Key words: HIV, prison system, epidemic model, equilibrium point, Newton s method, screening policy, quarantine policy.

Transcription

1 COMMUICATIOS I IFORMATIO AD SYSTEMS c 27 International Press Vol. 7, o. 4, pp , 27 1 O OPTIMAL SCREEIG AD QUARATIIG POLICY I A ETWORK OF PRISOS WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G Abstract. In tis paper, we propose matematical models for te spread of HIV in a network of prisons. We study te effect of bot screening prisoners and quarantining infectives. Efficient algoritms based on ewton s metod are ten developed for computing te equilibrium values of te infectives in eac prison. We also give an optimization formulation for obtaining te optimal screening and quarantine policy. Te models and algoritms developed can be extended to model te spread of a disease in a general network of connected zones. Key words: HIV, prison system, epidemic model, equilibrium point, ewton s metod, screening policy, quarantine policy. 1. Introduction. Modelling te spread of HIV is an important and interesting topic for a lot of researcers. Many matematical models ave been proposed by different researcers, see for instance Daley and Gani [4], Greenalg and Lewis [8], Huang and Villasans [9], Wang [15] and Ma et al. [11]. A serious problem of prison life is te spread of HIV by bot sexual contacts and needle saring activities among te prisoners. In a special report of te ew York Times, it was mentioned tat te number of HIV infectives in Argentine federal prisons is of te order 3% of all inmates [6]. Due to te new results in complex networks, now we understand better our social networks [12]. Moreover, people are now connected in efficient transportation networks, dangerous diseases suc as SARS can be spread very fast [3, 13]. Researcers ave paid more attention to te spread of diseases in a network [5]. Motivated by te report and te models in [6], Cing et al. [2] proposed a fast numerical algoritm for solving te equilibrium values of te spread of HIV in a network of prisons. However, teir model does not consider te impact of screening and quarantining polices. In tis paper, we develop models and numerical algoritms for studying suc policies as Advanced Modeling and Applied Computing Laboratory, Department of Matematics, Te University of Hong Kong, Pokfulam Road, Hong Kong. wcing@kusua.ku.k. Researc supported in part by RGC Grants 717/7P, HKU CRCG Grants, Hung Hing Ying Pysical Sciences Researc Fund and HKU Strategic Researc Teme Fund on Computational Pysics and umerical Metods. Advanced Modeling and Applied Computing Laboratory, Department of Matematics, Te University of Hong Kong, Pokfulam Road, Hong Kong. congyang@kusua.ku.k Department of Information and Computational Matematics, Xiamen University, Xiamen 3615, People s Republic of Cina zjbai@xmu.edu.cn. Tis autor s researc was partially supported by te ational atural Science Foundation of Cina Grant and Program for ew Century Excellent Talents in Xiamen University. Advanced Modeling and Applied Computing Laboratory, Department of Matematics, Te University of Hong Kong, Pokfulam Road, Hong Kong. ntw@mats.ku.k. 313

2 314 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G tey are useful in reducing te transmission of te disease like HIV in a social network. We remark tat te models and algoritms developed can be easily extended to model a spread of a disease in a general network of connected zones. Te paper is structured as follows. In Section 2, we present bot discrete time and continuous time models for te spread of HIV in one prison system wit screening prisoners and quarantining infectives. We give a sufficient condition for te existence of te system equilibrium. umerical examples are given to demonstrate te effect of screening prisoners and quarantining infectives. We give a matematical formulation for finding optimal screening and quarantining policy. In section 3, we extend te model and te teory to a network of prisons. Bot discrete time and continuous time models are discussed. ewton s metod is ten applied to solving te equilibrium values. A sufficient condition is derived for te existence of te inverse of te Jacobian matrix in ewton s metod. We also give a discussion on bot te stability of te equilibrium and te convergence rate of te ewton s metod. umerical experiments sow tat ewton s metod converges very fast. We ten present a matematical formulation for finding optimal screening and quarantining policy. Finally concluding remarks are given in Section Te One-Prison Quarantine Model. We consider a prison consisting of nmates and it is subject at time t (t =, 1, 2,...) to a simultaneous inflow and outflow of n < prisoners. Ten screening and quarantining are taken, after wic tere are y(t) prisoners wo are infected but not identified, m(t) prisoners wo are detected HIV positive but not quarantined, q(t) prisoners wo are quarantined and x(t) prisoners are susceptible. It is clear tat x(t) = y(t) m(t) q(t). Here we assume tat omogeneous mixing occurs in te prison during te interval (t, t + 1) and we furter assume tat te new infectives produced is proportional to x(t)(y(t) + m(t)) wit a constant β. Tis means tat te number of new infectives produced is βx(t)(y(t) + m(t)). Tus at time t + 1, tere are y(t) + m(t) + βx(t)(y(t) + m(t)) infectives not quarantined and tere are y(t) + βx(t)(y(t) + m(t)) infectives not detected. For tose detected HIV positive, tey will be recorded witin te prison and will not be screened next time. We also assume tat an inflow of n new prisoners from te outside world suc tat te proportion of infectives is µ. Tis means tat tere are nµ infectives added to te prison. ow a proportion of

