A Fast Algorithm for the Spread of HIV in a System of Prisons

Transcription

1 A Fast Algorith for the Spread of HIV in a Syste of Prisons Wai-Ki Ching Yang Cong Tuen-Wai Ng Allen H Tai Abstract In this paper, we propose a continuous tie odel for odeling the spread of HIV in a network of prisons We give soe sufficient conditions for the equilibriu points of the syste to be stable We also develop an efficient algorith based on Newton s ethod and the Sheran-Morrison-Woodbury Forula for coputing the equilibriu values of the infectives in each prison Key Words: HIV, Prison Syste, Epideic Model, Equilibriu Point, Newton s Method, Sheran-Morrison-Woodbury Forula 1 Introduction The spread of HIV has captured the attention of a lot of researchers There have been any research papers on the odeling of the spread of HIV in the counity Greenhalgh and Lewis [6] develop a odel for the spread of HIV aongst a population of injecting drug users The odel they discuss focuses on the transission of HIV through the sharing of containated drug injection equipent and in particular they exaine the ixing of addicts and needles when the AIDS incubation period is divided into distinct infectious stages Huang and Villasans [10] extended the Kerack-Mckendrick odel for the spread of HIV Their odel is an SI type epideic odel which takes preventive education into account Global behavior of an SEIRS epideic odel and SIR odel with tie delays have been studied by Wang [21] and Ma et al [14] respectively A discrete branching process odel was proposed by Knolle [13] for odeling the spread of HIV via steady sexual partnerships Advanced Modeling and Applied Coputing Laboratory, Departent of Matheatics, The University of Hong Kong, Pokfula Road, Hong Kong E-ail: wching@hkusuahkuhk Research supported in part by RGC Grants, HKU CRCG Grants, Hung Hing Ying Physical Sciences Research Fund and HKU Strategic Research Thee Fund on Coputational Physics and Nuerical Methods Departent of Applied Matheatics, Dalian University of Technology, Dalian, China Eail:congyang@studentdluteducn Advanced Modeling and Applied Coputing Laboratory, Departent of Matheatics, The University of Hong Kong, Pokfula Road, Hong Kong E-ail: ntw@athshkuhk Departent of Industrial and Manufacturing Systes, The University of Hong Kong, Pokfula Road, Hong Kong E-ail:allenhtai@graduatehkuhk 1

2 A serious proble of prison life is the spread of HIV by both sexual contacts and needle sharing activities aong the prisoners In [4], it is entioned that a special report in the New York Ties notes that the nuber of HIV infectives in Argentine federal prisons is of the order 30% of all inates and suggests that local and rural prisons have siilar prevalence, as do prisons in other Latin Aerican countries, including Brazil Motivated by the report, Gani et al [4] proposed deterinistic epideic odels in both continuous tie and discrete tie and also stochastic odels in discrete tie for odeling the spread of HIV in a prison syste Recently, due to the new results in coplex networks, we understand better our social networks and the Internet [15] It is entioned in [3] that by 2030, ore than 60% people in the world will live in cities They are connected by different transportation ethods Dangerous diseases such as SARS can be spread very fast [1, 17] Researchers have paid ore attention to the spread of diseases and coputer viruses through a network [3] With these otivations, we extend the deterinistic continuous tie odel of [4] to the case of a network of prisons We first review a continuous deterinistic odel for one prison syste in [4] We then give the outline of the paper The odel in [4] considers a prison of N prisoners in continuous tie At tie point t, there are siultaneous inflows and outflows of n prisoners where n < N After that there are yt) prisoners who are HIV + and N yt)) susceptibles We then assue that hoogeneous ixing occurs in the prison The odel further assues that the new infectives produced is proportional to yt) N yt) ) with a proportioned constant β, for instance [2, p 9] This eans the nuber of new infectives produced is β yt) N yt) ) They also assue an inflow of n new prisoners joining the prison fro the outside world, where the proportion of infectives in the outside world is µ0 < µ < 1) Thus there are nµ infectives added to the prison At the sae tie, after ixing with the prisoners, an outflow of n prisoners leaves the prison of which ) yt) n N are infectives Thus the differential equation governing the infectives in the syste is given by dy = nµ n N yt) + βyt) N yt) ) = β η 1 yt) ) η 2 + yt) ) where η 1 = 1 2 { A + A 2 + 4nµ }, η 2 = 1 { A A β nµ } β 2

