81-103
Working Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria
THE AUTHOR
Leslie Mayhew is an IIASA research scholar working within the Health Care Task of the Human Settlements and Services Area.
He is on secondement from the Operational Research Unit of the Department of Health and Social Security, UK.
FOREWORD
The p r i n c i p a l aim of h e a l t h c a r e r e s e a r c h a t I I A S A h a s been t o d e v e l o p a f a m i l y of submodels of n a t i o n a l h e a l t h c a r e systems f o r u s e by h e a l t h s e r v i c e p l a n n e r s . The modeling work i s p r o c e e d i n g a l o n g t h e l i n e s proposed i n t h e I n s t i t u t e ' s
c u r r e n t Research P l a n . I t i n v o l v e s t h e c o n s t r u c t i o n o f l i n k e d submodels d e a l i n g w i t h p o p u l a t i o n , d i s e a s e p r e v a l e n c e , r e s o u r c e need, r e s o u r c e a l l o c a t i o n , and r e s o u r c e s u p p l y .
The work i n t r o d u c e d h e r e i s an approach t h a t i s d e s i g n e d
t o a i d t h e l o c a t i o n and s c h e d u l i n g of emergency m e d i c a l f a c i l i t i e s i n l a r g e c i t i e s . Although t r a v e l t i m e t o a medical c e n t e r i s
n o t c r i t i c a l l y i m p o r t a n t f o r r o u t i n e m e d i c a l s e r v i c e s , it i s i n t h e c a s e o f emergencies. Thus h e a l t h a u t h o r i t i e s must pay c l o s e
a t t e n t i o n t o t h e s p a c i n g o f f a c i l i t i e s . U n f o r t u n a t e l y , b e c a u s e o f c o n g e s t i o n e f f e c t s , s e a s o n a l f a c t o r s , s t a f f s h o r t a g e s , and o t h e r r e a s o n s , it i s d i f f i c u l t t o s u p p l y t h e s e s e r v i c e s e f f i - c i e n t l y and c h e a p l y . The i d e a developed i n t h i s p a p e r i s based on Csochrones--the l o c u s o f p o i n t s around a c e n t e r t h a t c a n be r e a c h e d i n a g i v e n t i m e . I s o c h r o n e s d e l i m i t t h e emergency r e s p o n s e a r e a s under v a r y i n g t r a f f i c c o n d i t i o n s f o r d i f f e r e n t s e t s o f c e n t e r s , s o t h a t g e o g r a p h i c a l gaps i n p r o v i s i o n may be d e t e r m i n e d and, i f n e c e s s a r y , remedied.
R e l a t e d p u b l i c a t i o n s i n t h e H e a l t h Care Systems Task a r e l i s t e d a t t h e end of t h i s p a p e r .
Andrei Rogers Chairman
Human S e t t l e m e n t s and S e r v i c e s Area
ABSTRACT
T h i s p a p e r d e s c r i b e s a method f o r t h e a u t o m a t i c c a l c u l a t i o n and g r a p h i c a l r e p r o d u c t i o n o f i s o c h r o n e s t h a t a r e s e t f o r d i f f e r - e n t t i m e s t a n d a r d s and f o r v a r y i n g t r a f f i c c o n d i t i o n s a r o u n d
emergency m e d i c a l c e n t e r s i n l a r g e c i t i e s . The t e c h n i q u e i s b a s e d on t h e c o n c e p t o f a v e l o c i t y f i e l d . I t p e r m i t s a r a p i d e v a l u a t i o n o f c o v e r a g e s t a n d a r d s u n d e r d i f f e r e n t o p e r a t i n g c o n d i t i o n s .