3 O OPTIMAL SCREEIG AD QUARATIIG POLICY 315 τ( < τ < 1) of te prisoners including tose incoming prisoners but excluding tose quarantined and detected HIV positive will be screened. Ten a proportion of κ of tose detected HIV positive will be quarantined. ow at time t+1, after omogeneous mixing, tere are y(t) + βx(t)(y(t) + m(t)) new infectives but not detected prisoners, m(t) detected but not quarantined prisoners. After excanging wit te outside world, tere are (1 n )(y(t) + βx(t)(y(t) + m(t))) + nµ infected but not detected prisoners, and (1 n )m(t) detected but not quarantined prisoners. We ten screen a proportion of τ of te prisoners. Terefore we get (1) y(t + 1) = (1 τ)((1 n )(y(t) + βx(t)(y(t) + m(t))) + nµ) = (1 τ)(1 n )(y(t) + βx(t)(y(t) + m(t))) + (1 τ)nµ infected but not detected, and (1 n )m(t) + τ((1 n )(y(t) + βx(t)(y(t) + m(t))) + nµ) detected prisoners. A proportion of κ of te detected prisoners is ten quarantined. We ave (2) m(t + 1) = (1 κ)(1 n )m(t) +(1 κ)τ(1 n )(y(t) + βx(t)(y(t) + m(t))) + (1 κ)τnµ detected but not quarantined prisoners and (3) q(t + 1) = (1 n )q(t) + κ(1 n )m(t) +κτ(1 n )(y(t) + βx(t)(y(t) + m(t))) + τκµn quarantined prisoners. Tus we ave te following difference equations: y(t + 1) = (1 τ)(1 n )(y(t) + βx(t)(y(t) + m(t))) + (1 τ)nµ m(t + 1) = (1 κ)(1 n )m(t) (4) +(1 κ)τ(1 n )(y(t) + βx(t)(y(t) + m(t))) + (1 κ)τnµ q(t + 1) = (1 n )q(t) + κ(1 n )m(t) +κτ(1 n )(y(t) + βx(t)(y(t) + m(t))) + τκµn x(t + 1) = y(t + 1) m(t + 1) q(t + 1) umerical Demonstration of te Effect of Screening and Quarantining. We give an example to illustrate te effect of screening and quarantining infected prisoners. We assume = 5, β =.5, n = 5, y() = 5, µ =.1, τ =.1 and κ =.1.

4 316 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G Fig. 1. A Comparison. In Figure 1, te upper curve represents te number of infectives wen tere is no screening and quarantining of prisoners (282 infectives in equilibrium state). Suppose a screening policy of 1% is implemented and among tose detected infectives 1% of tem are quarantined. Te second curve represents te total number of infectives but not quarantined (145 in equilibrium) and te lowest curve represents te total number of quarantined prisoners (55 in equilibrium). Tus te policy of screening 1% of te prisoners and quarantining 1% of te infectives can reduce 82 infectives provided tat around 1% of te space is available for quarantining infected prisoners. Ten in equilibrium, we ave lim y(t) = y, lim t m(t) = m, lim t q(t) = q, lim t x(t) = x. t Here we ave (5) y m n q = (1 τ)(1 n )(y + βx(y + m)) + (1 τ)nµ = (1 κ)(1 n )m + (1 κ)τ(1 n )(y + βx(y + m)) + (1 κ)τnµ = κ(1 n )m + κτ(1 n )(y + βx(y + m)) + τκµn x = y m q. Let z = y + βx(y + m), we ave (6) y (1 τ)(1 n )z = (1 τ)nµ (κ + n κn )m (1 κ)τ(1 n )z = (1 κ)τnµ n q κ(1 n )m κτ(1 n )z = τκµn q + x + y + m =.

5 O OPTIMAL SCREEIG AD QUARATIIG POLICY 317 Simplifying te equations in (6), we ave (7) y = (1 τ)nµ + (1 τ)(1 n )z a 1 + b 1 z ( (1 κ)τnµ (1 κ)τ(1 n m = κ + n κn + ) ) κ + n κn z a 2 + b 2 z q = τκµ + τµκ(1 κ)(1 n ) κ + n κn a 3 + b 3 z x = z y β(y + m). + n ( κτ(1 κ)(1 n )2 κ + n κn + κτ(1 n ) ) z Using te fact tat x + y + m + q =, we ave (a 1 + a 2 + a 3 ) + (b 1 + b 2 + b 3 )z + We get te following equation: z a 1 b 1 z β((a 1 + a 2 ) + (b 1 + b 2 )z) =. (8) β(a 1 + a 2 ) + β(b 1 + b 2 )z = β(a 1 + a 2 + a 3 )((a 1 + a 2 ) + (b 1 + b 2 )z) +β(a 1 + a 2 )(b 1 + b 2 + b 3 )z +β(b 1 + b 2 + b 3 )(b 1 + b 2 )z 2 + (1 b 1 )z a 1. Tus we obtain a quadratic equation: (9) Az 2 + Bz + C = were A = β(b 1 + b 2 )(b 1 + b 2 + b 3 ) ( ) B = 1 b 1 + β (a 1 + a 2 )(b 1 + b 2 + b 3 ) + (b 1 + b 2 )( a i ) C = β(a 1 + a 2 )( a i ) a 1. Solving z, one can solve te equilibrium solution. We ave te following proposition and te proof can be found in Appendix. Proposition 1. A sufficient condition for te quadratic equation (9) to ave a unique positive root (terefore te existence of te equilibrium point of (4)) is µ < 2 + n. From te results of te discrete time model, it is straigtforward to develop te

