Crossing paths in 2D Random Walks

CR OSSING P A THS IN 2D RANDOM W ALKS MARC AR TZR OUNI DEP AR TMENT OF MA THEMA TICS UNIVERSITY OF P AU; 64000 P AU; FRANCE Abstract. W e inv estigate crossing path probabilities for tw o agent s that mov e randomly in a bounded region of the plane or on a sphere (denoted R ). At eac h discrete tim e- step the agent s mov e, i ndependent ly , fixed distances d 1 and d 2 at angles that are u niformly distributed in (0 , 2 π ). If R i s large enough and the initial posi tions of the agen ts are uniformly distributed in R , then the probability of paths crossing at the fir st time-step is close to 2 d 1 d 2 / ( πA [ R ]), where A [ R ] is the area of R . Simulat ions suggest that the l ong-run rate at which paths cross i s also close to 2 d 1 d 2 / ( πA [ R ]) (despite mark ed departures fr om uniformity and indep endence conditions needed for suc h a conclusion). 1. Introduction. Rando m w alks hav e bee n studied in abstract settings such as int eger lattices Z d or Riemannian manifolds ([4], [7], [10]). In applied settings ther e are many s pa tially explicit individua l-based mo dels (IBMs) in which the b ehavior of the sys tem is deter mined by the meeting o f ra ndomly moving a gents. The trans mis- sion of a pathog e nic agent, the spread of a rumor, o r the sharing of so me prop erty when randomly moving particles meet are examples that come to mind in biology , so ciolog y , or physics ([3], [8], [6 ], [5], [2]). In many of these mo dels the movemen t o f agents is conceptualized as discrete tra nsitions betw een squar e or hexagonal cells ([3]). How e ver, such a stylized repr e s entation of individua l mov emen ts may not a lwa ys b e ent irely realis tic. Although IBMs are powerful to ols for the description of complex systems, they suffer from a shortag e of analy tical res ults. F or example, if a susc e ptible and an infectiv e ag ent move randomly in some b ounded space, what is the probabilit y of them mee ting , and hence o f the transmiss io n of the infection? What is the average time unt il the meeting takes place? In the pre s ent pap er we beg in to answer these ques tions by considering a random walk in a b ounded region of the pla ne or on the sphere, which we denote by R . The mo del evolves in discre te time. At each time-step an agent leaves its cur rent p osition at a uniformly distributed angle in the (0 , 2 π ) interv al. On the plane the ag e nt mov es a fixe d distance d in a straight line. On a sphere the ag ent moves a fixed distance d on a g eo desic. Here we will consider tw o such a gents who mov e different distances d 1 and d 2 at each time-step. W e assume that the agents’ initial p ositions a re unifor mly distributed in R . The pap er’s central results concer n the proba bilit y that the paths of the t wo agents cross during the fir st time-step. If R is either a sufficiently lar ge b o unded region of the plane or a sphere, this ” first-step” proba bility of intersection is close to 2 d 1 d 2 / ( π A [ R ]) wher e A [ R ] is the a rea o f R . In applied settings w e are o ften interested in the long-run aver age r ate at whic h the paths cr oss. In o rder to extend results on the ” first-step” probability of intersection and apply the law of larg e num bers we would need the following assumptions : • The p ositio ns of the tw o agents are uniformly distributed at every time step (whic h is the case on the sphere but not on the plane b eca use of reflectio n problems at the b oundary of R ), • The cr ossing- pa th e ven ts a re indep endent ov er time (which is the c ase in neither setting b ecaus e o f the strong spa tial depe ndence at consec utive time- steps). 1 Fig. 1.1 . E xample of t wo tra je ctories V 1 ( k ) and V 2 ( k ) with p aths that cr osse d sy nchr onously at m 1 for k = 2 . (We ar e not int er este d in the asynchr onous cro ssing at m 2 which to ok plac e at differ e nt times for the t wo agents). Numerous simulations have shown that despite marked depar tures fro m these assumptions the long- run rate at which the pa ths cross is also close to 2 d 1 d 2 / ( π A [ R ]). Section 2 contains the results b o th for the plane and the sphere. Se c tion 3 is devoted to the numerical simulations. Extensions are discussed in Section 4. Thr ee techn ical app endices ca n b e found in Section 5. 