3 and A = N Solving the ordinary differential equation, we get n βn yt) = η 1Ke βη 1+η 2 )t η Ke βη 1+η 2 )t where K = η 2 + y0) η 1 y0) and li yt) = η 1 1) t The reainder of the paper is organized as follows In Section 2, we consider a continuous tie two-prison syste In Section 3, we extend our odeling to a general network of prisons We develop a fast algorith based on Newton s ethod and the Sheran- Morrison-Woodbury forula for coputing the liiting values of the infectives in each prison Nuerical exaples are given in Section 4 Finally, there are soe concluding rearks in Section 5 2 The Two-Prison Case In this section, before we present a general network of prisons, let us first consider a two-prison syste A discrete tie odel for a two-prison syste has been proposed in [4] Here we consider a continuous tie odel for a two-prison syste Both prisons are allowed to have interactions with the outside world In the odel, we assue that the death rate is relatively insignificant We define the following notations for our odel 1 N i, the nuber of prisoners in Prison i, i = 1, 2 2 y i t), the nuber of infectives in Prison ii = 1, 2) at tie t 3 β i, the infection rate in Prison ii = 1, 2) 4 µ, the ean proportion of infectives in the outside world 5 n i, the nuber of prisoners oving in and out fro the outside world to Prison ii = 1, 2) 6, the nuber of prisoners exchanged between the two prisons With the above notations, the syste of ordinary differential equations governing the nuber of infectives in each prison is given by dy 1 t) dy 2 t) = n 1 µ + β 1 y 1 t) y 1 t)) n 1y 1 t) = n 2 µ + β 2 y 2 t) y 2 t)) n 2y 2 t) y 1t) y 2t) + y 2t) + y 1t) 2) 3

4 Re-arranging the ters we have dy 1 t) dy 2 t) In equilibriu, we have = n 1 µ + β 1 n 1 + )y 1 t) β 1 y N 1t) 2 + y 2t) 1 = n 2 µ + β 2 n 2 + )y 2 t) β 2 y2 2 N t) + y 1t) 2 0 = F 1 y1, y 2 ) = n 1µ + β 1 n 1 + )y1 N β 1y y = F 2 y1, y2) = n 2 µ + β 2 n 2 + )y2 β 2 y y 1 3) 4) To solve for the equilibriu points, one ay apply Newton s ethod [12, p 586] Let y 10) and y 20) be the initial guesses, then the iterative schee reads: y1 n + 1) y2 n + 1)! = y 1 n) y 2 n)! 2β 1y1 n) + β 1 n 1+ 2β y 1 2 n) + β 2 n 2+! 1 F 1 y 1 n), y 2 n)) F 2 y 1 n), y 2 n))! To classify the equilibriu points, one ay follow the analysis in [19, p 261] and [8, 92] Proposition 1 Let y 1,y 2 ) be the non-negative equilibriu point, if = and then the equilibriu point is a stable one β i < n i i Proof: We consider the atrix 2β 1 y 1 + β 1 n 1+ 2β 2 y2 + β 2 n 2+ ) By applying the Gershgorin disc theore [5, p 341] to the atrix above, the real part of its eigenvalues lie in the union of the two intervals: and 2β 1 y1 + β 1 n 1 +, 2β 1 y1 N + β 1 n ), 2β 1 y1 2 N ) 2 2β 2 y2 + β 2 n 2 +, 2β 2 y2 + β 2 n ), 2β 2 y N 2) 1 Therefore the real part of the eigenvalues are negative and the equilibriu point is stable [8] We reark that one can derive siilar results for the case 4