6 318 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G model to te continuous case as follows: dy(t) = ((1 τ)(1 n ) dt ) 1 y(t) (1) dm(t) dt dq(t) dt dx(t) dt = +(1 τ)(1 n )βx(t)(y(t) + m(t)) + (1 τ)µn ((1 κ)(1 n ) ) 1 m(t) +τ(1 κ)(1 n )(y(t) + βx(t)(y(t) + m(t))) + (1 κ)τµn = n q(t) + κ(1 n )m(t) +κτ(1 n )(y(t) + βx(t)(y(t) + m(t))) + τκµn = dy(t) dt In equilibrium, we ave dy(t) dt dm(t) dt It is straigtforward to sow tat = dm(t) dt dq(t). dt = dq(t) dt = dx(t) dt =. Proposition 2. Te equilibrium solutions of (1) satisfy te equations in (5). Since te analysis of te continuous model in our context is similar to te discrete model, we will focus on te discrete model only. For te stability of te equilibrium, we ave te following result and te proof can be found in Appendix. Proposition 3. Under te condition of Proposition 1, te equilibrium point of (4) determined by (9) is asymptotically stable. We end tis subsection by presenting te numerical results of equilibrium solutions wen = 5, β =.5, n = 5, and µ =.1 for different values of τ and κ in Figures 2 and Optimal Screening and Quarantining Policy. In tis subsection, we consider te problem of finding te optimal screening and quarantining policy. In order to reduce te number of infectives in te system, screening prisoners and quarantining infectives are effective strategies. However, tere are costs associated wit tese strategies. Here we assume tat te screening cost C s (τ, ) depends on τ and only and te quarantining cost C q (κ, ) depends on κ and only. Ten te total cost is given by (11) C(τ, κ, ) = C s (τ, ) + C q (κ, ). For te screening cost and te quarantining cost, one may assume tat C s (τ, ) = aτ and C q (κ, ) = bκ for some positive constants a and b. At te same time, tere are some constraints to be met. One would expect tat te number of quarantined prisoners q in equilibrium

7 O OPTIMAL SCREEIG AD QUARATIIG POLICY 319 y tau kappa.4.2 m tau kappa.2 Fig. 2. Total infected prisoners but not (detected (Left)) (quarantined (Rigt)). cannot exceed an upper pysical limit Q and te total number of infectives y + m+q is less tan or equal to a certain tolerant level I. Terefore one may consider te following optimization problem: min C(τ, κ, ) = (aτ + bκ) s.t. (12) q Q y + m + q I τ, κ 1. Here q, m, y are te equilibrium solutions wen τ, κ, are given. To find te optimal values of τ and κ, one may perform a grid searc, say wit grid size.1 for bot parameters. Tis means tat we are going to try all possible pairs (τ, κ) of te form: {(.1i,.1j) : i, j =, 1,..., 1}. For eac pair of (τ, κ), we solve for te equilibrium values and terefore te constraints can be cecked and te cost function can be evaluated. Hence we can obtain te optimal pair (τ, κ) up to two decimal places.

8 32 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G q tau kappa.8 y+m+q tau kappa.4.2 Fig. 3. Total quarantined prisoners (Left) and infectives (Rigt). In te following, we suppose tat = 5, β =.5, n = 5, µ =.1 and a = 1, b = 5, Q = 3, I = 1. Te first optimal pair is (.38,.15), te equilibrium values are given as follows: (y, m, q, y + m + q) = (7.5, 16.7, 29.6, 53.8) and te second optimal pair (.43,.14) and te equilibrium values are given as follows: (y, m, q, y + m + q) = (6.4, 18.4, 3., 54.8). Te optimal cost in bot cases are te same as A etwork of Prisons. In tis section, we consider te case of a network of s prisons. We assume tere are interactions among te prisons and also wit te outside world. Let us give te notations of te model as follows: (i), te number of prisoners in Prison i. (ii) y i (t), te number of prisoners wo are infected but not tested at time t in Prison i.

9 O OPTIMAL SCREEIG AD QUARATIIG POLICY 321 (iii) m i (t), te number of prisoners wo are detected HIV positive but not quarantined at time t in Prison i. (iv) q i (t), te number of prisoners wo are quarantined at time t in Prison i. (v) β i, te infection rate in Prison i. (vi) µ, te mean proportion of infectives in te outside world. (vii) n i, te number of prisoners in Prison i, excanging wit te outside world. (viii), te number of prisoners excange between te two prisons. It is clear tat for i = 1, 2,...,s x i (t) = y i (t) m i (t) q i (t) is te number of susceptible prisoners in Prison i. Te discrete model is ten given as follows: y i (t + 1) = (1 τ)(1 ni+(s 1) )(y i (t) + β i x i (t)(y i (t) + m i (t))) s +(1 τ) (y j (t) + β j x j (t)(y j (t) + m j (t))) + (1 τ)n i µ s m i (t + 1) = (1 κ)(1 ni+(s 1) )m i (t) + (1 κ) m j (t) +(1 κ)τ(1 ni+(s 1) )(y i (t) + β i x i (t)(y i (t) + m i (t))) +(1 κ)τ s (y j (t) + β j x j (t)(y j (t) + m j (t))) + (1 κ)τn i µ q i (t + 1) = (1 ni+(s 1) )q i (t) + s +κ +κτ s q j (t) + κ(1 ni+(s 1) m j (t)) + κτ(1 ni+(s 1) s )m i (t) )(y i (t) + β i x i (t)(y i (t) + m i (t))) (y j (t) + β j x j (t)(y j (t) + m j (t))) + κτn i µ For te equilibrium point, similarly we use z i = y i + β i x i (y i + m i ), i = 1, 2,..., s.