2. Mo del. 2.1. Geom etric description in the pl ane and o n the sphere. A b ounded region o f the pla ne is the most na tur al se tting for a gents moving in a 2 D e nvironmen t. How e ver w e then need to sp ecify how agents are reflected when they hit the b oundar y of the region. There is no s uch problem on a sphere. In what follows R is e ither a bo unded regio n of the plane or a spher e. The initial p ositio ns V 1 (0) and V 2 (0) of the t wo agents ar e assumed uniformly distributed in R . At ea ch time-step the tw o age nt s mov e distances d 1 and d 2 (whic h are fixed positive parameters) in a stra ight line (or along a geodesic on a sphere). They depart a t r andom a ngles α 1 and α 2 that a re unifor mly and indep endently distributed ov er (0 , 2 π ). The endp oints after the k − th time-step ar e V 1 ( k ) a nd V 2 ( k ) (Figure 1.1). The definition of a meeting in such a model is tr icky b ecause the probability of the tw o agents b e ing in exactly the s ame p osition at any g iven p erio d is 0. There are how ev er different wa ys of a pproximating such a meeting. One could say that the a g ents meet if the distance b etw een tw o po int s V 1 ( k ) and V 2 ( k ) is less than some ǫ . In such a definition res ults would depend on ǫ , which is undesirable. F o r this reason we choose to define a meeting during the k − th time- step when paths c r oss be tw een k and k + 1. This means that the segments (or the ”geo desic arcs ”) V 1 ( k ) V 1 ( k + 1) and V 2 ( k ) V 2 ( k + 1) intersect. W e recog nize that V 1 ( k ) and V 2 ( k ) can then b e clo se without the paths cro ssing, but at least the definitio n do es not dep end on some arbitr a ry ǫ . The po int m 1 in Figure 1.1 is an example of such a synchronous cross ing of paths. Of course the a gents ar e not at m 1 at the s ame time. In the figure the paths cross 2 asynchronously at m 2 . Much will dep end on whether V 1 ( k ) and V 2 ( k ) are uniformly distributed on R for every k . This w ill b e the c a se if R is a sphere b eca use V 1 (0) a nd V 2 (0) a re themselves uniformly distributed. If on the other hand R is a b ounded r egion o f the pla ne, then the uniformit y in the distributio ns of V 1 ( k ) and V 2 ( k ) is compromised for k > 0 by the vexing problem of the b ehavior of the agents when they hit the b oundar y of R . F or this reas on we fo cus on the probability of paths cr ossing at the first time- s tep only . W e simplify notations by letting V 1 def . = ( x 1 , y 1 ) a nd V 2 def . = ( x 2 , y 2 ) b e the initial po sitions of the tw o agents. The co rresp onding endp oints are denoted W 1 and W 2 . W e next pr o ceed with calculations when R is a bo unded regio n of the plane. 2.2. Cross ing-path probabili t y in a b ounded region of the plane. In the plane the endpo ints W 1 and W 2 are W 1 def . = V 1 + d 1 ( cos ( α 1 ) , sin ( α 1 )) (2.1) W 2 def . = V 2 + d 2 ( cos ( α 2 ) , sin ( α 2 )) (2.2) where α 1 and α 2 are p olar ang le s that ar e uniformly distributed in (0 , 2 π ). W e will finesse the reflection problem at the b oundary o f R by considering the po ssibility o f intersection on the basis of E qs. (2.1) - (2.2) even if W 1 or W 2 is outside R . In such a ca se we exa mine first w hether the intersection has o ccurred. W e then mov e, in some unsp ecified manner, the wayw ard p oint(s) back inside R . W e now define the fe asible domain F D ( V 1 , α 1 ) a s the set of p oints V that are within d 2 of the segment V 1 W 1 : F D ( V 1 , α 1 ) def . = { V = ( x, y ) such that d ( V , ( V 1 W 1 )) ≤ d 2 } (2.3) where d ( V , ( V 1 W 1 )) is the distance b etw een a po int V and the segment V 1 W 1 . The p oint V 2 m ust b e in F D ( V 1 , α 1 ) in or der for the segments V 1 W 1 and V 2 W 2 to intersect - a lthough the co ndition is not sufficient. Figure 2.1 depicts a segment V 1 W 1 and the corresp o nding feas ible domain F D ( V 1 , α 1 ). This domain is bo unded by dotted lines (a rectangle with sides d 1 and 2 d 2 with a half c ir cle of radius