5 Exaple 1 Consider the following exaple with paraeters = 500, = 200, n 1 = 50, n 2 = 20, = 10, µ = 01, β 1 = β 2 = 00003, y 1 0) = 250, y 2 0) = 100 Newton s ethod converges in a few steps and the equilibriu point is y1,y 2 ) = ,319999) 59,32) Moreover, we have tried a nuber of different initial guesses, and Newton s ethod still converges very fast to the sae solution 3 A Network of Prisons In this section, we consider a network of prisons A siple epideic odel for interacting groups has been studied by Rushton and Mautner [18], see also [2, pp 23-27] We assue there are interactions aong the prisons and also with the outside world Let us give the notations of our odel as follows: 1 N i, the nuber of prisoners in Prison ii = 1,2,,s) 2 y i t), the nuber of infectives in Prison ii = 1,2,,s) at tie t 3 β i, the infection rate in Prison ii = 1,2,,s) 4 µ, the ean proportion of infectives in the outside world 5 n i, the nuber of prisoners oving in and out fro the outside world to Prison ii = 1,2,,s) For siplicity of discussion, we assue that the prisoners are exchanged aong the s prisons so that the nuber of prisoners oving out to other prisons is equal to the nuber of prisoners received fro other prisons We assue that there is a echanis to ix up the prisoners uniforly and we have for i = 1,2,,s dy i t) = n i µ + β i y i t)n i y i t)) n iy i t) N i s 1)y it) N i + k=1,k i y k t) N k 5) In equilibriu, we have 0 = F i y 1,y 2,,y s) = n i µ + β i N i n i + s 1) N i )y i β i y i 2 + k=1,k i y k N k 6) We reark that by Bezout s theore [7, p 87], there are at ost 2s equilibriu points Newton s ethod is a popular and effective ethod for solving non-linear syste of equations [11, 16] Other ethods such as inexact Newton s ethod and Broyden s ethod can be found in [11] 5

6 Since for our proble, Newton s ethod works very well, we will only focus on Newton s ethod When Newton s ethod is applied to solve equations 6), the atrix syste to be solved is given by D + E) where D is a diagonal atrix D ii = 2β i y i + β in i n i + s N i 7) We reark that if β i < n i / i then the atrix D is invertible The atrix E is a rank one atrix given by 1,1,,1) t,,, N s ) Since E is a rank one atrix, the linear syste can be solved by using the Sheran-Morrison- Woodbury Forula Proposition 2 Sheran-Morrison-Woodbury Forula) [5, p 51] Let M be an non-singular r r atrix, u and v be two r l l r) atrices such that the atrix I l +v t Mu) is non-singular Then we have M + uv t ) 1 = M 1 M 1 u I l + v t M 1 u ) 1 v t M 1 Thus by using Proposition 2, we have D + E) 1 = D 1 D 1 1,1,,1) t 1 + i N i D ii ) 1,,, N s )D 1 or D + E) 1 = D ) 1 1,, N i D ii D 11 1 ) t,, D ss D 11 For Newton s ethod, we have the following convergence theore N s D ss ) 8) Proposition 3 Kantorovich Theore [9, p 249]) Let a be a point in R K, U be an open neighborhood of a in R K and F : U R K be a differential apping with its derivative [DFa)] being invertible Define h = [DFa)] 1 Fa), a 1 = a + h and U 0 = B h a 1 ) If U 0 U and the derivative [DFx)] satisfies the condition DFu 1 ) DFu 2 ) M u 1 u 2 for all points u 1 and u 2 and if the inequality Fa) DFa) 2 M 1 2 is satisfied, then the equation Fx) = 0 has a unique solution in U 0 and Newton s ethod converges with an initial guess a Reark 1 If we set a = 1 2,,,N s ) t as the initial guess then it can be shown that we 6

