Tuberculosis is a dangerous infectious disease caused by the mycobacteria of the M. ... Mathematical modeling of the spread and treatment of tuberculosis.
c Pleiades Publishing, Ltd., 2007. ISSN 0005-1179, Automation and Remote Control, 2007, Vol. 68, No. 9, pp. 1604–1617. c K.K. Avilov, A.A. Romanyukha, 2007, published in Avtomatika i Telemekhanika, 2007, No. 9, pp. 145–160. Original Russian Text
CONTROL IN BIOLOGICAL SYSTEMS AND MEDICINE
Mathematical Modeling of Tuberculosis Propagation and Patient Detection1 K. K. Avilov and A. A. Romanyukha Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow, Russia Received March 13, 2007
Abstract—A mathematical model of the spread of tuberculosis taking into account the impact of the antituberculosis programs was considered. The singularity of the proposed model lies in making explicit discrimination between the detected and overlooked patients and taking into account their migration. Studies demonstrated insufficiency of the standard accounting form of the regional Russian antituberculosis institutions. Proposed was a submodel enabling objective estimation of the patient detection system on the basis of individualized database. These estimates corroborated the assumption of a substantial nonuniformity of the Russian regions in terms of tuberculosis morbidity and efficiency of patient detection. For example, the rates of patient detection can differ by the factor of five through six. The model can be used to forecast the tuberculosis morbidity and efficiency of the antituberculosis programs. PACS number: 87.23.Cc DOI: 10.1134/S0005117907090159
1. INTRODUCTION Tuberculosis is a dangerous infectious disease caused by the mycobacteria of the M. tuberculosis and M. bovis kinds. In the case of the untreated respiratory organs tuberculosis (ROT), mortality reaches 50%. The patients discharging bacteria with grave forms of ROT are the main sources of infection for other individuals, the infection being transmitted by aerogenic means. ROT averages 80–90% of the total morbidity. Medication of the patients detected by the antituberculosis institutions is now the main means of tuberculosis control. It not only increases the probability of clinical patient recovery, but in a short time2 dramatically reduces or completely stops the discharge the tuberculosis infecting agents by patients, which breaks the chain of infection transmission and reduces the infection strength λ, that is, the probability of infecting one individual in unit time. This reduces the tuberculosis morbidity both in short-term owing to the reduction in the number of cases caused by superinfection of the already infected individuals and fast development of disease in the newly infected individuals and in long-terms owing to the reduction in the general level of population contamination. Therefore, fast detection of an appreciable part of the ROT patients is the prerequisite for efficiency of the modern antituberculosis programs. Mathematical modeling of the spread and treatment of tuberculosis offers one of the methods of data analysis and estimation of the efficiency of antituberculosis actions. Several dozens of mathematical models were developed to describe the processes of tuberculosis spread and treatment, as well as to seek the optimal strategy of controlling this disease. The structure of models varied together with the notions of tuberculosis pathogenesis (mechanisms of 1 2
The work was supported by the Grant of the Presidium of Russian Academy of Sciences “Fundamental Sciences for Medicine,” 2006. From several weeks to two months (in the case of efficient treatment). 1604
MATHEMATICAL MODELING OF TUBERCULOSIS PROPAGATION α
Π S Noninfected
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λ
L Infected
δ
T Diseased
μ
μ
μ+μt
Fig. 1. Block diagram of the model [1]. Π is the inflow of the young people, μ is the mortality, μt is the additional, disease-induced mortality, λ is the infection strength, δ is the constant of the rate of development of the active disease, and α is the recovery rate constant.