d 2 at each end of the rectangle ). In the figur e a p oint V 2 is in the feasible domain and the t wo s egments V 1 W 1 and V 2 W 2 int ersect. Whether there is an intersection dep ends on the angle α 2 . The r egion R is partitioned into an inner r egion R i and a bo rder region R b char- acterized by a distance to the o utside world that is either larger (for R i ) o r smaller (for R b ) than d 1 + d 2 . (The p oint is that F D ( V 1 , α 1 ) is entirely in R if V 1 ∈ R i ). W e define A [ S ] as the area of a b ounded subset S o f R 2 . With V 1 = ( x 1 , y 1 ) and V 2 = ( x 2 , y 2 ) uniformly and indep endently distr ibuted on R , we will no w calculate the probability that V 1 W 1 and V 2 W 2 int ersect. This first-step probability of in tersection dep ends only on d 1 , d 2 and R , and is noted P p ( d 1 , d 2 , R ). (The subscript p indicates a pro ba bility in the plane). W e first need the pr obability G p ( x 1 , y 1 , α 1 ) of the intersection conditionally on V 1 = ( x 1 , y 1 ) and α 1 ; G p ( x 1 , y 1 , α 1 ) is the proba bility (denoted by p 1 below) that V 2 falls in F D (( x 1 , y 1 ) , α 1 ) T R multiplied by the probability (denoted by p 2 below) of the seg ments int ersecting conditionally on V 2 falling in F D (( x 1 , y 1 ) , α 1 ) T R . (In the F D function we hav e repla ced V 1 by its co ordina tes ( x 1 , y 1 )). 3 Fig. 2.1 . Example of two int erse cting se gments V 1 W 1 and V 2 W 2 with fe asible domain F D ( V 1 , α 1 ) . Inner and b or der r e gions R i and R b ar e at distanc es to the outside world that ar e lar ger than and less than the sum d 1 + d 2 . (The distanc e d 1 + d 2 b e twe en the two b oundaries i s not to sc ale). Given the uniformity as sumption o n V 2 , the probability p 1 is then p 1 = A [ F D (( x 1 , y 1 ) , α 1 ) T R ] A [ R ] . (2.4) In order to calcula te p 2 we need to define the proba bility f p ( x 1 , y 1 , α 1 , x 2 , y 2 ) tha t V 2 W 2 int ersects V 1 W 1 , co nditionally o n V 1 = ( x 1 , y 1 ) , V 2 = ( x 2 , y 2 ) and the angle α 1 . (This proba bilit y , der ived in App endix A, is obtained by ca lculating the ma gnitude of the angle β within which α 2 m ust fall for the intersection to o ccur, a nd then dividing by 2 π ). The conditiona l pro bability p 2 of the intersection is then p 2 = R R ( x 2 ,y 2 ) ∈ F D (( x 1 ,y 1 ) ,α 1 ) T R f p ( x 1 , y 1 , α 1 , x 2 , y 2 ) dx 2 dy 2 A [ F D (( x 1 , y 1 ) , α 1 ) T R ] . (2.5) The probability G p ( x 1 , y 1 , α 1 ) is now the pro duct p 1 p 2 which s implifies to G p ( x 1 , y 1 , α 1 ) = RR ( x 2 ,y 2 ) ∈ F D (( x 1 ,y 1 ) ,α 1 ) T R f p ( x 1 , y 1 , α 1 , x 2 , y 2 ) dx 2 dy 2 A [ R ] . (2.6) 4 When V 1 = ( x 1 , y 1 ) is in R i (i.e. the feasible domain is entirely in R ) then the double integral o n the right-hand s ide of Eq. (2.6) is independent of V 1 and is noted I p ( d 1 , d 2 ), i.e. I p ( d 1 , d 2 ) def . = Z Z ( x 2 ,y 2 ) ∈ F D (( x 1 ,y 1 ) ,α 1 ) f p ( x 1 , y 1 , α 1 , x 2 , y 2 ) dx 2 dy 2 . (2.7) The qua nt ity I p ( d 1 , d 2 ) is an upp e r b o und for the double in tegral in E q. (2.6) when V 1 = ( x 1 , y 1 ) is in the b or der region R b (beca use the integration is then ov er an area smaller than the feasible doma in F D ( x 1 , y 1 )). W e now have the following result on I p ( d 1 , d 2 ), G p ( x 1 , y 1 , α 1 ), and P p ( d 1 , d 2 , R ). Proposition 2.1. We have I p ( d 1 , d 2 ) = 2 d 1 d 2 /π (2.8) and when V 1 = ( x 1 , y 1 ) is in R i , G p ( x 1 , y 1 , α 1 ) = 2 d 1 d 2 π A [ R ] . (2.9) When V 1 = ( x 1 , y 1 ) is in R b then G p ( x 1 , y 1 , α 1 ) ≤ 2 d 1 d 2 π A [ R ] . (2.10) The first-st ep pr ob abil ity of interse ct ion P p ( d 1 , d 2 , R ) s atisfi es p ℓo def . = 2 d 1 d 2 A [ R i ] π A [ R ] 2 ≤ P p ( d 1 , d 2 , R ) ≤ p hi def . = 2 d 1 d 2 π A [ R ] (2.11) which le ads to the mid-p oint appr oximation P p ( d 1 , d 2 , R ) ≅ p ∗ def . = d 1 d 2 A [ R i ] + A [ R ] π A [ R ] 2 . (2.12) The absolute value of t he maximum p er c entage err or (A VMPE) made with the appr oximation of (2.12) is AV M P E = 1 00 A [ R ] − A [ R i ] A [ R ] + A [ R i ] . (2.13) Pr o of . See Appendix A. R emark. The approximation of E q. (2 .12) is of in terest and the error