10 322 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G Ten we can get (13) y i = (1 τ)(1 n i + (s 1) )z i + (1 τ) m i = (1 κ)(1 n i + (s 1) )m i + (1 κ) +(1 κ)τ(1 n i + (s 1) )z i +(1 κ)τ s q i = (1 n i + (s 1) )q i + +κ s +κτ s z i = y i + β i x i y i + β i x i m i =. s s z j + (1 τ)n i µ = m j z j + (1 κ)τn i µ = s q j + κ(1 n i + (s 1) )m i m j + κτ(1 n i + (s 1) z j + κτn i µ = )z i To find te condition suc tat te equilibrium solution determined by (13) is asymptotically stable, we first discuss te case of te network of s prisons witout prisoners excange between te two prisons, i.e., =. By setting = in (13), we obtain (14) y i = (1 τ)(1 n i )z i + (1 τ)n i µ m i = (1 κ)(1 n i )m i + (1 κ)τ(1 n i )z i + (1 κ)τn i µ q i = (1 n i )q i + κ(1 n i )m i + κτ(1 n i )z i + κτn i µ z i = y i + β i x i y i + β i x i m i. By a simplification, (14) becomes (15) y i = (1 τ)n i µ + (1 τ)(1 n i )z i a o i1 + b o i1z i i m i = (1 κ)τn n iµ (1 κ)τ(1 i + i ) z κ + ni κni κ + ni κni i a o i2 + b o i2z i q i = κτµ + κ(1 n i )a o i2 + κ(1 n i )(b o i2 + τ)z i a o i3 + b o n i n i i3z i i z i = y i + β i x i y i + β i x i m i. On te existence of an equilibrium solution to (14), we ave te following sufficient condition. Based on (15), te proof is similar to tat of Proposition 1, we omit it ere. Proposition 4. Tere exists an equilibrium solution to (14) if te following condition olds µ < 2 + n i for i = 1, 2,...,s.

11 O OPTIMAL SCREEIG AD QUARATIIG POLICY 323 For te stability of te equilibrium point determined by (14), we ave te following result wose proof can be found in Appendix. Proposition 5. Under te conditions of Proposition 4, te equilibrium point of te discrete model for a network of s prisons witout prisoners excange between te two prisons is asymptotically stable. ow, we consider te general case for te network of s prisons, were tere exists prisoners excange between te two prisons, i.e., >. For i = 1,..., s, let c i = κ + ni+(s 1) κ(ni+(s 1)), d i = ni+(s 1). By a simple calculation, we can reduce (13) to te following form (16) y i = (1 τ)n iµ + (1 τ)(1 m i = a i1 + b i1z i + (1 τ) sè (1 κ)τniµ (1 κ)τ(1 + + (1 κ) c i c i È s a i2 + b i2z i + e i sè q i = κτniµ d i + + (1 κ)κ(1 di ) ni + (s 1) )z i + (1 τ) z j m j + (1 κ)τ c i κ(1 di) a i2 + d i c i d i + κ d i È s + 1 d i sè a i3 + b i3z i + g i sè n i +(s 1) ) z i c i È s m j + f i sè z j z j κ(1 di) (b i2 + τ)z i d i q j + (1 κ)κτ(1 di ) c i d i z i = y i + β ix iy i + β ix im i. m j + i m j + κτ d i sè sè z j q j + w i s sè z j z j Based on (16), we can establis te stability of te equilibrium point determined by (13). Te proof can be found in Appendix. Proposition 6. Suppose tat µ < i 2+n i for i = 1,...,s. Te equilibrium point of te discrete model for te network of s prisons is asymptotically stable if for i = 1,...,s. Remark 1. Under te conditions of Proposition 6, te Jacobian matrix is nonsingular. Tus one can use te ewton s metod for our problem and te quadratic converge is guaranteed if te initial guess is close to te equilibrium solution sufficiently. In te following numerical experiments, we assume tat 1 = 2 = 5; n i = 1i; = 1; y 1 = y 2 = 1; µ =.1.

12 324 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G y tau kappa.4.2 m tau kappa.4.2 Fig. 4. Total infected prisoners but not (detected (Left)) (quarantined (Rigt)) in Prison 1. Table 3.1 τ =.3, κ =.7 β (y 1, y 2 ) (2.2, 2.7) (2.2,2.7) (2.4,3.4) (m 1, m 2 ) (.7,.7) (.7,.7) (.8,.9) (q 1, q 2 ) (2.1, 1.6) (2.2,1.6) (2.5,1.9) For non-symmetric prisoner system we present te numerical results. We first consider te case wen β 1 =.3. We ten present te numerical result wen we ave a bigger β 1 =.3. From Table 3.1 to Table 3.6, wit te increase of β 2, for bot prisons, te infected not detected, detected not quarantined and te quarantined all increase. Wit te increase of te infective rate in Prison 2, te HIV prisoners generated by omogeneous mixing are increased for Prison 2. As a result, y 2, m 2, q 2 are increased. Due to te excange of prisoners between te two prisons, te HIV infectives brougt in from Prison 2 increase.

