Heroin epidemics, treatment and ODE modelling

Transcription

1 Mathematical Biosciences 208 (2007) Heroin epidemics, treatment and ODE modelling Emma White *, Catherine Comiske Department of Mathematics, UI Manooth, Co., Kildare, Ireland Received Ma 2006; received in revised form 6 October 2006; accepted 23 October 2006 Available online 7 ovember 2006 Abstract The U [United ations Office on Drugs and Crime (UODC): World Drug Report, 2005, vol. : Analsis. UODC, 2005.], EU [European Monitoring Centre for Drugs and Drug Addiction (EMCDDA): Annual Report, and WHO [World Health Organisation (WHO): Biregional Strateg for Harm Reduction, HIV and Injecting Drug Use. WHO, 2005.] have consistentl highlighted in recent ears the ongoing and persistent nature of opiate and particularl heroin use on a global scale. While this is a global phenomenon, authors have emphasised the significant impact such an epidemic has on individual lives and on societ. ational prevalence studies have indicated the scale of the problem, but the drug-using career, tpicall consisting of initiation, habitual use, a treatment-relapse ccle and eventual recover, is not well understood. This paper presents one of the first ODE models of opiate addiction, based on the principles of mathematical epidemiolog. The aim of this model is to identif parameters of interest for further stud, with a view to informing and assisting polic-makers in targeting prevention and treatment resources for maximum effectiveness. An epidemic threshold value, R 0, is proposed for the drug-using career. Sensitivit analsis is performed on R 0 and it is then used to examine the stabilit of the sstem. A condition under which a backward bifurcation ma exist is found, as are conditions that permit the existence of one or more endemic equilibria. A ke result arising from this model is that prevention is indeed better than cure. Ó 2006 Elsevier Inc. All rights reserved. Kewords: ODE models; Mathematical epidemiolog; Heroin use; Reproduction ratio R 0 ; Drug-using career; Treatment * Corresponding author. Tel.: ; fax: address: emma.white@nuim.ie (E. White) /$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:0.06/j.mbs

2 E. White, C. Comiske / Mathematical Biosciences 208 (2007) Background to heroin use in Ireland The use of heroin and other drugs in Europe and more specificall Ireland and the resulting prevalence are well documented [2,4,5]. However, in spite of the extensive developments in the mathematical and statistical techniques applied to modelling infectious diseases, little has been done to appl this work to the emerging heroin epidemics. In this paper, attempts to extend dnamic disease modelling to the drug-using career are introduced and applied in a general model. A drug user is defined here as an individual whose habitual usage of opiate drugs harms the phsical, mental or social wellbeing of the individual, their famil or peer group, or societ [6]. Opiate use in Ireland appears to have been minimal until 979 [7]. Police reports, hospital records and an increase in the numbers presenting for treatment from 5 people in 979 to 239 people in 983, made the increase in heroin use ver clear [8]. In fact, the period is seen as defining the first heroin epidemic in Ireland. Subsequent epidemic periods have been noted, alternated with periods of relative stabilit in the numbers using the drug. Illicit drug-use, b its nature, is a hidden activit. That said, independent prevalence estimates obtained in 996 and produced broadl similar results for the total size of the drug-using population. These prevalence estimates of drug-users in Ireland are shown in Table. While prevalence has remained approximatel constant (although the individuals who comprise the drug-using population ma have changed), the total number of individuals seeking treatment has been growing [9]. At the same time, the percentage of individuals seeking treatment for the first time, as a proportion of the total seeking treatment, is dropping [9]. Eight-seven percent of the respondents in the ROSIE stud [0] reported prior treatment episodes. This return to treatment accounts for some increase in treatment demand and also suggests that rather than being a once-off event, a treatment-relapse ccle of some kind exists. aturall this has important implications for the provision of treatment services and for the perception of successful treatment episodes. The Irish government has committed itself to the ational Drugs Strateg Building on Experience, Agencies with primar responsibilit include the ational Advisor Committee for Drugs (ACD) and the Drug Misuse Research Division (DMRD) of the Health Research Board (HRB). With the aims of these bodies in mind, this paper provides an initial framework, in the mathematical epidemiolog context, within which certain characteristics of the opiate-using career can be identified. Table Estimated prevalence of opiate use Year Age group Est. o. Rate per 000 population Source , Comiske a , Kell et al. b a Ref. [4]. Ref. [5].