of Eq. (2 .13) is small only if A [ R ] is large eno ugh in the sense that A [ R i ] is r elatively close to A [ R ]. This means there is a lar ge subset o f R within which the feasible doma in F D ( V 1 , α 1 ) is e nt irely in R . Supp ose for example that R is a c ir cle of r adius r > d 1 + d 2 , then AV M P E = 1 00 1 −  1 − d 1 + d 2 r  2 1 +  1 − d 1 + d 2 r  2 (2.14) which is a pproximately 100( d 1 + d 2 ) /r when d 1 + d 2 is muc h smaller than r . Therefore if d 1 + d 2 is one p er cent of the radius then the maximum error made with the estimate of E q. (2.12) is also approximately one p er cent. W e next tur n o ur attent ion to the situation in which R is a sphere . 5 2.3. Cross ing-path probability on a sphe re. When the domain R is a sphere of r a dius ρ we use the spherica l s ystem o f co ordinates wher e a p oint V is defined by the triplet ( ρ, θ , φ ) of radial, azimuthal, a nd zenithal co ordina tes. Given initial p oints V k def . = ( ρ, θ k , φ k ) , ( k = 1 , 2) ea ch endp oint W k is on the circle at a geo des ic distance d k from V k . The p osition of W k on the cir c le is determined by an ang le α k uniformly distr ibuted in (0 , 2 π ). See App endix B fo r the exact deriv a tion of the endp oints W k . W e let P s ( d 1 , d 2 , R ) b e the proba bilit y that the arcs V 1 W 1 and V 2 W 2 int ersect. Because the initia l po int s V k are uniformly distr ibuted, the endpo int s will also b e uniformly distributed on the spher e. Therefore P s ( d 1 , d 2 , R ) is the probability of paths crossing at every time-step. In order to calculate P s ( d 1 , d 2 , R ) we pro ceed as b efor e except tha t there is no bo rder ar e a a nd the double integrals are calculated in s pherical co or dinates. The feasible domain is denoted F D ( θ 1 , φ 1 ) and is defined with ge o desic distances . The differential area element is now ρ 2 sin ( φ ) dθdφ . W e let f s ( θ 1 , φ 1 , α 1 , θ 2 , φ 2 ) b e the pr obability of intersection conditionally o n the arc V 1 W 1 (defined by θ 1 , φ 1 and α 1 ) a nd on V 2 (defined by θ 2 and φ 2 ). W e a lso define the double int egral I s ( d 1 , d 2 , ρ ) def . = Z Z ( θ 2 ,φ 2 ) ∈ F D (( θ 1 ,φ 1 ) ,α 1 ) f s ( θ 1 , φ 1 , α 1 , θ 2 , φ 2 ) ρ 2 sin ( φ 2 ) dθ 2 dφ 2 . (2.15) The proba bilit y o f the intersection conditiona lly o n V 1 = ( ρ, θ 1 , φ 1 ) and α 1 is independent of ρ, θ 1 , φ 1 and is denoted G s ( d 1 , d 2 , ρ ). W e then hav e G s ( d 1 , d 2 , ρ ) = I s ( d 1 , d 2 , ρ ) A [ R ] (2.16) where the area A [ R ] o f the spher e a pp e a ring in the denominato r is now 4 π ρ 2 . The probability of paths cro s sing a t each time-s tep is then P s ( d 1 , d 2 , R ) = RR R ( θ 1 ,φ 1 ) ∈ R,α 1 ∈ (0 , 2 π ) G s ( θ 1 , φ 1 , α 1 ) ρ 2 sin ( φ 1 ) dθ 1 dφ 1 dα 1 2 π A [ R ] = I s ( d 1 , d 2 , ρ ) A [ R ] = I s ( d 1 , d 2 , ρ ) 4 π ρ 2 . (2.17) When ρ → ∞ the integral I s ( d 1 , d 2 , ρ ) on the spher e approaches the c o rresp onding int egral I p ( d 1 , d 2 ) = 2 d 1 d 2 /π on the plane (Eq. (2.8)). In the next pr op osition we show numerically that I s ( d 1 , d 2 , ρ ) is extr emely c lo se to 2 d 1 d 2 /π even when ρ is not par ticularly large co mpared to d 1 and d 2 . W e will just assume that d 2 ρ < π 2 , d 1 2 ρ + d 2 ρ < π (2.18) which insures that the feasible do main do es not ”wra p aro und” the spher e . 6 Proposition 2.2. With (2.18) we have I s ( d 1 , d 2 , ρ ) ≅ 2 d 1 d 2 /π (2.19) and t he pr ob ability of p aths cr ossing at e ach time-step is P s ( d 1 , d 2 , R ) ≅ 2 d 1 d 2 π A [ R ] = d 1 d 2 2 π 2 ρ 2 . (2.20) The differ enc es b etwe en b oth sides of ( 2.19) and (2.20) ar e so smal l that t hey ar e within the mar gins of err or when c alculating I s ( d 1 , d 2 , ρ ) ( and P s ( d 1 , d 2 , R ) ) numeric al ly. Pr o of . See Appendix C. The next se ction is devoted to simulations aimed at a ssessing the quality of the approximations derived ab ov e. 3. Si m ulations. W e consider a reg ion R that is a circle of r adius r = 10 . At each time-step the tw o a gents move distances d 1 = 1 and d 2 = 0 . 