A Modeling Approach for Assessing the Spread of Tuberculosis and Human Immunodeficiency Virus Co-Infections in Thailand

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1 Kasetsart J. (at. Sci.) 49 : 99 - (5) A Modeling Approach for Assessing the Spread of Tuberculosis and uman Immunodeficiency Virus Co-Infections in Thailand Kornkanok Bunwong,3, Wichuta Sae-jie,3,* and atdanai Boonsri,3 ABSTRACT Mathematical models were formulated to describe the transmission dynamics of tuberculosis (TB) and human immunodeficiency virus (IV) and the relationship between them. For TB infections, the population was separated into: ) a susceptible non-infected group, ) a latent asymptomatic infected group and 3) an active symptomatic infected group. For IV infections, it was assumed that each of the three TB groups could be: ) not infected with IV or ) actively infected with IV. The model was used to study the spread of TB through the non-iv and IV groups. Formulas for the basic reproduction numbers and threshold criteria for the spread of TB through the two IV groups were derived using the next generation method. Using population statistics (number of population, birth rate, and death rate) and health statistics (death by leading cause group, and number of persons affected with tuberculosis and acquired immune deficiency syndrome) reported by the Ministry of Interior, the Ministry of Public ealth, and the World ealth Organization, parameter values and estimate values were obtained for the basic reproduction numbers for the spread of TB. umerical simulations confirmed the threshold criteria for the existence of a stable, disease-free equilibrium point and of a stable, endemic equilibrium point. Keywords: basic reproduction number, epidemic model, tuberculosis (TB) and human immunodeficiency virus (IV) co-infection, transmission dynamics ITRODUCTIO Detailed information on the tuberculosis disease and worldwide tuberculosis infection has been given in a recent report (World ealth Organization, 4a). Tuberculosis (TB) is an airborne infectious disease caused by bacteria (Mycobacterium tuberculosis). Pulmonary TB is the most common type of TB infection. Once a person with pulmonary TB coughs, sneezes, speaks or sings, TB germs are released to the air. They can survive in the air for a period of time, depending on moisture and temperature conditions. Consequently, susceptible individuals have a chance to inhale TB germs and become a TB reservoir. Approximately, one-third of the world s human population has latent TB and additionally, people with a latent TB infection have a lifetime risk of % of falling ill with an active TB infection (World ealth Organization, 4a). Field investigators face considerable Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 4, Thailand. Department of Applied Mathematics and informatics, Faculty of Science and Industrial Technology, Prince of Songkla University, Suratthani 84, Thailand. 3 Centre of Excellence in Mathematics, CE, Si Ayutthaya Road, Bangkok 4, Thailand. * Corresponding author: wichuta.sa@psu.ac.th Received date : 4/4/5 Accepted date : 3//5