3 34 E. White, C. Comiske / Mathematical Biosciences 208 (2007) Methods 2.. Mathematical epidemiolog The literature and development of mathematical epidemiolog is well documented and can be found in the works of Baile [], Anderson and Ma [2], Murra [3] and Brauer and Castillo- Chavez [4]. Yet, while social problems such as alcohol and drug use have been referred to in terms of epidemics, little has been published on the application of mathematical modelling methods to such problems Approach On the premise that drug use follows a process that can be modelled in a similar wa to the modelling of disease, a mathematical epidemiological treatment of drug use ma ield insights on the progression through the drug-using career, from initiation to habitual use, treatment, relapse and eventual recover. It is of course critical to understand, insofar as it is possible, the process being modelled. Information from the ROSIE stud [0] and feedback from professionals in addiction-related areas were fundamental in developing the model shown in Fig.. The approach used was to fit the characteristics of the drug-using career to a susceptible-infectious disease model. Each compartment in Fig. represents a stage in the drug-using career. The arrows show the paths that can be taken between compartments. Following standard methods [,2], once this simple compartment model and its corresponding dnamics were identified, ordinar differential equations were derived to mathematicall represent the sstem. A value for R 0, the basic reproduction number, is then proposed for this sstem. This number tells us how man secondar infections will result from the introduction of one infected individual into a susceptible population. R 0 = means that each infected person will infect one susceptible person. Usuall, R 0 < implies that an epidemic will not result from the introduction of one infected individual, whereas R 0 > implies that an epidemic will occur and R 0 = requires further investigation. In the drug-using context, R 0 > implies that, on average, during the drug-using career, each single drug user will infect, that is, introduce to drug use, at least one other individual. However, as will be seen, the model () (3) ma impl something further, namel that the threshold value of R 0 must be brought far below one in order to avoid an epidemic and if this does not happen, an endemic equilibrium of drug-users ma be established in the population and eradication of drug-use could become more difficult. R 0 is also useful for establishing the existence of equilibrium points and in performing stabilit analsis for the sstem. Fig.. A model of the opiate-using career.

4 E. White, C. Comiske / Mathematical Biosciences 208 (2007) A model of the opiate-using career and its parameters The model presented in Fig. ma be represented b the set of Eqs. () (3) below: ds dt ¼ K b U S ls; ðþ du ¼ b U S dt pu þ b 3U U 2 ðl þ d ÞU ; ð2þ du 2 ¼ pu dt b 3U U 2 ðl þ d 2 ÞU 2 : ð3þ The parameters in () (3) are as follows: : Size of the total population. S: The number of susceptible individuals in the population. Here, all individuals range from age 5 64 [7]. U : The number of drug users not in treatment; initial and relapsed drug users. U 2 : The number of drug users in treatment. K: The number of individuals in the general population entering the susceptible population, i.e. the demographic process of individuals reaching age 5 in the modelling time period. l: The natural death rate of the general population. d : A removal rate that includes drug-related deaths of users not in treatment and a spontaneous recover rate; individuals not in treatment who stop using drugs but are no longer susceptible. d 2 : A removal rate that includes the drug-related deaths of users in treatment and a rate of successful cure that corresponds to recover to a drug free life and immunit to drug addiction for the duration of the modelling time period. b : The probabilit of becoming a drug user. p: The proportion of drug users who enter treatment. b 3 : The probabilit of a drug user in treatment relapsing to untreated use. Clearl the model involves certain assumptions. These consist of the following: = S + U + U 2 ; the population is assumed to be of constant size within the modelling time period, that is, K = ls +(l + d )U +(l + d 2 )U 2. A proportion of drug users enter treatment in each modelling time period. Per field research, individuals in treatment are using drugs [0]. A proportion of users who are not in treatment stop using in each modelling time period; this again corresponds to what is observed in practice. Drug users not in treatment are infectious to susceptibles and to users in treatment. Users in treatment most commonl relapse due to contact with users who are not in treatment [0]. Drug users in treatment are not infectious to susceptibles. Homogeneous mixing is assumed; ever individual in the population has an equal chance of encountering an other individual.