development) and the strategy of antituberculosis measures. The first published mathematical model of the tuberculosis spread [1] considered it as an ordinary infectious disease with a very long latent period and without the possibility of complete excretion of the mycobacteria from the human organism (Fig. 1). However, as the authors of [1] note, this simplistic approach serves just as a base for more sophisticated models taking into account the peculiarities of tuberculosis and its distinctions from other infectious diseases. Among the most important features reflected in the mathematical models of spread and treatment of tuberculosis, one may cite the following: Specificities of pathogenesis: • very long latent period (its expected mean duration exceeds very much the mean duration of the human life; that is why the active forms of the disease develop only in a small part of the infected persons (10–30%)); • possibility of fast disease progress (higher probability of disease development after the primary infection); • superinfection (higher probability of disease development in the first several years after the recontamination of the already infected individual); • dependence of the parameters of interaction with the mycobacterium (sensitivity to infection, risk of disease development, rate of the spontaneous self-recovery, and so on) on the age and sex of the individual; • discrimination between the noninfectious and infectious forms of disease which may be understood either as two alternatives of disease development or as two successive stages of a single process; • possibility of the development of bacterioexcretion (infectiousness) in noninfectious patients; Specificities of treatment and prophylaxis: • treatment of patients (and carriers of the latent infection); • immunization (mostly by the Calmette–Gurin bacillus (BCG) reducing the risk of disease development, but actually not protecting against the infection); • discrimination between the detected and overlooked patients (consequently, getting treatment or not); • inefficient treatment leading to selection of the drug-resistant cultures of the tuberculosis infectious agent; • distinction of the group of recovered patients from other groups (the recovered ones can be regarded both as getting the life-long immunity or having a higher risk of disease recurrence than the ordinary infected individuals); Other specificities: • demographic processes, including migration (they are more affected by the demographic factors because of the slowness of the processes of tuberculosis spread); AUTOMATION AND REMOTE CONTROL
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method of describing the transmission of infection (depends on the assumptions about the structure of constants between the individuals); • interaction with other diseases (mostly HIV/AIDS that accelerate development of the tuberculosis and reduce immunity to the tuberculosis infection, but at the same time reduce the fraction of patients developing bacterioexcretion). The majority of the published mathematical models of tuberculosis spread and treatment focus on a narrow class of phenomena from the aforementioned list and estimation of the efficient of the antituberculosis program (lying in treatment and immunization) with regard for the above effects. Some publications [2–4] attempted to develop models reflecting simultaneously the majority of the specific feature of tuberculosis spread. However, both the overwhelming majority of the simpler models and more sophisticated models have a disadvantage in common, that is, they cannot be adjusted to the actual data. Efficiency of the antituberculosis programs is estimated using the mathematical models adjusted either to the assumed data (expert estimates) about the current tuberculosis situation or to some model population which, strictly speaking, has no grounds to be regarded as an equivalent of the actual population. Additionally, some works estimated the current tuberculosis situation either by adjusting the mathematical model to a stationary solution or by modeling the historical development of epidemy.3 Among the studies relying on the real data one may cite the works of H.T. Waaler (for example, [1, 5, 6]) and E. Vynnycky and P.E.M. Fine (for example, [7, 8]) who developed the concepts of I. Sutherland [9]. Waaler used in his models the data of the World Health Organization acquired in Bangalore (India) [10] which describe comprehensively the tuberculosis situation in a population of over 60 thousand persons over five years. The Vynnycky–Fine model from the very outset was constructed to estimate the rates and relative contributions to morbidity of the different ways of tuberculosis development (endogenous activation, exogenous superinfection and fast progression). The data about the strength of infection in 1880–1988 and tuberculosis morbidity (with age stratification) in 1953–1988 in England and Wales were used to adjust the model. Nevertheless, none of the publications could estimate the effect of patient underdetection. At adjusting the models which do not differentiate between the detected and overlooked patient, the detected morbidity, that is, the number of patients newly detected by the medical institutions, was usually assumed to be equal to the full morbidity, and for differentiation between these types of patients, some fixed fraction or constant of the rate of detection was usually defined on the basis of publications or expert estimates. However, the studies of the present authors [11–13] demonstrated that the regions of the European part of the Russian Federation are heterogeneous in quality and rate of detecting the patients. Therefore, without independent qualitative estimation of the parameters of patient detection for each of the regions under consideration, the adjustment of the mathematical models of tuberculosis spread to the data for Russia will be incorrect. •
2. MATHEMATICAL MODEL OF TUBERCULOSIS SPREAD AND DETECTION IN RUSSIA The mathematical model used below was already published in [14]. It is based on the division of the modeled population into six time-dependent groups (Fig. 2): • S(t)—noninfected by the tuberculosis infectious agent, • L(t)—infected individuals (carries of the latent infection), 3
It was assumed that at some time instant (for Europe, it is the early XIXth century), one patient was introduced into a fully noninfected population, which was followed by an epidemic accompanied by the advent of efficient medication in the 1950’s and spread of HIV/AIDS in the 1980–1990’s. The current situation was estimated as the state of the mathematical model by the current official year with regard for all aforementioned impacts whose parameters were defined by means of the expert estimates. AUTOMATION AND REMOTE CONTROL
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inflow of the young people S Noninfected
fast progress
infection TnU Nondetected BE--- patients
L Infected
detection of BE--clinical recovery
Δ TiU detected BE + patients detection of BE +
Δ TnT detected
Δ TiT detected BE + patients
BE--- patients
Fig. 2. Model block diagram [14]. Flows of mortality and migration are not shown.