CONTEMTS
I N T R O D U C T I O N 1 . METHOD 2 . A N A L Y S I S C O N C L U S I O N R E F E R E N C E S
AUTOMATED ISOCHRONES AND THE LOCATION OF EMERGENCY MEDICAL SERVICES I N C I T I E S : A NOTE
INTRODUCTION
I n t h e p l a n n i n g and o p e r a t i o n o f a c c i d e n t and emergency m e d i c a l c e n t e r s , t h e d i s t a n c e o f t r a v e l between t h e s i t e o f a n a c c i d e n t and t h e n e a r e s t t r e a t m e n t c e n t e r i s r e c o g n i z e d a s b e i n g i m p o r t a n t . The t y p e s o f c o v e r a g e problems t h a t t h i s t r a v e l
c o n s t r a i n t g e n e r a t e s have been t h e s u b j e c t o f s e v e r a l s t u d i e s ( f o r example, Church and R e v e l l e 1974; Mayhew 1979; T o r e g a s e t a l . 1971; and T o r e g a s and R e v e l l e 1 9 7 3 ) . I n u r b a n a r e a s t h e t i m e o f t r a v e l i s c e r t a i n l y more i m p o r t a n t t h a n d i s t a n c e b e c a u s e o f t h e v a r i a t i o n i n c o n g e s t i o n l e v e l s on d i f f e r e n t r o a d s i n t h e t r a n s p o r t network. A l s o , a s t u d y o f a c i t y ' s emergency m e d i c a l c e n t e r s o f t e n i n d i c a t e s a dimension i n t h e i r b e h a v i o r t h a t p e r h a p s i s n o t r e f l e c t e d i n t h e l o c a t i o n models d e v e l o p e d t o d a t e . Very i m p o r t a n t , f o r i n s t a n c e , a r e t h e o p e n i n g h o u r s , which a r e d i f f e r e n t ( a n d sometimes u n s c h e d u l e d ) among t h e c e n t e r s d u r i n g t h e day and n i g h t , t h u s a l l o w i n g , i n a c o m p l i c a t e d way, t h e s u b s t i t u t i o n o f one c e n t e r f o r a n o t h e r . The o b j e c t o f t h i s n o t e i s t o s u g g e s t how t h e n a t u r e o f t r a v e l i n a c i t y and t h e o p e r a t i n g b e h a v i o r o f t h e m e d i c a l c e n t e r s i n t e r a c t and how t h e y may be m o n i t o r e d . The r e s u l t s g i v e n a r e p r e l i m i n a r y and a r e
intended as an introduction to a more general approach that will be developed in subsequent publications.
The idea developed here is based on the notion of automated isochrones to be drawn on a visual display unit for any time standard and set of locations. Isochrones are the locus of points about an emergency center that can be reached in a given time. The areas the isochrones encompass are called response areas, because they can be reached by emergency vehicles in less than the given time. When knowledge of these areas is centrally located in a command and control center, it becomes possible to obtain a prompt indication of which parts of the city are
adequately covered by accident and emergency services and which are not.
The time standard defined by the isochrone is simply the maximum desirable travel time taken to reach the site of an
emergency. Typically, different operating authorities will have different views on what constitutes an adequate time standard. Once selected, however, the standard strongly influences the character of the system, particularly the
scheduling of services and the number of centers that are open at any time. Nevertheless, with the response areas easily displayed on a screen, any significant change that would alter the accident and emergency coverage would then be quickly
identified and an alternative plan could be devised using this method.
The premise for mapping the isochrones in the way to be discussed is that published data on journey times are often unreliable and do not reflect the daily variations in traffic conditions, which in most cities are an important factor
influencing journey time. The results shown from this study are exploratory, but they are encouraging and capable of further development. The first part of this note gives an outline of the methods; the second part details the mathematics for a particular example.