7 resp ectively . If an agent mov es o utside the circle we fir st c heck whether pa ths hav e crossed. W e then mov e the wa yward age nt to a p o int diametr ically opp os ed to its current po sition, a t a dista nce inside the circle e qual to the distance b etw ee n the circle and the curr ent po sition. This algor ithm co uld b e considered a 2 D version of the wrapping a round that tak es place on a sphere. The goa l is to try to k eep the distribution of the agents a s uniform as p ossible. W e need this in order for the crossing -path pr obability to rema in as close as p ossible to the ”firs t-step” probability derived under the assumption that initial p ositions are uniformly distr ibuted. The b ounds of (2.11) a nd the approximation o f (2.1 2) are now used to calculate the low, high and mid-p oint approximations p ℓo , p hi , p ∗ for the first-step pr obability of intersection P s ( d 1 , d 2 , ρ ): p ℓo = 2 d 1 d 2 A [ R i ] π A [ R ] 2 = 0 . 000 9772 , p hi = 2 d 1 d 2 π A [ R ] = 0 . 001 418 , (3.1) p ∗ = d 1 d 2 A [ R i ] + A [ R b ] π A [ R ] 2 = 0 . 00119 8 (3.2) which tr a nslate into a ma ximum er ror AV M P E on p ∗ of 18 .42% (Eq. (2.14)). T his error is relatively large b ecause the sum d 1 + d 2 is 1.7, which is not pa r ticularly sma ll compared to the radius 10 of the cir cle. W e let F ( k ) b e the r andom v ariable equal to the av erage cro ssing path frequency ov er the firs t k time-steps. If the pro ba bility of paths cr ossing at every time-s tep were p ℓo (or p hi ) and if paths cross ing w ere indep endent even ts (which they are no t) then the Central Limit Theorem would insur e that for la rge k the corre s p o nding ”hypothetical freque ncy ” F h ( k ) would b e a pproximately nor mally distributed with mean p ℓo (or p hi ) and standard deviation p p ℓo (1 − p ℓo ) /k (or p p hi (1 − p hi ) /k ). Figure 3.1 depicts a simulated tra jectory o f the freq uency F ( k ). W e als o plotted the hypothetical low a nd high int erv a ls within which each F h ( k ) would fall with probability 0 . 95 . These ba nds shed light on the exp ected fluctuatio ns o f the frequency , in the case of indep endent even ts tak ing place with proba bility p ℓo (or p hi ). This and other simulations sugg est that p ∗ gives at lea st an idea o f the (long -run) crossing - path probability . 7 0 5000 10000 15000 0 0.5 1 1.5 2 2.5 3 x 10 −3 k Frequency p ℓo − 1 . 96 p p ℓo (1 − p ℓo ) /k p ℓo + 1 . 96 p p ℓo (1 − p ℓo ) /k p hi − 1 . 9 6 p p hi (1 − p hi ) /k p hi + 1 . 9 6 p p hi (1 − p hi ) /k p ∗ Fig. 3.1 . Aver age cro ssing-p ath fr e quenc y F ( k ) (over t he first k time- steps) for two r andom walks in a cir cle of r adius r = 10 with d 1 = 1 and d 2 = 0 . 7 (15,000 time-steps). At e ach time-st e p the inte rvals wit hin which t he ”hyp othetic al fr e quency” F h ( k ) fal ls with pr ob ability 0 . 95 ar e g iven for the low and high appr oximations p ℓo and p hi of P ( d 1 , d 2 , ρ ) . The mid-p oint appr oximation p ∗ is also plotte d. There is less uncertaint y o n the spher e as we hav e s hown that the one-step prob- ability of intersection P s ( d 1 , d 2 , R ) is extr emely c lo se to d 1 d 2 / (2 π 2 ρ 2 ). Sim ulations per formed o n the sphere (not shown) yield long- r un crossing -path r ates that are close to d 1 d 2 / (2 π 2 ρ 2 ) even though the law of larg e n umbers canno t b e inv oked b ecause o f spatial dep endence ov er time. 4. Dis cussion. The expr ession of E q. (2.12) for the cro ssing-path probability in the plane can be relatively crude. How ever it ha s the merit of s implicit y and it improv es if the are a of the region R increases . On the spher e the cr ossing- path probability can b e approximated very closely by the simple expressio n d 1 d 2 / (2 π 2 ρ 2 ). W e derived this ex pression on the bas is o f an analytical result for the plane (App e ndix A), exp ecting it to b e a g o o d approximation only for a sphere of infinitely la rge radius. It is of some in terest to note that b ecause of the complexity o f the multiple int egrals in spherical co ordina tes we saw now wa y o f obtaining this approximation fro m the calculations p erfo rmed directly o n the sphere (Appendix C). Because o f marked depa r tures fro m required a ssumptions the law of larg e num b er s could b e applied neither