2 Kasetsart J. (at. Sci.) 49(6) 99 difficulties in collecting reliable data on the number of people with latent and active TB infections. Consequently, the approximation of the spread of TB using mathematical models is necessary. Porco and Blower (998) developed a model of the intrinsic transmission dynamics of tuberculosis in the form of a flow diagram. Castillo-Chavez and Song (4) introduced the slow-fast model for TB transmission and also modeled the number of new cases of TB per year in the the United States from 953 to. Fortunately, most TB patients can be treated so the number of new cases is decreasing (World ealth Organization, 4a). Detailed information on human immunodeficiency virus (IV) and worldwide IV infection has also been given in a recent report by the World ealth Organization (World ealth Organization, 4b). At the present time, there is no known cure for IV infection. The most advanced stage of IV infection is acquired immune deficiency syndrome (AIDS). There are many transmission routes such as having sex or sharing needles with someone who has IV as well as receiving an IV-contaminated blood transfusion. Moreover, infants may acquire IV at delivery or through breast feeding if the mother is IV-positive. Once infected, individuals may have latent infection and not appear clinically ill. Of course, since most IV-infected people feel depressed and want to keep their status secret, there can be difficulty in obtaining reliable information on IV infection. Again mathematical models can be used to understand the AIDS epidemic (yman and Stanley, 988). Due to medical advances, there are now effective treatments and therapies to prolong the life of an IV-infected individual. Unfortunately, IV weakens the immune system and therefore, IV-infected people usually suffer from other diseases. TB is one of the leading causes of death in an IV-infected population. On the other hand, the number of IV-infected people carrying TB bacteria can be greater than in the non IV-infected population; therefore, understanding TB-IV coepidemics has become increasingly important. Long et al. (8) have constructed and analyzed TB-IV co-infection dynamics. Their parameter values were based on information from India. Bacaër et al. (8) proposed a system of ordinary differential equations with six compartments, combining two states for IV (IV- and IV+) with three states for TB infection (susceptible, latent TB and active TB). System parameters were obtained by either reviewing the medical literature or approximating from data from South Africa. Finally, they studied the impact of various control methods on these epidemics. In epidemiology, the basic reproduction number (R ) is one of the most important quantities used to predict invasion of a disease into a population. The basic reproduction number is defined as the expected number of secondary cases produced, in a completely susceptible population, by a typical infected individual during their entire period of infectiousness. Moreover, the threshold criterion states The disease can invade if R > whereas it cannot if R < (Diekmann et al., 99; Diekmann et al., ), that is, a diseasefree equilibrium point is locally stable if R < and unstable if R >. R can be observed in the field or calculated from mathematical models. A classical method for calculating the stability of a disease-free equilibrium point or an endemic equilibrium point is to calculate the eigenvalues of the Jacobian matrix of the linearized system about the equilibrium point. Alternative approaches for testing local stability are finding an intrinsic growth rate, a survivor function, or the next generation matrix (effernan et al., 5). Thus the basic reproduction numbers from different models and different approaches are unable to be compared. The objective of the current study was to compute and compare the basic reproduction numbers for the spread of TB through the two IV groups derived using the next generation method.

3 99 Kasetsart J. (at. Sci.) 49(6) MATERIALS AD METODS Model formulation The motivation for this study came from demographic statistics reported by the Thai Ministry of Interior and epidemiological information reported by the Thai Ministry of Public ealth (Ousvapoom et al., 3; Official Statistics Registration Systems, 5). These data provide sufficient TB and IV information to construct a TB-IV co-infection model. The main aim of the model is to compare the spread of TB through a population that is not infected with IV and through a population that is infected with IV. A compartment model was considered consisting of six populations classified either as infected or not infected with IV and either as being TB susceptible (not having TB) or as having either a latent or active TB infection as shown in Figure. The notations for the populations in Figure are as follows. S umber of people who are IV susceptible and TB susceptible E umber of people who are IV susceptible and have latent TB infection I umber of people who are IV susceptible and have active TB infection S umber of people who have IV infection and are TB susceptible E umber of people who have IV infection and latent TB infection I umber of people who have IV infection and active TB infection Clearly, S, E, I are subgroups of the non-iv group and S, E, I are subgroups of the IV group. Thus, Figure can be translated into the differential equations as follows. Equation is assumed to describe the rate of change of the population who are IV and TB susceptible. ds = Λ+ qri β S I µ S ΩS. () where Λ represents the natural birth rate, q r I represents the rate of recovery from active TB infection due to successful treatment, β S I represents the rate at which the TB susceptible pβ qr Λ S ( p ) β E µ µ d Ω Ω π α ( q) r I µ ( p ) β E S µ d α qr ( q) r I pβ Figure Flow diagram of tuberculosis (TB) and human immunodeficiency virus (IV) model (see Table for definition of terms).