13 O OPTIMAL SCREEIG AD QUARATIIG POLICY 325 Table 3.2 τ =.5, κ =.5 β (y 1, y 2 ) (1.5, 2.1) (1.5,2.1) (1.6,2.6) (m 1, m 2 ) (1.1, 1.2) (1.1,1.2) (1.2,1.4) (q 1, q 2 ) (2.4, 1.8) (2.4,1.8) (2.7,2.1) Table 3.3 τ =.7, κ =.3 β (y 1, y 2 ) (1.2, 1.7) (1.2,1.7) (1.2,2.1) (m 1, m 2 ) (1.7, 1.7) (1.7,1.8) (1.9,2.1) (q 1, q 2 ) (2.1, 1.6) (2.2,1.6) (2.5,1.9) Table 3.4 τ =.3, κ =.7 β (y 1, y 2 ) (3., 2.9) (3.,2.9) (3.3,3.7) (m 1, m 2 ) (.9,.8) (1.,.8) (1.1,1.) (q 1, q 2 ) (2.7, 1.9) (2.8,1.9) (3.2,2.3) Table 3.5 τ =.5, κ =.5 β (y 1, y 2 ) (2.1, 2.2) (2.1,2.2) (2.3,2.7) (m 1, m 2 ) (1.5, 1.3) (1.5,1.3) (1.6,1.6) (q 1, q 2 ) (3., 2.1) (3.,2.1) (3.4,2.5) Table 3.6 τ =.7, κ =.3 β (y 1, y 2 ) (1.7, 1.7) (1.7,1.8) (1.8,2.2) (m 1, m 2 ) (2.3, 1.9) (2.3,2.) (2.6,2.4) (q 1, q 2 ) (2.7, 1.9) (2.8,1.9) (3.2,2.3)

14 326 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G q tau kappa overall tau kappa.4.2 Fig. 5. Total quarantined prisoners (Left) and infectives (Rigt) in Prison 1. Table 3.7 τ =.3, κ = (y 1, y 2 ) (2.17,2.68) (2.22,2.65) (2.26,2.62) (2.28,2.6) (2.31,2.59) (m 1, m 2 ) (.71,.72) (.72,.71) (.73,.71) (.73,.71) (.73,.71) (q 1, q 2 ) (2.14,1.61) (2.8,1.66) (2.3,1.68) (2.,1.7) (1.98,1.72) We now discuss te effect of te excange between te two prisons. In te following numerical experiments, we also assume tat 1 = 2 = 5, n i = 1i, y 1 = y 2 = 1, β 1 = β 2 =.3 and µ =.1. From Table 3.7 to Table 3.9, te increase of stands for te increase of prisoners excanged between te two prison. Tis would of course lead to te increase of mixing of te prisoners in te two prisons. Tus te differences between te infectives but not detected and te quarantined infectives for bot prisons all decrease.

15 O OPTIMAL SCREEIG AD QUARATIIG POLICY 327 y tau kappa.4.2 m tau kappa.4.2 Fig. 6. Total infected prisoners but not (detected (Left)) (quarantined (Rigt)) in Prison 2. Table 3.8 τ =.5, κ = (y 1, y 2 ) (1.53,2.5) (1.57,2.2) (1.6,1.99) (1.62,1.97) (1.64,1.96) (m 1, m 2 ) (1.12,1.16) (1.14,1.14) (1.15,1.14) (1.16,1.14) (1.16,1.13) (q 1, q 2 ) (2.37,1.81) (2.31,1.85) (2.26,1.88) (2.23,1.9) (2.21,1.92) 3.1. Optimal Screening and Quarantining Policy. In tis subsection, we consider te two-prison case wen applying optimal screening and quarantining policy. Here we assume tat te screening cost C si (τ i, ) depends on τ i and only, te quarantining cost C qi (κ i, ) depends on κ i and only,i = 1, 2. Terefore te total cost is C = C(τ i, κ i, ) = C si (τ i, ) + C qi (κ i, ). We also assume tat for i = 1, 2, C si (τ i, ) = aτ i and C qi (κ i, ) = bκ i

16 328 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G q tau kappa overall tau kappa.4.2 Fig. 7. Total quarantined prisoners (Left) and infectives (Rigt) in Prison 2. Table 3.9 τ =.7, κ = (y 1, y 2 ) (1.16,1.65) (1.19,1.63) (1.22,1.6) (1.24,1.59) (1.25,1.57) (m 1, m 2 ) (1.72,1.75) (1.75,1.73) (1.77,1.73) (1.78,1.72) (1.78,1.72) (q 1, q 2 ) (2.14,1.61) (2.8,1.66) (2.3,1.68) (2.,1.7) (1.98,1.72) for some positive constants a and b. We would like to minimize te above cost but at te same time, tere are some constraints to be met. We would expect tat te number of quarantined prisoners q i in equilibrium does not exceed an upper pysical limit Q i and te total number of infectives y i + m i + q i is less tan or equal to a tolerant level I i, i = 1, 2. To simplify te case, we assume tat τ 1 = τ 2 = τ, and