7 can take see [9, p 246]) Moreover, we have Fa) 2 = M 2 = 4 βi 2 n i µ 1 2 ) + β ini 2 ) 2 4 By applying the Gershgorin disc theore to 7), we have DFa) 2 Thus by Proposition 3, a sufficient condition for Newton s ethod to be convergent with the initial guess a is β 2 i ) s n i µ 1 2 ) + β i i 4 Ni n i ) 2 ) 2 ) s In view of Proposition 1, it is straightforward to show that Ni n i ) Proposition 4 Let y 1,y 2,,y s) be the non-negative equilibriu point, if = = = N s and β i < n i i then the equilibriu point is a stable one 4 Nuerical Exaples In this section, we present soe nuerical exaples on the equilibriu points for different syste paraeters In all the nuerical tests, we assue each N i = 500 and n i = i 10 for i = 1,2,,8 The initial guess is taken to be ,500,,500) for all cases The stopping criterion for all the nuerical results is the following: Fy 1,y 2,,y s) We reark that Newton s ethod converges in a few steps in all the tested cases In the first three tables, β i = and for each value of µ = 001,005 and 01, we solve for the following 9 cases of equilibriu points s = 2,4,8 and = 10,20,40 7

8 Table 1: The Equilibriu Points of the Infectives when µ = 001 = 10 81,72) 68,64,62,61) 59,58,58,57,56,56,56,55) = 20 78,73) 65,64,62,61) 58,57,57,57,56,56,56,56) = 40 77,74) 64,63,62,62) 57,57,57,56,56,56,56,56) Table 2: The Equilibriu Points of the Infectives when µ = 005 = ,351) 333,318,307,300) 295,290,286,283,280,278,276,275) = ,356) 321,314,308,303) 287,285,283,282,280,279,278,276) = ,359) 315,311,308,305) 284,283,282,281,280,279,278,278) the results are reported in Tables 1, 2 and 3 We observe that when µ increases, all the nubers of infectives in each of the prisons in equilibriu are also increased When we consider the connection with the outside world, µ increases eans the inflow of infectives by exchanging with the outside world is also increased For each fixed s, when is increased, the variation of the infectives in equilibriu aong the prisons is decreased When is increased, the interactions between prisons are ore frequent and we expect this will decrease the variation of the infectives in equilibriu For the nuerical tests in Table 4, we assue s = 4,µ = 01 and we calculate the equilibriu points for the cases where β i = , and for = 10,20,40 We observe that for each fixed β i, when is increased, the variation of the infectives in equilibriu aong the prisons will be decreased Siilar to the previous arguent, an increase in enhance the exchanges between the prisons Moreover, there is a significant jup in the nuber of infectives when β i increases fro to One ay further consider the case that β i to be different fro one each other To siplify this case, for each, we calculate the equilibriu points for the following three cases three rows): β 1 changes fro , ,000003, while β i = for i 1 Then we get Tables 5, 6 and 7 for µ = 001,005 and 01 respectively We observe the following phenoenon For each fixed β i, when is increased, the less variation of the infectives in equilibriu aong the prisons is observed The reason for this is when is is increased, the interactions between prisons are ore frequent Moreover, the infectives in equilibriu increases when µ increases and this is expected 5 Concluding Rearks In this paper, we have proposed a continuous tie odel for odeling the spread of HIV in a network of prisons We give soe sufficient conditions for the equilibriu points of the syste 8