TnU (t)—overlooked patients without bacterioexcretion (BE−), • T U (t)—overlooked patients with bacterioexcretion (BE+), i • T T (t)—detected patients without bacterioexcretion (BE−), n • T T (t)—detected patients with bacterioexcretion (BE+). i Dynamics of the groups is described by the system of ordinary nonlinear differential Eqs. (1)–(6), the parameters used being described in the table: •
dS = fS (t) − λ(t, S, . . . , TiT ) + μ + e S, dt
(1)
dL = fL (t, S, . . . , TiT ) + (1 − p)λ(t, S, . . . , TiT )S + γL TnU + γL0 TnT dt − δ + pλ(t, S, . . . , TiT ) + μ + e L,
(2)
dTnU = fnU (t) + pλ(t, S, . . . , TiT )S + δ + pλ(t, S, . . . , TiT ) L + γn TiU dt −(ϕn + γL + δi + μnU + enU )TnU ,
(3)
dTiU dt
= fiU (t) + δi TnU − (ϕi + γn + μiU + eiU )TiU ,
dTnT = fnT (t) + γn0 TiT + ϕn TnU − (γL0 + δi0 + μnT + enT )TnT , dt dTiT = fiT (t) + δi0 TnT + ϕi TiU − (γn0 + μiT + eiT )TiT . dt
(4) (5) (6)
Equation (1) describes the inflow of the noninfected young people and noninfected migrants (fS (t)), the process of infection (term −λS), as well as the death (−μS) and migration outflow (−eS). Equation (2) describes the inflow of the infected young people and infected migrants (fL (t)), the flow of recently infected individuals who passed into the stage of latent infection ((1 − p)λS), the flows of recovered (γL0 TnT ) and spontaneously self-recovering (γL TnU ) individuals, disease development as the result of endogenous activation (−δL) and exogenous superinfection (−pλL), death (−μL) and migration outflow (−eL). In Eq. (3) which describes the dynamics of the group of overlooked patients without bacterioexcretion, in addition to the above morbidity flows there is the flow pλS modeling rapid progression of the disease in part of the recently infected individuals. The function fnU (t) defines the migration inflow to the group TnU , the flow −δi TnU corresponds to the disease progress, the flow −ϕn TnU , the detection of the patients without bacterioexcretion, and the flows −μnU TnU and −enU TnU , to death and migration outflow. Similar notation is used for the group of the overlooked patients with bacterioexcretion TiU : the flow −γn TiU models spontaneous cessation of bacterioexcretion, the flow −ϕi TiU , detection of the patients of AUTOMATION AND REMOTE CONTROL
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Coefficients and variables used in the model [14]. Abbreviation “r.c.” stands for “rate constant,” note “varies” means that the value is estimated from the statistical data independently for each region Symbol Description Value U U T T fi (t) rate of the migration inflow to the group i, i = S, L, Tn , Ti , Tn , Ti ; varies fS (t) and fL (t) include the inflows of young people infection strength calculated λ μ mean population mortality rate for the groups S and L varies 0.07 1/year μnU mortality rate in the group of overlooked BE− (TnU ) μiU mortality rate in the group of overlooked BE+ (TiU ) 0.5 1/year μnT mortality rate in the group of detected BE− (TnT ) varies μiT mortality rate in the group of detected BE+ (TiT ) varies r.c. migration outflow from the groups S and L 0.002 1/year e ei r.c. migration outflow from the group of patients i, i = TnU , TiU , TnT , TiT varies p probability of fast disease progress 5% r.c. of endogenous activation 1.43 × 10−3 1/year δ δi r.c. of development of disease infectious forms 0.79 1/year γL r.c. of spontaneous self-recovery 0.14 1/year r.c. of spontaneous cessation of bacterioexcretion 0.5 1/year γn ϕi r.c. of detection of patients with bacterioexcretion varies r.c. of detection of patients without bacterioexcretion varies ϕn δi0 r.c. of development of infectious forms of disease against the background of varies treatment varies γL0 r.c. of clinical recovery γn0 r.c. of cessation of bacterioexcretion against the background of treatment varies τi mean duration of BE+ stage of disease in the absence of treatment 1 year mean duration of BE− stage of disease in the absence of treatment 1 year τn rB cumulative probability of detecting a person discharging bacteria 80% relative infection of the treated persons discharging bacteria 0.1 k Ic fraction of the infected at infancy 30% Im fraction of infected among migrants 50% 2% M C cumulative infant mortality rate AM fraction of adults among migrants 80% ρt spatial density of urban population 1975 persons/km2
the group TiU , and the flows −μiU TiU and −eiU TiU , death and migration outflow. Equations (5) and (6) for the groups of detected patients on the whole are similar to those for the overlooked patients, the difference lying in the fact that the detection flows are directed to the groups of the detected patients, there are no morbidity flows (all infected persons are regarded as unknown to the medical institutions) and all rate constants take into consideration the effect of treatment. This model block-diagram allows one to describe three classes of processes: • Demographic—nativity–mortality balance and migration (they act mostly on the groups S and L because they comprise up to 99% of the population). • Epidemiological—the processes of infection spread and development (modeled mostly by the groups L, TnU , and TiU ). • Effect of the medical institutions—detection and treatment of the patients (modeled by the groups TnT and TiT ). Owing to the practically general BCG immunization of the Russian population which results in an appreciable relaxation of the processes of tuberculosis spread among children, the modeled part of the population includes only persons over fifteen who are all of them regarded as vaccinated. Additionally, consideration is given only to the pulmonary tuberculosis as defined the spread of tuberculosis infection. AUTOMATION AND REMOTE CONTROL
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The form of infection strength λ vs. the model variables deserves special discussion. Other authors most frequently used the following two approaches: (1) “Full mix model” λ = βg Tieff .