4 Kasetsart J. (at. Sci.) 49(6) 993 population becomes infected with either latent TB or active TB due to contact with the actively infected TB population, µ S represents the natural death rate and ΩS represents an IV infection rate. Equation is assumed to describe the rate of change of the population who are IV susceptible and have a latent TB infection. de = ( p ) β S I + ( q ) ri ( α + µ + Ω ) E. () where ( p )β S I represents the rate at which the population of TB and IV susceptible people become infected with latent TB due to contact with the actively infected population, ( q ) r I represents the rate at which the non-iv population with active TB is partially cured and reverts to a state of latent TB infection, (α + µ + Ω) E represents a decrease in the non-iv, latent TB population due to transition to an active TB infection at rate α E, dying at death rate µ E or becoming IV-infected at rate ΩE In practice, it is difficult to identify a person with a latent TB infection and therefore it is difficult for them to receive proper treatment. Equation 3 is assumed to describe the rate of change of the non-iv, active TB population. di = p β S I + α E ( r + d + π ) I. (3) where p β S I represent the rate at which the non-iv, TB susceptible population becomes actively infected with TB due to contact with the actively infected population, α E represents the rate at which the latent TB of the non-iv population changes to the active form of TB, (r + d + π) I represents the rate of decrease of the non-iv, active TB population due either to: total cure to the susceptible TB population or partial cure to the latent TB population at rate r I, dying at death rate d I, or becoming infected with IV at rate ΩI. It is assumed that the rates of change of the IV-infected populations (S, E, I ) for the susceptible, latent and active TB states can be described by equations similar to Equations 3. Consequently, Equations 4 6 are: ds de di = S + q ri β S I µ S (4) Ω = ( p ) β S I + ( q ) ri + E ( α + µ ) E Ω (5) = p β S I + α E + π I ( r + d ) I. (6) In these equations, the meanings of the parameters d, p, q, r, α, β and µ for the IV infected populations are the same as the meanings of the parameters d, p, q, r, α, β and µ for the non-iv populations. Basic reproduction number The basic reproduction number R for a disease is defined so that a disease-free equilibrium point for a population is locally asymptotically stable if R < and unstable if R >. This can also be interpreted to mean that if one infected individual enters a population that is disease free, then R is the average number of secondary infections that will occur. TB-only model The TB-only model in Equations 3 has two infected states (E, I ) and one uninfected state (S ). At the TB-infection-free equilibrium point E = I = and hence the equilibrium population for S is S * = Λ. The appearance of µ S in Equations 3 can now be replaced by Λ µ. For small (E, I ), the following linearized system is obtained about the TB infection-free equilibrium point: de di Λ = ( p ) β I + ( q ) ri ( α + µ ) E µ (7) Λ = pβ I + α E ( r + d ) I. (8) µ R can be calculated from these equations by using the next generation method as described by Diekmann et al. (). Let X = (E, I )'

5 994 Kasetsart J. (at. Sci.) 49(6) where the prime denotes transpose. Following Diekmann s recipe, the linearized infection subsystem can be rewritten in the form: X = ( T + Σ ) X (9) where the matrix T corresponds to transmissions: ( p ) β Λ TTB = µ () p βλ µ and the matrix Σ corresponds to transitions: Σ TB q r = ( α + µ ) ( ). () α ( r+ d) ence, the next generation matrix (GM) with large domain is two-dimensional and given by K K TB L TB L = = T TB Σ () TB αβ Λ βλ ( p ) ( p ) ( α + µ ) µ µ (3) A αpβ Λ βλp ( α + µ ) µ µ where A = det Σ TB = (α + µ ) (r + d ) α r ( q ). The spectral radius of a matrix A is defined by ρ( A): = sup { λ : λ σ( A) }, where σ(a) denotes the spectrum of A. By definition, the basic reproduction number is the largest eigenvalue of the GM. Thus, R = ρ(k). Consequently, ( TB L ) TB TB TB TB R = ( KL )= trace KL + trace KL 4 K ρ ( ) ( ) det( ) (4) Finally, the reproductive number for the TB only model is ( ( )) TB R = ( p )α + p α + µ βλ. (5) Aµ ere some parameters have superscripts or subscripts to indicate the model they belong to. For example, R TB is the basic reproduction number for the TB-only model. IV-only model From Equations and 4 6, this system has two infected states (E, I ) and two uninfected states (S, S ) At the TB-infection-free equilibrium point E = I =. ence, Λ Ω Λ and. (6) + Ω * * S = S = µ + Ω µ µ For small (E, I ), the following linearized system is obtained: de di Ω Λ = ( p) β I + ( q) ri ( α + µ ) E µ µ + Ω (7) pβω Λ = I + αe ( r + d) I (8) µ µ + Ω Let X = (E, I )'. Then the following results are obtained: T IV = ( p )βω Λ µ µ + Ω pβω Λ µ µ + Ω q r Σ IV = ( α + µ ) ( ) α ( r + d ) ( p) αβω Λ ( p) βω Λ ( α + µ ) + K IV µ µ Ω µ µ + Ω L = B pαβω Λ pβω Λ ( α + µ ) µ µ + Ω µ µ + Ω (9) () () where B = det ΣIV = ( α + µ )( r + d) α ( q ) r. Therefore, the basic reproductive number IV for the IV-only model is R ρ K L. IV R = ( ) ( p) αβω Λ pβω Λ = + ( α + µ ) Bµ µ + Ω Bµ µ + Ω () TB and IV co-epidemic model From Equations 6, this system has four infected states (E, E, I, I ) and two uninfected states (S, S ). At the TB infection free equilibrium point E = I = E = I =. ence,