17 O OPTIMAL SCREEIG AD QUARATIIG POLICY 329 κ 1 = κ 2 = κ. Tus te optimization is min C(τ, κ, ) = (aτ + bκ)( ) s.t. q i Q i, i = 1, 2 (17) 2 y i + m i + q i I τ, κ 1. Here q i, m i, y i, i = 1, 2 are equilibrium values wen τ, κ,, i = 1, 2 are given. To find te optimal values of τ and κ, we perform a grid searc, say wit grid size.1 for bot parameters. Tis means we are going to try all possible pairs (τ, κ) of te form {(.1i,.1j) : i, j =, 1,...,1}. For eac pair of (τ, κ), we ten solve for te equilibrium values and terefore te te constraints can be cecked and te cost function can also be evaluated. Hence we can obtain te optimal pair (τ, κ). In te following numerical example, we suppose tat = 5, β i =.3, y(i) = 1, i = 1, 2 n 1 = 1, n 2 = 2, µ =.1, = 1 and a = 1, b = 5, Q 1 = Q 2 = 4, I = 14. We ave two optimal pairs (.7,.4) and (.12,.3). Wen te optimal pair is (.7,.4) is applied, te equilibrium values are given as follows: (y 1, m 1, q 1, y 1 + m 1 + q 1 ) = (1.7, 2.1, 3.4, 7.3) and (y 2, m 2, q 2, y 2 + m 2 + q 2 ) = (2.2, 2., 2.6, 6.7). Wen (.12,.3) is applied te equilibrium values are given as follows: (y 1, m 1, q 1, y 1 + m 1 + q 1 ) = (1., 2.8, 3.4, 7.3) and (y 2, m 2, q 2, y 2 + m 2 + q 2 ) = (1.4, 2.8, 2.6, 6.7). Te optimal cost in bot cases are te same as Concluding Remarks. We propose matematical models for modeling te spread of HIV in a network of prisons wit te effect of bot screening prisoners and quarantining infectives. Efficient algoritm based on ewton s metod was developed for computing te equilibrium values of te infectives in eac prison. Optimal screening and quarantine policy can be obtained by solving a simple optimization problem. Acknowledgment: Te autors would like to tank te anonymous referee for many elpful comments, corrections and constructive suggestions.

18 33 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G 5. Appendix Proof of Proposition 1. In te following, we analyze te roots of te quadratic equation Az 2 + Bz + C =. First, we note tat κ + n κn = ( (1 κ)(1 n ) 1 ). Tus we know tat κ + n κn >. Hence we can get (18) a 1 = (1 τ)nµ >, < b 1 = (1 τ)(1 n ) < 1, (1 κ)τnµ a 2 = κ + n κn >, b 2 = (1 κ)τ(1 n ) κ + n κn >, a 3 = τκµ + τµκ(1 κ)(1 n ) κ + n κn >, b 3 = ( κτ(1 κ)(1 n )2 n κ + n κn + κτ(1 n ) ) >, A = β(b 1 + b 2 )(b 1 + b 2 + b 3 ) >. ow we discuss te roots of te quadratic equation Az 2 + Bz + C = by considering B 2 4AC. B 2 = (1 b 1 ) 2 + β 2 (a 1 + a 2 ) 2 (b 1 + b 2 + b 3 ) 2 + β 2 (b 1 + b 2 ) 2 ( a i ) 2 +2(1 b 1 )β(a 1 + a 2 )(b 1 + b 2 + b 3 ) + 2(1 b 1 )β(b 1 + b 2 )( (19) +2β 2 (a 1 + a 2 )(b 1 + b 2 + b 3 )(b 1 + b 2 )( a i ) 4AC = 4β(b 1 + b 2 )(b 1 + b 2 + b 3 )(β(a 1 + a 2 )( = 4β 2 (b 1 + b 2 )(b 1 + b 2 + b 3 )(a 1 + a 2 )( a i ) a 1 ) a i ) a i ) 4a 1 β(b 1 + b 2 )(b 1 + b 2 + b 3 )