in the plane nor on the sphere. Despite that, the long-run av erage crossing -path proba bilities a pp ea r to b e clos e to the first-step probabilities. This sug gests that a weak er version of the law of larg e n umbers ma y b e applica ble . F or exa mple results on ” weakly depe ndent” random v ariables (i.e. v ar iables that are ”m” (or ” ϕ ”)-dep endent ([9])) may pr ovide more insights into the long-r un b ehavior of the system. W e note some obvious and some less o bvious extensions that can b e of use to po pulation biologists (and p erhaps other s): 8 Fig. 5.1 . Interse cting se g ments V 1 W 1 and V 2 W 2 in co or dinate system in which V 1 W 1 lies on the x -axis and the origin i s at the midd le of V 1 W 1 . • If the crossing of paths ta kes place b etw ee n a n infected and a susc e ptible agent, then the transmissio n o f the infection may o ccur with o nly a probability τ . In such a cas e the cr ossing- path pro babilities found here need simply b e m ultiplied by τ in or der to o btain the pr obability of transmissio n at each time-step. • An imp or tant extensio n would hav e I such infectives and S such s usceptibles. Epidemiologis ts would b e keenly int erested in analytic al r esu lts on the rate at w hich the infection would then spr ead. • The assumption that an age nt moves at an ang le that is uniformly dis tr ibuted in (0,2 π ) may no t be realistic. F or example a nimals may mov e only within a limited a ngle in the contin ua tion of the pr evious direction. Preliminary inv es tigations suggest tha t the results obtained here may still b e applicable. The r esults g iven in this pap er are merely starting p oints for more in-depth theo- retical inv estigations. They also provide practitioners with s o me answers concerning the dy na mics of a pro cess that dep ends on r a ndomly moving agents meeting in a spatially explicit environment. 5. App endices. 5.1. App endix A: Pro of of Prop o sition 2. 1. The double integral I p ( d 1 , d 2 ) in Eq. (2.6) is calculated in the or thonormal co ordinate system ( x, y ) for which the segment V 1 W 1 lies o n the x -axis a nd the or ig in is a t the middle of the s egment (Figure 5.1). In the new co or dinate sys tem, the probability f p ( x 1 , y 1 , α 1 , x 2 , y 2 ) tha t V 1 W 1 and V 2 W 2 int ersect dep ends o nly on the comp onents ( x 2 , y 2 ) of V 2 . If this probability is denoted F p ( x 2 , y 2 ), then I p ( d 1 , d 2 ) is I p ( d 1 , d 2 ) = Z y = d 2 y = − d 2 Z x = d 1 / 2+ √ d 2 2 − y 2 x = − d 1 / 2 − √ d 2 2 − y 2 F p ( x, y ) dxdy . (5.1) The probability F p ( x 2 , y 2 ) is obtained by calculating the magnitude of the angle β within which V 2 W 2 (defined by the a ngle α 2 ) must fall a nd then dividing b y 2 π (see 9 Figure 5.1). In the Fig ure the p oint V 2 is in neither c ircle a nd the angle β is the vertex angle in the iso sceles triangle with a p ex V 2 and tw o sides o f leng th d 2 . Therefore, when V 2 is in neither circle the proba bility F p ( x 2 , y 2 ) of an intersection is given by the function F p, 1 ( x 2 , y 2 ) def . = 2 arccos  | y 2 | d 2  2 π . (5.2) Similar geo metric consideratio ns show that if the tw o circles ov erlap (i.e. d 1 / 2 < d 2 ) then for a p oint ( x 2 , y 2 ) in the intersection o f the tw o c ircles, the probability of int ersection F ( x 2 , y 2 ) is given by the function F p, 2 ( x 2 , y 2 ) def . = arctan  x 2 + d 1 / 2 | y 2 |  − a rctan  x 2 − d 1 / 2 | y 2 |  2 π . (5.3) Finally , if ( x 2 , y 2 ) is in only one o f the circles, then one of the vertices of the triangle with ap ex V 2 will b e the center V 1 or W 1 of that circle. The pr obability of int ersection F ( x 2 , y 2 ) is then F p, 3 ( x 2 , y 2 ) def . = arccos  | y 2 | d 2  − a rctan  x 2 − d 1 / 2 | y 2 |  2 π . (5.4) Long but elementary calculations show that the innermost integral in (5.1), co n- sidered a function H ( y ) of y , is now equal to H ( y ) def . = Z x = d 1 / 2+ √ d 2 2 − y 2 x = − d 1 / 2 − √ d 2 2 − y 2 F p ( x, y ) dx = d 1 × a rccos  | y | d 2  π . (5.5) W e therefo re hav e I p ( d 1 , d 2 ) = Z y = d 2 y = − d 2 H ( y ) dy = 2 d 1 d 2 π (5.6) which is Eq. (2.8) and yields Eq. (2.9). The inequality