6 Kasetsart J. (at. Sci.) 49(6) 995 S Λ Ω Λ = and S =. (3) µ + π µ µ + Ω * * For small (E, E, I, I ), the following linearized system is obtained: de βλ = ( p ) I + ( q) ri ( α + µ + ) E µ + Ω Ω (4) de βω Λ = ( p ) I + ( q) ri + E ( α + µ ) E µ µ + Ω di Ω (5) pβλ = I + αe ( r+ d + π) I (6) µ + Ω di pβω Λ = I + αe + πi r + d I µ µ + Ω ( ) (7) Let X = ( E, E, I, I). Then the following results are obtained: T TB IV = βλ ( p ) µ + Ω βω Λ ( p ) µ µ + Ω pβλ µ + Ω pβω Λ µ + Ω µ ( α + µ + Ω) ( q) r Ω ( α + µ ) ( q ) r ΣTB IV = α ( r+ d + π) α π ( r + d ) (8) (9) and KL = TΣ. Since det K L =, the high dimensional matrix can be reduced to the GM with small domain which is given by K = T Σ where R = L ( p) βλ µ + Ω C = pβλ µ + Ω ( p) βωλ µ ( µ + Ω) pβωλ µ ( µ + Ω) (3) (3) Therefore, the basic reproductive number for TB-IV co-epidemic model can be obtained easily from the trace and the determinant of the corresponding two-dimensional matrix. ( ) R = K = trace K + trace K 4 K ρ( ) ( ) ( ) det( ). (3) S S S S It is difficult to show the long expression of the basic reproductive number, so the numerical value of R will be presented in the next section instead. RESULTS AD DISCUSSIO In, the Registration Administration Bureau, Department Local Administration, Ministry of Interior reported that the Thai population size was approximately 63,7,73 and the number of births and deaths were 766,37 and 44,888, respectively (Official Statistics Registration Systems, 5). Additional information from Bureau of Epidemiology, Department of Disease Control, Ministry of Public ealth, provided insight into the causes of death. owever, the current study is concerned with TB and IV only. The numbers of deaths-by-cause groups were 4,46 cases of TB and 3,63 cases of IV (Ousvapoom et al., 3). The death rate for IV-infected TB patients was about 4%. From clinical data in, there were 6,365 TB patients and 77,656 IV patients. About 8,564 people or 6% of TB patients were detected IV-positive (Bureau of Tuberculosis, 4). Therefore, the correspondence between the information and the model are: S + E + I + S + E + I + = 63,7,73 S + E + I + = 7,656 I + I = 6,365 6% I = 8,564 As a result, in, the Thai population can be classified into six compartments and the numbers in each compartment can be calculated step by step. Firstly, I = 53,55 and I = 8,84.

7 996 Kasetsart J. (at. Sci.) 49(6) ational Tuberculosis Control Programme Guidelines, Thailand, (3) also suggested that 9% of people who get infected with TB will develop latent TB. Otherwise, they become active TB (ateniyom, 3). To simplify the problem, this concept was used for the IV group and non- IV group separately. Therefore,.(E + I ) = I.(E + I ) = I Then E = 48,75 and E = 79,56. Finally, S = 89,56 and S = 6,888,797. Moreover, about one in ten latent TB will eventually progress to active TB. World ealth Statistics reported that the incidence rate of TB in Thailand was 37 per, population per year (Ousvapoom et al., 3). It can be seen that many epidemiological quantities are reported in the form of rate per, individuals. Thus, the whole population is considered as, individuals and proportions are used for the remainder. Table shows the meaning of parameters and their estimated values. Figures 4 are the numerical simulations carried out in support of the analytical results with the same set of parameters and initial conditions. Parameter values are shown in Table. The initial condition is (5,,,,, 5,,,,) Figure series of people with human immunodeficiency virus (IV) susceptible and TB susceptible (solid curve) and people with IV infected and TB susceptible (dashed curve). S S Table Description of the parameters, their values (per year) and references with regard to human immunodeficiency virus (IV) and tuberculosis (TB). Model Description Without IV Parameter value With IV Parameter value Mortality rate µ.64% e µ.88% e TB mortality rate d 5.95% e d 4.% d TB detection rate β.% e β 67.4% e Proportion of active TB p. b p. b TB recovery rate r 89.% a r 89.% a TB successful treatment q.97 a q.97 a TB activation rate α.% b α.% b Model Without infectious TB With infectious TB Description Parameter value Parameter value IV detection rate Ω.6% e π 6.% a Birth rate Λ, per, individuals a,c a = Bureau of Tuberculosis, Department of Disease Control, Ministry of Public ealth, Thailand (4). b = ateniyom S (3). c = Official Statistics Registration Systems (5). d = Tuberculosis Center Region (5). e = calculated from data of a, b, c and Official Statistics Registration Systems. 5.