19 O OPTIMAL SCREEIG AD QUARATIIG POLICY 331 Ten we ave B 2 4AC = (1 b 1 ) 2 + β 2 (a 1 + a 2 ) 2 (b 1 + b 2 + b 3 ) 2 + β 2 (b 1 + b 2 ) 2 ( (2) +2(1 b 1 )β(a 1 + a 2 )(b 1 + b 2 + b 3 ) + 2(1 b 1 )β(b 1 + b 2 )( 2β 2 (a 1 + a 2 )(b 1 + b 2 + b 3 )(b 1 + b 2 )( +4a 1 β(b 1 + b 2 )(b 1 + b 2 + b 3 ) a i ) = ((1 b 1 ) 2 + β 2 (a 1 + a 2 ) 2 (b 1 + b 2 + b 3 ) 2 + β 2 (b 1 + b 2 ) 2 ( = 2(1 b 1 )β(a 1 + a 2 )(b 1 + b 2 + b 3 ) + 2(1 b 1 )β(b 1 + b 2 )( 2β 2 (a 1 + a 2 )(b 1 + b 2 + b 3 )(b 1 + b 2 )( a i )) a i ) 2 a i ) a i ) 2 a i ) +4(1 b 1 )β(a 1 + a 2 )(b 1 + b 2 + b 3 ) + 4a 1 β(b 1 + b 2 )(b 1 + b 2 + b 3 ) ( β((a 1 + a 2 )(b 1 + b 2 + b 3 ) (1 b 1 ) β(b 1 + b 2 )( a i ) +4(1 b 1 )β(a 1 + a 2 )(b 1 + b 2 + b 3 ) + 4a 1 β(b 1 + b 2 )(b 1 + b 2 + b 3 ) > provided tat τ < 1 and κ < 1. Tus bot roots of te quadratic equation Az 2 + Bz + C = are real. Let te roots of te equation be z 1 and z 2. Ten we ave z 1 = B + B 2 4AC 2A and z 2 = B B 2 4AC. 2A We will sow sortly tat under te condition µ < /(2 + n) we ave C <.. Let us assume tat C < and terefore we ave AC < as A >. Tere are tree cases to be discussed. (i) If B >, ten z 1 >, z 2 <. (ii) If B <, ten z 1 >, z 2 <. (iii) If B =, from B 2 4AC >, we know AC <, ten we ave z 1 >, z 2 <. ) 2 In general, wen C < we ave z 1 = B + B 2 4AC 2A > being positive (te root as a practical meaning). Ten we ave te expression of y, m, q, x in te equilibrium point: y = a 1 + b 1 z 1 m = a 2 + b 2 z 1, q = a 3 + b 3 z 1, x = y m q.

20 332 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G ow te following is te condition we need to guarantee a unique positive root. C = β(a 1 + a 2 )( a i ) a 1 <. In fact, (21) a 2 = (1 κ)τnµ κ + n κn = (1 κ)τnµ (1 κ)τnµ = κ + n κn κ( n) + n (1 κ)τµ τµ = (1 κ)τµ κ ( n) n + 1 and (22) τµκ(1 κ)(1 n ) κ + n κn = τµκ(1 κ)( n) τµ(1 κ)( n) κ + n (1 κ) = 1 + n 1 κ κ < τµ(1 κ)( n) < τµ(1 κ). Tus, we get a 3 τκµ + τµ(1 κ). As a summary, we ave a 1 = (1 τ)nµ (1 κ)τnµ a 2 = κ + n κn τµ a 3 = τκµ + τµκ(1 κ)(1 n ) κ + n κn τκµ + τµ(1 κ). Hence Tus if we ave ten and terefore C <. a 1 + a 2 + a 3 nµ + 2µ µ < 2 + n a i < 5.2. Proof of Proposition 3. It is obvious tat te equilibrium point of (4) satisfies (5) or (23) F = y + a 1 + b 1 z = G = m + a 2 + b 2 z = H = q + a 3 + b 3 z = K = z + y + βxy + βxm =,

21 O OPTIMAL SCREEIG AD QUARATIIG POLICY 333 were te numbers a 1, a 2, a 3, b 1, b 2, b 3 are defined in (7). Te Jacobian matrix of te above equations is given by 1 b 1 J = 1 b 2 1 b βx βx 1 Te caracteristic polynomial of J is reduced to λ + 1 b 1 Det(λI J) = (λ + 1) Det λ + 1 b 2 1 βx βx λ + 1 (24) Since x = = (λ + 1) 2 [(λ + 1) 2 b 1 (b 1 + b 2 )βx]. z y β(y+m) and a 1, a 2, b 1, b 2 >, it follows from (7) tat (25) b 1 + (b 1 + b 2 )βx = b 1 + (b 1 + b 2 )β = b 1 + (b 1 + b 2 ) < b 1 + z b1z a1 z < b b 1 = 1. z y β(y+m) z b 1z a 1 (a 1+a 2)+(b 1+b 2)z From (25) and (24), we observe tat all te eigenvalues of te Jacobian matrix J are real and negative. By [14, Teorem 1.6], we know tat te equilibrium point of (4) determined by (9) is asymptotically stable Proof of Proposition 5. From (15), it is easy to ceck tat te equilibrium solution of (14) is determined by Fi o = y i + a o i1 + bo i1 z i = G o i = m i + a o i2 + bo i2 z i = H o i = q i + a o i3 + bo i3 z i = K o i = z i + y i + β i x i y i + β i x i m i = for i = 1, 2,..., s. Te Jacobian matrix of te above equations is given by 1 b o i1 (26) J o = diag(j1 o,...,jo s ), Jo i = 1 b o i2 1 b o i β i x i β i x i 1 For i = 1,...,s, we can sow tat all te eigenvalues of J o i are real and negative by following te similar proof of Proposition 3. Terefore, all te eigenvalues of te Jacobian matrix J o are real and negative. By [14, Teorem 1.6], we know tat te equilibrium point of (14) is asymptotically stable. Tis completes te proof.