in (2.10) results fro m the fact that I p ( d 1 , d 2 ) is a n upper b ound for the double integral in E q . (2.6) when V 1 = ( x 1 , y 1 ) is in the b or der re g ion R b . With V 1 and V 2 uniformly distributed on R , the pro bability o f intersection P p ( d 1 , d 2 , R ) is now obtained by int egrating G p ( x 1 , y 1 , α 1 ) ov er ( x 1 , y 1 ) in R = R i S R b and ov er α 1 in (0 , 2 π ), a nd then dividing by 2 π A [ R ]: P p ( d 1 , d 2 , R ) = R R R ( x 1 ,y 1 ) ∈ R,α 1 ∈ (0 , 2 π ) G p ( x 1 , y 1 , α 1 ) dx 1 dy 1 dα 1 2 π A [ R ] = RR R ( x 1 ,y 1 ) ∈ R i ,α 1 ∈ (0 , 2 π ) G p ( x 1 , y 1 , α 1 ) dx 1 dy 1 dα 1 2 π A [ R ] 10 + RR R ( x 1 ,y 1 ) ∈ R b ,α 1 ∈ (0 , 2 π ) G p ( x 1 , y 1 , α 1 ) dx 1 dy 1 dα 1 2 π A [ R ] = 2 d 1 d 2 A [ R i ] π A [ R ] 2 + RR R ( x 1 ,y 1 ) ∈ R b ,α 1 ∈ (0 , 2 π ) G p ( x 1 , y 1 , α 1 ) dx 1 dy 1 dα 1 2 π A [ R ] . (5.7) The first term is easily calculated while the second one is a complicated triple int egral of a double int egral that ca n only b e ca lculated numerically . The inequa lity in (2.10) implies how ev er that 2 d 1 d 2 A [ R i ] π A [ R ] 2 ≤ P p ( d 1 , d 2 , R ) ≤ 2 d 1 d 2 ( A [ R i ] + A [ R b ]) π A [ R ] 2 = 2 d 1 d 2 π A [ R ] . (5.8) which is (2.11). The approximation of (2.1 2) is simply the mid-point o f the interv al in (5.8). The erro r term of Eq. (2.14) is based on the low and high b ounds of (5.8). 5.2. App endix B: Deri v ation of W 1 , W 2 . In or der to find each endp oint W k ( k = 1 , 2) we will first determine a particular p oint Q k at a geo desic distance d k from V k . Each W k is then obtained by rotating Q k by a uniformly dis tributed angle α k ab out the − − → OV k axis. Each Q k is defined as having the same azimuthal co o rdinate θ k as V k = ( ρ, θ k , φ k ) and a zenithal co ordina te φ that puts Q k at a geo desic dista nc e d k from V k . T o calculate Q k we define the function z ( φ, d ) def . =  φ + d/ ρ if φ < π − d/ρ ; φ − d/ ρ otherwise . (5.9) Then Q k def . = ( ρ, θ k , z ( φ k , d k )) . (5.10) T o obtain W k from Q k we define the normalized vectors u k def . = − − → OV k / k − − → OV k k with Cartesian co ordinates u k,x , u k,y , u k,z . W e next define the a nt isymmetric matrix A k def . =   0 − u k,z u k,y u k,z 0 − u k,x − u k,y u k,x 0   . (5.11) W e let I 3 denote the 3 × 3 identit y matrix. W e will also need the C art ( ρ, θ , φ ) and S ph ( x, y , z ) functions that transform spherical co o rdinates in to Ca rtesian ones, and vice-versa, i.e. C ar t ( ρ, θ , φ )) = ρ (cos( θ ) sin( φ ) , sin( θ ) sin( φ ) , cos( φ )) (5.12) and S ph ( x, y , z ) =  p x 2 + y 2 + z 2 , θ ( x, y , z ) , arcco s  z x 2 + y 2 + z 2  (5.13) 11 where θ ( x, y , z ) def . =    arcsin y √ x 2 + y 2 if 0 ≤ x π − arcs in y √ x 2 + y 2 otherwise . (5.14) Each W k , expres sed in spher ical c o ordinates, is now obtained by using Ro drig ues’ rotation formula, ([1]) i.e. W k def . = S ph  [ I 3 + A k sin( α k ) + A 2 k (1 − cos ( α k ))] C art ( Q k )  (5.15) where the α k ′ s are uniformly distributed in (0 , 2 π ). 5.3. App endix C: Pro of of Prop ositi on 2. 2. W e will derive an expre s sion for I s ( d 1 , d 2 , ρ ) in the spherical co o rdinate system in which the ar c V 1 W 1 lies on the equator and its middle is at the po int ( ρ, 0 , π / 2), i.e. V 1 = ( ρ, − d 1 / (2 ρ ) , π / 2) , W 1 = ( ρ, d 1 / (2 ρ ) , π / 2) . (5.16) The fea sible domain now consists of tw o ar eas on the sur face of the sphere. First the ”spherical rectang le” c ent ered on ( ρ, 0 , π / 2) with sides o f geo desic lengths d 1 and 2 d 2 ; and se cond a t b oth ends of the rectang le the half-circle s centered at V 1 and W 1 and of (g eo desic) ra dius d 2 (Figure 5.2). The feasible domain now depends only o n d 1 , d 2 and is denoted F D ( d 1 , d 2 ). 