8 Kasetsart J. (at. Sci.) 49(6) x Figure 3 series of people with human immunodeficiency virus (IV) susceptible and latent tuberculosis (TB) infected (solid curve) and people with IV infected and latent TB infected (dashed curve). E E I 5 I Figure 4 series of people with human immunodeficiency virus (IV) susceptible and active tuberculosis (TB) infected (solid curve) and people with IV infected and active TB infected (dashed curve). From Figures 4, the endemic equilibrium point ((S *, S *, E *, E *, I *, I * ) = (96.58,.8, 39,85.88, 6,79.39, 4,.95, 3,463.67)) clearly exists and is locally stable and is associated with the basic reproduction numbers: R TB IV =,39.98 and R = R = 7,87.57 which satisfy R >. This implies that TB, IV, and TB-IV infections will remain in the system. Consequently, TB and IV still spread in Thai society. These simulations confirm the theoretical result that the disease-free equilibrium is unstable. This model also reveals that birth rate plays an important role since more people lead to more chance to get infected. The detection rate (Ω) is also sensitive to R. ew numerical results with the previous parameter values except with the birth rate changed to Λ =.% show that the disease-free equilibrium point ((S *, S *, E *, E *, I *, I * ) = (.5,.3,,,, )) is locally stable associated with the basic reproduction numbers: R TB IV = R =.3 =.79, satisfied R <. and R It implies that the numbers of people with TB and IV decline and the infections have a strong tendency to disappear from the system as shown in Figures 5 7. After running several numerical calculations to investigate the trends of the basic reproduction number R when a related parameter changes, only four parameters (β, β, p, Λ) could provide the threshold of R =. Among them, TB detection rates (β, β ) could be used to compare the spread of TB in two groups. Figure 8 shows how the basic reproduction number R varies with TB detection rates. Clearly, the value of R increases linearly as β or β increases. The thresholds of R = for a non-iv group and an IV group occur at β = 7.73 and β = 37.7, respectively. From Table, β =. and β =.674 imply that the number of people in the two groups of TB decline and the infections have a strong tendency to disappear from the system. Finally, the threshold parameter R in the TB detection rate space is illustrated in Figure 9. From the literature review, many researchers have studied TB-IV. The current study was mainly focussed on the situation in Thailand.

9 998 Kasetsart J. (at. Sci.) 49(6) 8.5 x S S E E Figure 5 series of people with human immunodeficiency virus (IV) susceptible and tuberculosis (TB) susceptible (solid curve) and people with IV infected and TB susceptible (dashed curve) Figure 6 series of people with human immunodeficiency virus (IV) susceptible and latent tuberculosis (TB) infected (solid curve) and people with IV infected and latent TB infected (dashed curve) I I The basic reproduction number.5.5 beta beta Figure 7 series of people with human immunodeficiency virus (IV) susceptible and active tuberculosis (TB) infected (solid curve) and people with IV infected and active TB infected (dashed curve). Thus, most parameter values were obtained using information from the Epidemiological Information Section, Bureau of Epidemiology, Department of Disease Control, Ministry of Public ealth, Thailand (Epidemiological Information Section, 5). Unfortunately, the reported data is beta Figure 8 Relationship between the basic reproduction number R and tuberculosis detection rates. secondary data which can be very difficult to obtain and complicated to interpret and use. Moreover, the long period of latent TB infection implies that new cases of infection are not clinically apparent. Therefore, they are unobserved for some time. To take this delay into consideration, it would be necessary to develop a time-delay differential equation model.