5 36 E. White, C. Comiske / Mathematical Biosciences 208 (2007) All members of the population are assumed to be equall susceptible to drug addiction. In practice, some sub-populations are more highl susceptible due to environmental, behavioural and genetic factors. Here an average value for b over the general population is used. 3. Results Presented here are detailed analsis and interpretation of R 0 for the model. R 0 is then itself used to investigate the existence of equilibria for the dnamical sstem () (3). 3.. The basic reproduction number, R Explanation of R 0 The basic reproduction number; a threshold value representing how man secondar infections result from the introduction of one infected individual into a population of susceptibles [6]. The value that R 0 takes can indicate the circumstances in which an epidemic is possible. In the drugusing context, R 0 tells us, on average, the total number of people that each single drug user will initiate to drug use during the drug-using career R 0 for this model R 0 here is defined as the probabilit of becoming addicted to drugs multiplied b the mean amount of time spent using drugs without treatment. In terms of the model parameters, R 0 ¼ b p þ l þ d : ð4þ Interpretation of R 0 for this model In this case, the basic reproduction number is the probabilit of an individual becoming a drug user divided b the sum of the proportion of individuals who enter treatment, the natural death rate of the population and the death rate of drug-using individuals who are not in treatment. The interpretation is ver straightforward: when the probabilit of becoming a drug user (the numerator in R 0 ) is greater than the sum of the cessation probabilities in the R 0 denominator, prevalence will rise R 0 sensitivit analsis To examine the sensitivit of R 0 to each of its parameters, following Arriola and Hman [8], the normalised forward sensitivit index with respect to each of the parameters is calculated: or 0 R A b ¼ 0 ¼ b or 0 p þ l þ d ¼ b ob R b 0 ob ¼ : b p þ l þ d Thus, for example, an increase in b of 2% will result in an increase in R 0 of 2%. ð5þ

6 E. White, C. Comiske / Mathematical Biosciences 208 (2007) R A p ¼ p A l ¼ A d ¼ ¼ p or 0 R 0 op p ¼ p þ l þ d < 0 R l ¼ l or 0 R 0 ol l ¼ p þ l þ d < 0 R d ¼ d or 0 R 0 od d ¼ p þ l þ d < : It is concluded that R 0 is most sensitive to changes in b. An increase in b will bring about an increase of the same proportion in R 0 (equall, a decrease in b will bring about an equivalent decrease in R 0 ; the are directl proportional). p, l and d have an inversel proportional relationship with R 0 ; an increase in an of them will bring about a decrease in R 0, however, the size of the decrease will be proportionall smaller. Recall that l is the natural death rate of the population and d is a removal rate that includes drug-related deaths of users not in treatment and a spontaneous recover rate; individuals not in treatment who stop using drugs but are no longer susceptible. It is clear that increases in either of these rates is neither ethical nor practical (although spontaneous recover is welcome, of course!). Thus we choose to focus on one of two parameters: either p, the proportion of users who enter treatment or b, the probabilit of an individual becoming a drug user. Given R 0 s sensitivit to b and in the knowledge that a treatment ccle exists (individuals who enter treatment are likel to relapse and re-enter treatment), it seems sensible to focus efforts on the reduction of b. In other words, this sensitivit analsis tells us that prevention is better than cure; efforts to increase prevention are more effective in controlling the spread of habitual drug use than efforts to increase the numbers of individuals accessing treatment Stabilit of drug free equilibrium, R 0 < Having derived R 0 for the model, it is now used to determine the existence of equilibria for the sstem. In particular, it is known that the drug free equilibrium (DFE) of epidemiological models

7 38 E. White, C. Comiske / Mathematical Biosciences 208 (2007) is locall asmptoticall stable at R 0 <[6]. For this to hold, the eigenvalues of the Jacobian matrix of the model () (3) with substitutions ðs ; U ; U 2 Þ¼ K l ; 0; 0 ð6þ should have negative real parts. The Jacobian of the sstem () (3) is 0 b U l b S JðS; U ; U 2 Þ¼ b U 0 b S p þ b 3U 2 ðl þ d Þ 0 p b 3U 2 b 3U b 3 U C A : ðl þ d 2Þ Substituting (6) and recalling that at the DFE, S * =, the matrix becomes 0 J K l b 0 l ; 0; 0 0 b ðp þ l þ d Þ 0 A: ð7þ 0 p ðl þ d 2 Þ The eigenvalues of this matrix are: k ¼ l k 2 ¼ ðl þ d 2 Þ k 3 ¼ b ðp þ l þ d Þ: k and k 2 are clearl real and negative. Also as R 0 <, then b < p þ l þ d and k 3 meets the necessar criteria. The sstem () (3) shows local asmptotic stabilit at the DFE Analsis at R 0 = Castillo-Chavez and Song [9] and Song [20] describe the following method to examine the direction of the bifurcation at R 0 =: A ¼ D w f ð0; 0Þ ¼ of i ð0; 0Þ ow j is the linearisation matrix of a general sstem of ordinar differential equations around the equilibrium 0 with bifurcation parameter / evaluated at 0. Zero is a simple eigenvalue of matrix A and all the other eigenvalues of A have negative real parts. A has a right eigenvector x and a left eigenvector that correspond to the zero eigenvalue. Let f k be the kth component of f and a ¼ X o 2 f k k x i x j ð0; 0Þ; ð8þ ox i ox j k;i;j¼ b ¼ X o 2 f k k x i ð0; 0Þ; ox k;i¼ i o/ ð9þ