3.95 4 −0.5 0 0.5 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 W 1 W 1 W 1 h r h r h r β β β V 2 V 2 V 2 h ℓ h ℓ h ℓ V 1 V 1 V 1 Fig. 5 .2 . F e asible domain on the spher e and angle β within which V 2 W 2 must fal l for the ar cs V 1 W 1 and V 2 W 2 to interse ct. (When V 2 is in the right cir cle (r esp. the left ci rcle), h r (r esp. h ℓ ) wil l be at W 1 (r esp. V 1 ). W e need s everal functions in order to calcula te G s ( θ 1 , φ 1 , α 1 ): • The g eo desic distance function be t ween tw o p oints P 1 = ( ρ, θ 1 , φ 1 ) and P 2 = ( ρ, θ 2 , φ 2 ) on the sphere: g d ( P 1 , P 2 ) def . = 12 ρ arccos[sin( φ 1 ) sin( φ 2 )(cos( θ 1 ) cos( θ 2 ) + sin( θ 1 ) sin( θ 2 )) + co s( φ 1 ) cos ( φ 2 )] . (5.17) • If X and Y are tw o vectors (in Ca rtesian co ordina tes ) o n the sphere o f radius ρ , then the normed tangent vector at X to the g eo desic line b etw een X and Y : τ ( X , Y ) def . = Y − X T Y k X k 2 X k Y − X T Y k X k 2 X k (5.18) • The C ar t ( ρ, θ, φ ) function tha t transforms spherical co o rdinates into Carte- sian ones (Eq. (5.12)). Given V 2 = ( ρ, θ 2 , φ 2 ) in the feasible doma in F D ( d 1 , d 2 ), we let h r and h ℓ be the t wo p oints on the equator (on the right a nd on the left of V 2 ) that deter mine the magnitude o f the angle β within which V 2 W 2 m ust fall for the intersection to o c c ur (Figure 5.2). Bearing in mind V 1 and W 1 of Eq. (5 .16), the Cartesia n co o rdinates o f h r and h ℓ , co nsidered functions of V 2 = ( θ 2 , φ 2 ), are h r ( θ 2 , φ 2 ) def . = ( C ar t ( W 1 ) if g d ( V 2 , W 1 ) < d 2 ρ  cos  θ 2 + a rccos cos( d 2 /ρ ) sin( φ 2 )  , sin  θ 2 + a rccos cos( d 2 /ρ ) sin( φ 2 )  , 0  otherwise . (5.19) h ℓ ( θ 2 , φ 2 ) def . = ( C ar t ( V 1 ) if g d ( V 2 , V 1 ) < d 2 ρ  cos  θ 2 − a rccos cos( d 2 /ρ ) sin( φ 2 )  , sin  θ 2 − a rccos cos( d 2 /ρ ) sin( φ 2 )  , 0  otherwise . (5.20) In the new co or dinate system, the proba bility of intersection f s ( θ 1 , φ 1 , α 1 , θ 2 , φ 2 ) conditionally on the arc V 1 W 1 and on V 2 depe nds only on the co mpo nent s ( ρ, θ 2 , φ 2 ) of V 2 and is denoted F s ( θ 2 , φ 2 ). Given the uniformity assumption for the ang le α 2 at w hich the seco nd agent leav es V 2 to go to W 2 , the pr obability is equal to the magnitude of the angle β within which the ar c V 2 W 2 m ust fa ll for the int ersection to o ccur, divided by 2 π . The angle β is the a ngle b etw een the tangent vectors at V 2 in the directions of h ℓ ( θ 2 , φ 2 ) and h r ( θ 2 , φ 2 ). The probability of intersection F s ( θ 2 , φ 2 ) is ther efore F s ( θ 2 , φ 2 ) = arccos  τ [ C ar t ( ρ, θ 2 , φ 2 ) , h r ( θ 2 , φ 2 )] T τ [ C ar t ( ρ, θ 2 , φ 2 ) , h ℓ ( θ 2 , φ 2 )]  2 π (5.21) The double integral I s ( d 1 , d 2 , ρ ) is now calculated in the new co o rdinate system by integrating F s ( θ 2 , φ 2 ) over the feasible doma in F D ( d 1 , d 2 ). Becaus e o f s ymmetries the integral I s ( d 1 , d 2 , ρ ) is the sum of four times the integral over the upp er rig ht 13 T a ble 5.1 Per centage err or 100 × “ 2 d 1 d 2 /π I s ( d 1 ,d 2 ,ρ ) − 1 ” when appr oximating I s ( d 1 , d 2 , ρ ) as 2 d 1 d 2 /π , with d 1 = 3 and an il lustr ative r ange of values for d 2 and ρ . d 2 ↓ ; ρ → 2 3 4 5 1 4 . 866 × 1 0 − 4 − 1 . 68 × 10 − 4 − 2 . 527 × 1 0 − 4 − 8 . 471 × 1 0 − 4 2 1 . 288 × 1 0 − 3 5 . 595 × 10 − 4 9 . 269 × 10 − 4 6 . 658 × 10 − 4 3 − 1 . 506 × 10 − 5 7 . 529 × 10 − 4 9 . 412 × 10 − 5 8 . 954 × 10 − 4 quarter of the r ectangular ar ea of F D ( d 1 , d 2 ) and o f four times the integral ov er the upper half of the rig ht cir c le. W e thus hav e I s ( d 1 , d 2 , ρ ) = 4 ρ 2 Z d 1 / (2 ρ ) θ 2 =0 Z π / 2 φ 2 = π / 2 − d 2 /ρ F s ( θ 2 , φ 2 ) sin ( φ 2 ) dθ 2 dφ 2 + 4 ρ 2 Z d 1 / (2 ρ )+ d 2 /ρ θ 2 = d 1 / (2 ρ ) Z π / 2 φ 2 = φ ( θ 2 ) F s ( θ 2 , φ 2 ) sin ( φ 2 ) dθ 2 dφ 2 (5.22) where φ ( θ 2 ) def . = arcs in   cos d 2 ρ cos  θ 2 − d 1 2 ρ    (5.23) is the low er v a lue of φ 2 when integrating in the rig ht ha lf cir cle. Over a wide range of v alues for d 1 , d 2 and ρ we found that the r elative err or made when a pproximating I s ( d 1 , d 2 , ρ ) o f Eq. (5.22) a s 2 d 1 d 2 /π was o f the or der of 10 − 3 to 10 − 5 p er c ent . See T able 5.1 for an example of this relative error for a r ange of v a lue s of d 2 and ρ . In fact the erro r is so sma ll that it is within the mar g in o f error when calculating I s ( d 1 , d 2 , ρ ) numerically . The fact that I s ( d 1 , d 2 , ρ ) ≅ 2 d 1 d 2 /π combined with Eq. (2 .17) yields the r e sult o f (2.2 0). REFERENCES [1] S. 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