22 334 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G 5.4. Proof of Proposition 6. It is obvious tat te equilibrium point for te network of s prisons is determined by (16), i.e., F i = y i + a i1 + b i1 z i + (1 τ) G i = m i + a i2 + b i2 z i + e i s H i = q i + a i3 + b i3 z i + g i s + i s q j + w i s s z j = m j + f i m j K i = z i + y i + β i x i y i + β i x i m i =. z j = s Te Jacobian matrix of te above equations as te following form J 11 J 12 J 1s J 21 J 22 J 2s J =..,.... J s1 J s2 J ss z j = were J ii = 1 b i1 1 b i2 1 b i3 1 + β i x i β i x i 1 and J ij = (1 τ) e i f i g i i w i for i j. From (16) and (15), it follows tat for 1 i s, b i1 = b o i1 (1 τ)(s 1) b i2 = b o i2 (1 κ)τ (κ+ n i κn i i b i3 = b o i3 κτ d i(κ+ n i κn i ) (s 1) )c i b o i2 es i i ( ) i n i + (1 di)(1 κ) c i (s 1) b o i3 fs i Ten we ave (1 τ)(s 1) J ii = Ji o + W i, W i = e s i fi s,, 1 i s,

23 O OPTIMAL SCREEIG AD QUARATIIG POLICY 335 were Ji o is defined in (26). Terefore, we can rewrite te Jacobian matrix J as te form J = J o + R, were R = W 1 J 12 J 1s J 21 W 2 J 2s J s1 J s2 W s By Proposition 5, we know tat all te eigenvalues of J o are real and negative if µ < 2 + n i for i = 1, 2,..., s. By Teorem VIII.1.1 in [1], we ave tat for any two n n matrices E and F wit te eigenvalues φ 1,..., φ n and ψ 1,...,ψ n, respectively, max min φ i ϕ j ( E + F ) 1 1/n E F 1/n, j i were denote any matrix norm. ow, suppose tat te eigenvalues of te matrices J and J o are denoted by λ 1,..., λ 4s and λ o 1,..., λ o 4s, respectively. Ten, for te matrix 1-norm 1, we get (27) max j min λ i λ o j ( J 1 + J o 1 ) 1 1/(4s) R 1/(4s) 1 i (2 J o 1 + R 1 ) 1 1/(4s) R 1/(4s) 1, ext, we estimate R 1. Let ξ j = s,i j (e i + g i ) ζ j = s,i j i η j = s,i j (1 τ + f i + w i ) + (1 τ)(s 1) + e s j + fs j δ = max j {ξ j, ζ j, η j }. Ten (28) Let } R 1 = max j {ξ j, ζ j, η j δ max j δ. { } min j λ o j ǫ < min 1, 4s δ(2 J o 1 + δ) 4s 1.

24 336 WAI-KI CHIG, YAG COG, ZHEG-JIA BAI, AD TUE-WAI G By (28) and (27), we know tat if is small suc tat max j ǫ, ten te real part of eac eigenvalue λ j of te Jacobian matrix J are negative. By [14, Teorem 1.6], it follows tat te equilibrium point of te discrete model for te network of s prisons is asymptotically stable. REFERECES [1] R. Batia, Matrix Analysis, Springer-Verlag, ew York, [2] W. Cing, T. g, and S. Cung, On Modeling SARS in Hong Kong, International Journal of Applied Matematics, 13(23), pp [3] W. Cing, T. g, Y. Cong, and A. Tai, A Fast Algoritm for te Spread of HIV in a System of Prisons, Matematical and Computer Modeling, 46(27), pp [4] D. Daley and J. Gani, Epidemic modelling: An Introduction, Cambridge University Press, Cambridge, [5] S. Eubank, H. Guclu, V. Kumar, M. Marate, A. Srinivasan, Z. Toroczkai, and. Wang, Modelling Disease Outbreaks in Realistic Urban Social etworks, ature, 429(24), pp [6] J. Gani, S. Yakowitz, and M. Blount, Te Spread and Quarantine of HIV Infection in a Prisoner System, SIAM J. Appl. Mat., 57(1997), pp [7] G. Golub and C. van Loan, Matrix Computations, 2nd Edition, Te Jon Hopkins University Press, London, [8] D. Greenalg and F. Lewis, Te General Mixing of Addicts and eedles in a Variableinfectivity eedle-saring Environment, Journal of Matematical Biology, 44(22), pp [9] X. Huang and M. Villasana, An Extension of te Kermack-Mckendrick Model for AIDS Epidemic, Journal of te Franklin Institute, 342(25), pp [1] C. Kelly, Iterative Metods for Linear and on-linear Equations, SIAM, Piladelpia, [11] W. Ma, M. Song, and Y. Takeuci, Global Stability of an SIR Epidemic Model wit Time Delay, Appl. Mat. Letters, 17(24), pp [12] M. ewman, Te Structure and Function of Complex etworks, SIAM Review, 45(23), pp [13] T. g, G. Turinici, and A. Dancin, A Double Epidemic Model for te SARS Propagation, BMC Infectious Diseases, 3(23), pp [14] R. Seydel, Practical Bifurcation and Stability Analysis: From Equilibrium to Caos, 2nd Edition, Springer-Verlag, ew York, Berlin, Heidelberg, [15] W. Wang, Global Beavior of an SEIRS Epidemic Model wit Time Delays, Appl. Mat. Letters, 15(22), pp