1 . METHOD
The b a s i s f o r t h i s a p p r o a c h t o d e t e r m i n i n g r e s p o n s e a r e a s i s t h e c o n c e p t o f a v e l o c i t y f i e l d i n t r o d u c e d by Angel and
Hyman (1970, 1971, 1977) f o r o b t a i n i n g m e a s u r e s o f j o u r n e y t i m e s between p a i r s o f p o i n t s i n a n u r b a n a r e a . I n t h e i r work, it i s o b s e r v e d t h a t , i n c i t i e s , t h e a v e r a g e s p e e d o f t r a v e l by r o a d shows an i n c r e a s e from t h e c e n t e r t o t h e p e r i p h e r y , and it i s w i t h r e f e r e n c e t o t h i s p r o p e r t y t h a t t h e a u t h o r s s o l v e f o r
minimum j o u r n e y t i m e s . A t t h i s l e v e l o f a b s t r a c t i o n t h e t w i s t s and t u r n s o f t h e a c t u a l network a r e i g n o r e d , which seems r e a s o n - a b l e p r o v i d i n g t h e r o a d network i n t h e c i t y i s s u f f i c i e n t l y d e n s e . An a n a l y s i s shows t h a t , i f c e r t a i n c o n d i t i o n s a r e m e t , t h e minimum t i m e p a t h s between two n o n - c e n t r a l l y l o c a t e d p o i n t s w i l l r e p r o d u c e on a d i a g r a m a s smooth c u r v e s t h a t s p i r a l a r o u n d t h e c i t y c e n t e r , t r a d i n g o f f t h e i n c r e a s e d t r a v e l d i s t a n c e
a g a i n s t t h e d e l a y s c a u s e d by t r a f f i c c o n g e s t i o n . F i g u r e 1 compares t h i s e f f e c t f o r o n e c i t y a g a i n s t a n o t h e r where t h e s p e e d i s c o n s i d e r e d t o b e u n i f o r m f o r a j o u r n e y o r i g i n t h a t i s s o u t h o f t h e c i t y c e n t e r . I n t h e second c a s e t h e s h o r t e s t p a t h s a r e mapped a s s t r a i g h t l i n e s .
I n g e n e r a l a minimum t i m e p a t h between A and B c a n b e e v a l u a t e d by f i n d i n g t h e s m a l l e s t v a l u e o f t h e i n t e g r a l ,
where V ( r ) i s t h e a v e r a g e s p e e d o f t r a v e l e x p r e s s e d a s a f u n c t i o n o f r t h e d i s t a n c e from t h e c i t y c e n t e r . I f t h e p a r a m e t e r s o f V ( r ) a r e z and p , t h e n t h e t i m e t c a n b e w r i t t e n i n f u n c t i o n a l
form a s
where
riel
and r202 are the points A and B expressed in polar coordinates. If we re-express time as a function of r instead,2 then in principle we can obtain,
the equation of the isochrone. By fixing r 8 z and p and then 1 1'
allowing
e2
to range through 2II radians, the desired isochrones about a facility located atriel
can thus be mapped for any desired value of t.It remains to parameterize the function V(r) by empirically determining values for p and z under varying road and weather conditions, and then solving for r2 in the above equations.
The first part of this would be achieved using special surveys to determine average speeds under different traffic conditions;
the theoretical part is considered for one class of function in part two of this note. The accuracy obtained by our method depends on our ability to characterize journey speeds with the function we choose for V(r). If a city cannot be characterized this way at all (that is if speed is clearly not a function of r) then the method is, of course, invalid. In any event the results can never be perfect as there is always a stochastic component in the time attached to any journey. This is true regardless of how the measurement is made. There may also be barriers to travel such as rivers for which corrections will be necessary. Nevertheless, with the method's development, the measure of time necessary for the journey can be brought to a level of accuracy that is sufficient for the application intended and certainly superior to the alternative measure of distance.
The following example will show how the method works and what possibilities there are for extensions.
Assume that the relationship between distance and the average speed of travel is given by
This function is flexible, analytically convenient (see part two), and easily fitted to empirical data. There are two drawbacks in the performance of this application: firstly, the journey speed at the city center is defined as zero, which is arguably impossible even in the most congested of cities;
secondly, speeds can increase indefinitely with no limit for a large r. Often ambulances are not bound by local speed limits but only by their physical capabilities, so the second criticism may not apply over the range of r that we would want to consider.
Nevertheless, this specification of V(r) is not exhaustive, and it is given here simply as an example.