8 If a > 0 and b > 0, the bifurcation at / = 0 is backward. The existence of a backward bifurcation means that multiple endemic equilibria exist; in particular, an endemic equilibrium exists when R 0 < and substantial effort is required to reduce R 0 to a level small enough to eradicate the disease from the population (recall that the DFE has been shown to be locall asmptoticall stable onl). b is the obvious choice of bifurcation parameter (recall that it represents the rate at which individuals are initiated to drug use), particularl as it has been shown in Eq. (5) that R 0 is more sensitive to changes in b than in its other parameters. At R 0 =, b ¼ p þ l þ d so we define b ¼ p þ l þ d : ð0þ The Jacobian matrix is obtained as before in Eq. (7) and the identit (0) is used, leading to: 0 l b 0 K l ; 0; 0 B p ðl þ d 2 Þ C A: ðþ The matrix () has eigenvalues (0, l, (l + d )) T, which meets the requirement of a simple zero eigenvalue and the others having negative real part. The right eigenvector x corresponding to the zero eigenvalue is ðp þ l þ d Þðl þ d 2 Þ lp ; l þ d 2 ; p x T 3 with x 3 free and the left eigenvector corresponding to the zero eigenvalue is (0,,0) 2, with 2 free. The second derivatives are calculated, evaluated at the DFE ðs ; U ; U 2 Þ¼ðK ; 0; 0Þ and with l b ¼ b. The formulas for a and b are then used. It is found that b is positive and a is positive if the following inequalit holds: ðp þ l þ d Þ 2 ðl þ d 2 Þ lp E. White, C. Comiske / Mathematical Biosciences 208 (2007) < b 3 ð2þ (see Appendix A for details of the calculations). ote that if b 3 = 0, there is no treatment ccle as individuals do not relapse to untreated use, and no backward bifurcation exists. It is the lapse from treatment due to contact with other drug users that drives the possibilit of an endemic equilibrium when R 0 <. This reflects what is usuall concluded about backward bifurcations the are driven b exogenous re-infection, that is, re-infection from an external source [9,5] Existence of endemic equilibrium: R 0 > It has been shown that the DFE, corresponding to R 0 < is locall asmptoticall stable and that there is a condition under which an endemic equilibrium ma exist when R 0 <. The existence of one or more non-trivial equilibria is now investigated. At an endemic equilibrium, disease (or in this case, drug addiction) is present in the population and the following hold:

9 320 E. White, C. Comiske / Mathematical Biosciences 208 (2007) S P 0; U > 0; U 2 P 0; ds dt ¼ du ¼ du 2 ¼ 0: dt dt To begin, non-trivial equilibrium values for each of the variables S,U,U 2 must be found. First solve Eq. () equal to zero for equilibrium value, S * : S ¼ K b U þ l : ow, substituting S * into Eq. (2), setting equal to zero and solving for U 2 gives b U K þ pu b U l þ b 3U U 2 ðl þ d ÞU ¼ 0: Hence K þ b U K l pu þ b 3U U 2 ðl þ d ÞU ¼ 0 and re-arranging ields U K 2 ¼ þ b K b 3 U b 3 l þ p þ l þ d : Putting U 2 into Eq. (3), setting the equation equal to zero and solving for U gives pu þ K þ b K b 3 U b 3 l b3 U ðp þ l þ d Þ b 3 þ l þ d 2 ¼ 0: Further algebraic manipulation produces the following quadratic in U : U 2 b K l ðl þ d Þ þ U K þ lðp þ l þ d Þþb b K d 2 ðp þ l þ d Þþ d 2b K 3 l þ K ðl þ d 2 Þ¼0: b 3 A real positive value for U is required for an endemic equilibrium to exist. Obviousl the constant term is positive. However, the signs of the co-efficients of U 2 and U are not obvious although it is known that b > p þ l þ d as R 0 >. Thus four quadratic cases arise: au 2 þ bu þ c ¼ 0 in which case, using Descartes Rule of Signs [2], no sign changes mean that there are no real roots and there is no endemic equilibrium. Using the same rule,