9 Table 3: The Equilibriu Points of the Infectives when µ = 01 = ,681) 651,624,606,592) 583,575,568,562,557,553,549,546) = ,689) 631,618,607,597) 570,566,562,559,556,554,551,549) = ,695) 620,613,607,601) 563,561,559,557,556,554,553,552) Table 4: The Equilibriu Points of the Infectives when s = 4 β i = β i = β i = = ,502,501,501) 519,516,514,512) 753,707,675,651) = ,502,501,501) 517,515,514,513) 718,695,676,661) = ,501,501,501) 515,515,514,513) 697,686,676,667) Table 5: The Equilibriu Point of the Infectives when µ=001 50, 50) 50, 50, 50, 50) 50, 50, 50, 50, 50, 50, 50, 50) = 10 52, 51) 51, 50, 50, 50) 51, 50, 50, 50, 50, 50, 50, 50) 90, 63) 65, 55, 54, 53) 56, 51, 51, 51, 51, 51, 51, 51) 50, 50) 50, 50, 50, 50) 50, 50, 50, 50, 50, 50, 50, 50) = 20 52, 51) 51, 50, 50, 50) 50, 50, 50, 50, 50, 50, 50, 50) 79, 65) 60, 54, 54, 54) 54, 51, 51, 51, 51, 51, 51, 51) 50, 50) 50, 50, 50, 50) 50, 50, 50, 50, 50, 50, 50, 50) = 40 52, 51) 51, 50, 50, 50) 50, 50, 50, 50, 50, 50, 50, 50) 73, 66) 57, 54, 54, 54) 52, 51, 51, 51, 51, 51, 51, 51) Table 6: The Equilibriu Point of the Infectives when µ= ,251) 251, 251, 251, 251) 251, 251, 250, 250, 250, 250, 250, 250) = ,254) 256, 252, 252, 252) 253, 251, 251, 251, 251, 251, 251, 251) 426,310) 320, 271, 268, 266) 260, 258, 257, 256, 256, 255, 255, 254) 251,251) 251, 251, 251, 251) 251, 250, 250, 250, 250, 250, 250, 250) = ,255) 254, 252, 252, 252) 252, 251, 251, 251, 251, 251, 251, 251) 383,317) 296, 271, 269, 267) 268, 256, 256, 256, 255, 255, 255, 255) 251,251) 251, 251, 251, 251) 250, 250, 250, 250, 250, 250, 250, 250) = ,256) 253, 252, 252, 252) 251, 251, 251, 251, 251, 251, 251, 251) 357,322) 283, 270, 269, 268) 262, 256, 256, 256, 255, 255, 255, 255) 9

10 Table 7: The Equilibriu Point of the Infectives when µ=01 503, 502) 502, 502, 501, 501) 501, 501, 501, 501, 501, 501, 501, 501) = , 508) 511, 504, 504, 503) 506, 502, 502, 502, 502, 501, 501, 501) 805, 603) 628, 539, 533, 529) 555, 513, 512, 511, 510, 509, 509, 509) 503, 502) 502, 501, 501, 501) 501, 501, 501, 501, 501, 501, 501, 501) = , 510) 508, 504, 504, 504) 504, 502, 502, 502, 502, 501, 501, 501) 737, 619) 586, 539, 535, 532) 533, 512, 511, 511, 510, 510, 511, 501) 502, 502) 502, 501, 501, 501) 501, 501, 501, 501, 501, 501, 501, 501) = , 511) 507, 504, 504, 504) 503, 502, 502, 502, 502, 502, 502, 501) 693, 629) 562, 538, 536, 534) 522, 511, 511, 511, 510, 510, 510, 510) to be stable Newton s ethod with the Sheran-Morrison-Woodbury Forula are used to copute the equilibriu values of the infectives Nuerical results indicate that the ethod is efficient Fro the tested nuerical exaples, we observe that the odel has a unique positive equilibriu point We seek a atheatical proof for the existence and uniqueness of a positive equilibriu point in the network The odel proposed here is of the SI type; it is therefore interesting to extend it to the SIR type epideic odel [2] and also the case of quarantining infected prisoners Moreover, the setting of the proble can be a general urban social network [3] In our odel, we assue that the nuber of prisoners exchanged aong the prisons are the sae One ay further extend the result to the case when this restriction is reoved References [1] W Ching, T Ng and S Chung, On Modeling SARS in Hong Kong, International Journal of Applied Matheatics, 13, 1 7, 2003) [2] D Daley and J Gani, Epideic odelling : An Introduction, Cabridge University Press, Cabridge, 1999) [3] S Eubank, H Guclu, V Kuar, M Marathe, A Srinivasan, Z Toroczkai and N Wang, Modelling Disease Outbreaks in Realistic Urban Social Networks, Nature, 429, , 2004) [4] J Gani, S Yakowitz and M Blount, The Spread and Quarantine of HIV Infection in a Prisoner Syste, SIAM J Appl Math, 57, , 1997) [5] G Golub and C van Loan, Matrix Coputations, 2nd Edition, The John Hopkins University Press, London, 1989) [6] D Greenhalgh and F Lewis, The General Mixing of Addicts and Needles in a Variableinfectivity Needle-sharing Environent, Journal of Matheatical Biology, 44, , 2002) 10

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