Figure 2 shows how the results will be produced. The area covered by the map is the administrative area of the Greater London Council. The locations are drawn from an existing sub- set of hospitals currently providing emergency treatment facil- ities., The entire image is reproduced from a microfilm linked to a graphics package. The values of z and p are, respectively, 3.0 and 0.75 in example (a) and 10.0 and 0.33 in example (b), whereas the value of the plotted isochrone t has been set to ten minutes in both cases.
As is seen the response area served by each center increases with distance from the city center. Although each area appears
to be circular, a closer inspection shows that the side nearest the city center is, in fact, slightly compressed. For certain values of p, z, r l , and t those areas closest to the city center will distort to cardiods, the cusp lying astride a radial through r l e l This would simply be a reflection of the higher congestion effects near the center where velocities approach zero. AS will be shown, those facilities at the city center itself possess a circular response area; this is the only point in the city where this can occur.
The q u e s t i o n a r i s e s i n p r a c t i c e : which i s t h e optimum c o v e r i n g s e t f o r o u r p a r a m e t e r v a l u e s and t i m e s t a n d a r d ? To answer t h i s t h e a p p r o a c h must be e x t e n d e d and t h e n l i n k e d t o t h e e x i s t i n g methods o f s o l u t i o n w h i l e r e t a i n i n g t h e same g r a p h i c a l o u t p u t . I n t h e " s e t c o v e r a g e problem" ( T o r e g a s e t a l . 1 9 7 1 ) , f o r example, t h e o b j e c t i v e i n o u r f o r m u l a t i o n would b e t o i d e n t i f y from t h e f e a s i b l e s e t t h e minimum number o f c e n t e r s n e c e s s a r y t o c o v e r demand p o i n t s w i t h i n a g i v e n t i m e .
(Demand p o i n t s w i t h i n a c i t y c a n be r e p r e s e n t e d on a g r i d . ) F o r v e r y s t r i c t t i m e s t a n d a r d s , t h i s may b e i m p o s s i b l e . A l t e r - n a t i v e l y , w e m i g h t c h o o s e t o s o l v e t h e "maximum c o v e r i n g problem"
(Church and R e V e l l e 1 9 7 4 ) , which h a s t h e s l i g h t l y l e s s n o b l e g o a l o f maximizing t h e t o t a l p o p u l a t i o n c o v e r e d i n a t i m e t from a f i x e d number o f c e n t e r s . T h i s would be v a l u a b l e f o r n i g h t t i m e c o v e r a g e when s t a f f s h o r t a g e s o r b u d g e t a r y c o n s i d e r a - t i o n s g r e a t l y c o n s t r a i n t h e f e a s i b l e s e t .
ANALYSIS
I n t h i s f i n a l s e c t i o n w e d e r i v e t h e e q u a t i o n s f o r t h e i s o c h r o n e s a s s o c i a t e d w i t h t h e v e l o c i t y f i e l d r e p r e s e n t e d i n e q u a t i o n ( 4 ) . The t e c h n i q u e f o r d e t e r m i n i n g s h o r t e s t j o u r n e y t i m e s , a n d h e n c e t h e r e q u i r e d i s o c h r o n e s , i s b a s e d on t h e con- c e p t o f a two-dimensional t i m e s u r f a c e , s e t i n t h r e e d i m e n s i o n s , on which t h e d i s t a n c e between two p o i n t s A ' and B ' i s e q u a l t o t h e j o u r n e y t i m e between t h e image p o i n t s , A and B , i n t h e u r b a n p l a n e . T h i s d i s t a n c e i s a g e o d e s i c ( s h o r t e s t p a t h ) w i t h r e s p e c t t o t h e s u r f a c e on which it l i e s . Thus i t s r e f l e c t i o n i n t h e u r b a n plane--a smooth c u r v e c o n n e c t i n g A and B - - i s a l s o t h e r e q u i r e d s h o r t e s t p a t h between t h e s e two p o i n t s . I t i s shown by Angel and Hyman (1977) t h a t f o r t h e v e l o c i t y r e l a t i o n - s h i p d e s c r i b e d by V ( r ) = z rP t h e a p p r o p r i a t e t i m e s u r f a c e i s a cone whose apex i s t h e image p o i n t o f t h e c i t y c e n t e r . Using t h e t r a n s f o r m a t i o n g i v e n by them, it can b e shown t h a t t h e
j o u r n e y t i m e between two p o i n t s i n a c i t y c h a r a c t e r i z e d by t h i s f i e l d i s g i v e n by
wnere r a n d r 2 a r e t h e r e s p e c t i v e d i s t a n c e s f r o m t h e c i t y 1
c e n t e r a n d
e 1 2
i s t h e i r a n g l e o f s e p a r a t i o n ( 0 ie 1 2
5 ill.R e a r r a n g i n g ( 5 ) a n d making r 2 t h e s u b j e c t , w e o b t a i n
'2 =
Im
c o s [ ( I - p )e 1 2 ]
where m = r , E q u a t i o n ( 6 ) i s t h u s o f t h e form s p e c i f i e d i n e q u a t i o n ( 3 ) . W e now c o n s i d e r t h r e e s p e c i a l c a s e s o f i t .