10 Kasetsart J. (at. Sci.) 49(6) 999 give special thanks to Dr. Elvin J. Moore for his efforts to improve our English and to provide useful comments. LITERATURE CITED Figure 9 Invasion analysis in the tuberculosis detection rate space. COCLUSIO This model investigated the spread of TB through a non-iv group and an IV group. Each group consists of three compartments, namely TB susceptible, latent TB infected and active TB infected. The basic reproduction numbers predicting invasion of TB, IV, and TB-IV were determined by following Diekmann s recipe (Diekmann et al, ). The useful information from related disease control departments led to parameter estimation. Several numerical results also illustrated the existence of a stable, diseasefree equilibrium point and a stable, endemic equilibrium point. For Thailand, the infections still have a substantial effect on the Thai population because of the hidden TB-infected population. ACKOWLEDGEMETS This research was financially supported by ) the Department of Mathematics, Faculty of Science, Mahidol University, ) the Centre of Excellence in Mathematics, Commission on igher Education, Ministry of Education and 3) the Department of Applied Mathematics and Informatics, Faculty of Science and Industrial Technology, Prince of Songkla University, Surat Thani Campus, 6. The authors would like to Bacaër,., R. Ouifki, C. Pretorius, R. Wood and B. Williams. 8. Modeling the joint epidemics of TB and IV in a South African township. J. Math. Biol. 57: Bureau of Tuberculosis, Department of Disease Control, Ministry of Public ealth, Thailand 4. [Available from: images/present/pre pdf]. [Sourced: 9 February 5]. Castillo-Chavez, C. and B. Song. 4. Dynamical models of tuberculosis and their applications. Math. Biosci. Eng. : Diekmann, O., J.A.P. eesterbeek and J.A.J. Metz. 99. On the definition and the computation of the basic reproduction ratio R in models for infectious diseases in heterogeneous populations. J. Math. Biol. 8: Diekmann, O., J.A.P. eesterbeek and M.G. Roberts.. The construction of nextgeneration matrices for compartmental epidemic models. J. R. Soc. Interface. 7: Epidemiological Information Section. 5. Bureau of Epidemiology, Department of Disease Control, Ministry of Public ealth, Thailand. [Available from: moph.go.th/files/ report/6_ pdf]. [Sourced: 9 February 5]. effernan, J.M., R.J. Smith and L.M. Wahl. 5. Perspectives on the basic reproductive ratio. J. R. Soc. Interface. : yman, J.M. and E.A. Stanley Using mathematical models to understand the AIDS epidemic. Math. Biosci. 9: Long, E.F.,.K. Vaidya and M.L. Brandeau. 8. Controlling co-epidemics: analysis of IV and tuberculosis infection dynamics. Oper. Res. 56:

11 Kasetsart J. (at. Sci.) 49(6) ateniyom S. 3. ational Tuberculosis Control Programme Guidelines. The Agricultural Co-operative Federation of Thailand, Limited. Bangkok, Thailand. 8 pp. Official Statistics Registration Systems. 5. Department of Provincial Administration, Ministry of Interior, Thailand. [Available from: stattdd]. [Sourced: 8 October 5]. Ousvapoom, P, A. Suthiart, T. Aryukasam and S. Senpong. 3. Statistical Thailand 3. WVO Office of Printing Mill. Bangkok, Thailand. 8 pp. Porco, T.C. and S.M. Blower Quantifying the intrinsic transmission dynamics of tuberculosis. Theor. Popul. Biol. 54: 7 3. Tuberculosis Center Region. 5. Tuberculosis Center Region, Yala [Available from: uploads/4/6/. Thailand-Tuberculosis- Situation_Feb4_V4.pdf ]. [Sourced: February 5]. World ealth Organization. 4a.World ealth Organization, Media Centre, Factsheets, Tuberculosis [Available from: who.int/mediacentre/factsheets/fs4/en/]. [Sourced: February 5]. World ealth Organization. 4b. World ealth Organization, Media Centre, Factsheets, IV/AIDS [Available from: who.int/mediacentre/factsheets/fs36/en/]. [Sourced: February 5].