10 au 2 bu þ c ¼ 0 E. White, C. Comiske / Mathematical Biosciences 208 (2007) has two sign changes and two real positive values for U would be expected. au 2 þ bu þ c ¼ 0 has one sign change and therefore one real positive root as does the final case: au 2 bu þ c ¼ 0: The following inequalities must be investigated to determine the existence of the endemic equilibrium. The co-efficient of U 2 will be negative if b K l < j l þ d j: The co-efficient of U can be negative onl if K < b b K þ d 2 ðp þ l þ d Þðl þ d 2 Þ 3 l ; where b K þ d 2 l < j ðp þ l þ d Þðl þ d 2 Þj: Either the co-efficient of U 2 or U or both must be negative to guarantee the existence of at least one endemic equilibrium. 4. Conclusions It is found that b p þ l þ d is the basic reproduction ratio, R 0, for the model () (3). This is interpreted as follows: when the probabilit of becoming a drug user is greater than the sum of the cessation probabilities, prevalence will rise. Sensitivit analsis identifies b, the probabilit of becoming a drug user, as the most useful parameter to target for the reduction of R 0. For practical purposes, this corresponds to prevention being better than cure; efforts to increase prevention are more effective in controlling the spread of habitual drug use than efforts to increase the numbers of individuals accessing treatment. The model () (3) is locall asmptoticall stable when R 0 < (this is known as the drug free equilibrium). It is found that analsis at R 0 = ields the following result: if the inequalit ðp þ l þ d Þ 2 ðl þ d 2 Þ lp < b 3

11 322 E. White, C. Comiske / Mathematical Biosciences 208 (2007) holds, then a backward bifurcation can occur and although R 0 ma be less than, an endemic equilibrium exists. If this equilibrium is stable, substantial effort ma be required to reduce prevalence and avoid an epidemic. When R 0 >, analsis produces a quadratic equation in U. The existence of an endemic equilibrium (or endemic equilibria) depends on the existence of at least one real, positive value for U ; thus, either the co-efficient of U 2 or U or both must be negative to guarantee the existence of at least one endemic equilibrium. This paper introduces a universal model that identifies parameters of interest in the drug-using career. Clearl the collection of data relevant to the ke parameters b epidemiologists and treatment providers at a global level, would be of significant use from an implementation and polic perspective. 5. Future work Using this model within the Irish setting, it is intended to estimate parameters with a view to ascertaining from current available data if a backward bifurcation exists and if an endemic equilibrium exists for R 0 >. Stabilit analsis will be performed on an equilibria shown to exist. Further models will incorporate heterogeneous mixing and address the assumption that all individuals have the same level of susceptibilit. Acknowledgements This research is funded b the Health Research Board (HRB) of Ireland with the support of the ational Universit of Ireland (UI), Manooth. Particular thanks are also due to Baojun Song, Leon Arriola and all involved with the Mathematical and Theoretical Biolog Institute (MTBI), Arizona State Universit (ASU), USA. The authors wish to thank Barr Cipra and the Editor and reviewers of Mathematical Biosciences for their useful feedback and comments. Appendix A. Calculation of a and b for determination of backward bifurcation condition a ¼ X o 2 f k k x i x j ð0; 0Þ; ox i ox j k;i;j¼ ð3þ b ¼ X o 2 f k k x i ð0; 0Þ: ox k;i¼ i o/ ð4þ If a > 0 and b > 0, the bifurcation at / = 0 is backward. The second derivatives are calculated, evaluated at the DFE ðs ; U ; U 2 Þ¼ðK ; 0; 0Þ and with b l ¼ b. The formulas for a and b are then used. The non-zero derivatives are as follows:

12 E. White, C. Comiske / Mathematical Biosciences 208 (2007) o 2 f ¼ o2 f ¼ lðp þ l þ d Þ ; ox 2 ox ox ox 2 K o 2 f 2 ¼ o2 f 2 ¼ lðp þ l þ d Þ ; ox 2 ox ox ox 2 K o 2 f 2 ¼ o2 f 2 ¼ b 3l ox 3 ox 2 ox 2 ox 3 K ; o 2 f 3 ¼ o2 f 3 ¼ b 3l ox 3 ox 2 ox 2 ox 3 K ; o 2 f ¼ ; ox 3 ob o 2 f 2 ¼ ; ox 2 ob b ¼ o2 f ox 3 ob x 3 þ o2 f 2 ox 2 ob 2 x 2 ¼ð Þð0ÞðÞþðÞðÞ l þ d 2 p ¼ l þ d 2 p a ¼ o2 f ox 2 ox x 2 x þ o2 f ox ox x x 2 þ o2 f 2 2 ox 2 ox 2 x 2 x þ o2 f 2 ox ox 2 x x 2 þ o2 f 2 2 ox 3 ox 2 x 3 x 2 þ o2 f 2 2 ox 2 ox 2 x 2 x 3 3 þ o2 f 3 ox 3 ox 3 x 3 x 2 þ o2 f 3 2 ox 2 ox 3 x 2 x 3 ¼ 2l l þ d 2 ðp þ l þ d Þ 2 ðl þ d 2 Þ þ b 3 Kp lp 3!: > 0; As 2l l þ d 2 > 0; Kp the backward bifurcation (a, b > 0) will occur onl if ðp þ l þ d Þ 2 ðl þ d 2 Þ lp < b 3: References [] United ations Office on Drugs and Crime (UODC): World Drug Report, 2005, vol. : Analsis. UODC, [2] European Monitoring Centre for Drugs and Drug Addiction (EMCDDA): Annual Report, annualreport.emcdda.eu.int/en/home-en.html. [3] World Health Organisation (WHO): Biregional Strateg for Harm Reduction, HIV and Injecting Drug Use. WHO, [4] C. Comiske, ational prevalence of problematic opiate use in Ireland, EMCDDA Tech. Report (999). [5] A. Kell, M. Carvalho, C. Teljeur: Prevalence of Opiate Use in Ireland A 3-Source Capture Recapture Stud. A Report to the ational Advisor Committee on Drugs, Sub-committee on Prevalence. Small Area Health Research Unit, Department of Public.

13 324 E. White, C. Comiske / Mathematical Biosciences 208 (2007) [6] M. O Brien, R. Moran, Overview of Drug Issues in Ireland 997, Drugs Research Division, Health Research Board (997). [7] D. Corrigan, The identification of drugs of abuse in the Republic of Ireland during the ears , Bull. arcotics 3 (2) (979) 57. [8] G. Dean, A. O Hare, G. Kell, A. O Connor, M. Kell: The Opiate Epidemic in Dublin The Medico Social Research Board, 73 Lower Baggot Street, Dublin 2, 983. [9] J. Long et al., Trends in Treated Problem Drug Use in Ireland, 998 to 2002, Occasional Paper o. 7/2005. DMRD, HRB, [0] C. Comiske, G. Cox: Research Outcome Stud in Ireland (ROSIE): Evaluating Drug Treatment Effectiveness, Baseline Findings. March []. Baile, The Mathematical Theor of Infectious Diseases, Charles Griffin, 975. [2] R.M. Anderson, R.M. Ma, Infectious Diseases of Humans, Dnamics and Control, Oxford Universit Press, 99. [3] J.D. Murra, Mathematical Biolog I and II, Springer, [4] F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biolog and Epidemiolog, Springer, [5] P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 80 (2002) 29. [6] O. Diekmann, J.A.P. Heesterbeek, Mathematical Epidemiolog of Infectious Diseases, John Wile & Son, Ltd, [7] European Monitoring Centre for Drugs and Drug Addiction (EMCDDA): Population Surves Methodolog. Methods and definitions, general population surves, detailed notes. EMCDDA Statistical Bulletin, [8] L. Arriola, J. Hman, Lecture notes, forward and adjoint sensitivit analsis: with applications in Dnamical Sstems, Linear Algebra and Optimisation Mathematical and Theoretical Biolog Institute, Summer, [9] C. Castillo-Chavez, B. Song, Dnamical models of tuberculosis and their applications, Math. Biosci. Eng. (2) (2004). [20] B. Song, Seminar otes, Backward or Forward at R0 =, Mathematical and Theoretical Biolog Institute, Summer, [2] F.E. Su et al., Descartes Rule of Signs Mudd Math Fun Facts.