C ~ s e A
From ( 6 ) f o r a r e a l s o l u t i o n ,
2 2 2
z t (1-p12 t m2 s i n [ ( I - ~ )
e12]
T h u s , a t t h e l i m i t s A a n d B shown i n F i g u r e 3a
On t h e s i d e f u r t h e s t f r o m t h e c i t y c e n t e r ( 6 ) i s t a k e n i n t h e p o s i t i v e ,
e 1 2
r a n g i n g f r o m( e l -
4 ) t o( e l
+ g ).
On t h e o p p o s i t e s i d e ( 6 ) i s t a k e n i n t h e n e g a t i v e o v e r t h e same r a n g e f o re 1 2 .
( a ) a non-central facility
(b) a central facility
Figure 3. Case (a) and (b)
.
C a s e B
A l s o f o r a r e a l s o l u t i o n , i f t h e n e g a t i v e o f ( 6 ) i s t a k e n ,
2 2 2 2 2
m c o s [ ( I - P I t (1-pi - m s i n [ ( I - p )
e 1 2 ]
( 9 )i m p l y i n g t h a t
T h i s means t h a t t h e r a d i a l from a n emergency c e n t e r t o t h e c i t y c e n t e r m u s t c u t t h e i s o c h r o n e en r o u t e . I f n o t , t h e i s o c h r o n e c u t s a l l f o u r q u a d r a n t s a s shown i n F i g u r e 3b. I n t h i s i n s t a n c e ,
e l 2
i n e q u a t i o n ( 6 ) r a n g e s o v e r t h e i n t e r v a l 0<
0 5 il.12
C a s e C
Suppose r l = 0. Then t h e emergency c e n t e r i s l o c a t e d a t t h e c i t y c e n t e r . E q u a t i o n ( 6 ) t h e n s i m p l i f i e s t o
which i s a c o n s t a n t . Thus t h e a s s o c i a t e d r e s p o n s e a r e a f o r any v a l u e o f t i s a c i r c l e .
CONCLUSION
A method has been shown that allows the automatic delimi- tation of response areas in cities around emergency medical
centers for any time standards and different traffic conditions.
In principle, it can be extended to schedule the opening times of different centers and to develop "optimum" configurations for various time standards at different times of the day or year.
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A n g e l , S . , a n d G. Hyman ( 1 9 7 0 ) U r b a n V e l o c i t y F i e l d s . E n v i r o n - m e n t and P l a n n i n g 2 : 2 2 1 - 2 2 4 .
A n g e l , S . , a n d G. Hyman ( 1 9 7 1 ) U r b a n T r a v e l T i m e . P a p e r s o f t h e R e g i o n a l S c i e n c e A s s o c i a t i o n 2 6 : 8 5 - 9 9 .
A n g e l , S . , a n d G. Hyman ( 1 9 7 7 ) Urban F i e l d s . L o n d o n : P i o n . C h u r c h , R . , a n d C. R e v e l l e ( 1 9 7 4 ) T h e M a x i m a l C o v e r i n g P r o b l e m .
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