In this case, the coefficient βg defines efficient infection of one patient, and Tieff defines the efficient number of infectious patients in the population. This model corresponds to the assumption that any individual in the population has the same probability of contact with any other individual. (2) “Direct interaction model” λ = βl Tieff /N .
Here, Tieff is the efficient number of the infectious patients, N denotes the total number of population. This model is based on the assumption that in a unit time each infectious individual infects precisely βl individuals. The mathematical model in [14] was constructed using the “full mix model” and Tieff = TiU + kTiT , but later the form of the dependence of λ on the model variables was modified. The present paper makes use of the following relation: λ(t, S, . . . , TiT ) = βρeff (TiU + kTiT ),
(7)
where ρeff (TiU +kTiT ) is the efficient spatial density of the persons discharging bacteria. This enables us to allow efficiently in the model for the information about the variations of the population density in the regions under study. 3. PRIMARY ADJUSTMENT OF THE MODEL The accounting data from “Form 33”4 for the twenty-one region of the European Russia that are sufficiently close in their sociological and epidemiological indices were used for the primary adjustment of model (1)–(6). These data describe the patient in the “visibility zone” of the antituberculosis institutions associated within the framework of our model with the groups TnT and TiT . The statistical data define completely the dynamics and interaction of the groups TnT and TiT , the inflow of the detected patients to these groups from the groups of overlooked patient TnU and TiU , and the flow from the group TnT to the group L modeling the discharge of the clinically recovered individuals from the visibility zone” of the antituberculosis institutions. As for the rest of the groups, there exists no direct information: the total population, the mean population mortality rate, the inflow of the young people, and the migration flows are known only from the demographic data [15]. The task of model adjustment comes to estimating the sizes of the groups S, L, TnU , and TiU by means of the expert-estimated constants describing the properties of the individuals making up these four groups, as well as the parameters of the patient detection system (see Table). Since the characteristic durations of the processes related with the tuberculosis spread are usually comparable with the mean duration of the human life and are much longer than the time interval over which the mathematical model is considered, it was attempted to make use of the condition for stationarity of the sizes of the groups S, L, TnU , and TiU as the parameter estimation criterion. The calculations demonstrated that there exists no general stationary solution for all regions under consideration, that is, for the identical set of expert estimates (see table), for some regions the estimated variables have “nonphysical” values such as negative sizes of groups, which suggested that the stationarity assumption must be rejected. Nevertheless, with the assumption of time-constancy of the parameters of the patient detection system, the statistical data enable rough estimation of the dynamics of the sizes of the groups TnU and TiU of overlooked patients. The relative variations of the flows of yearly patient detection defined by ϕn TnU (t) and ϕi TiU (t) may estimate the relative variation of the quaistationary numbers 4
Standardized accounting form of the regional tuberculosis prophylactic centers to the head organization, the Sechenov Research Institute of Phthisiology and Pulmonology. AUTOMATION AND REMOTE CONTROL
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Bryansk Tula
Orel
Volgograd
Kaluga
Tambov
Lipetsk
Pskov
Voronezh
Ivanovo
Tver'
Nizhnii Novgorod
Ryazan'
Belgorod
Penza
Kursk
Kostroma
60
Vologda Yaroslavl'
80
100 thousands 100 thousands
Saratov
100
(a) TiU per TiU per
Vladimir
120
20 0
(b)
Bryansk
Pskov Volgograd
Kaluga
Tula
Belgorod
Ryazan'
Penza
Orel
Tambov
Saratov
Lipetsk
Ivanovo
Kursk
Voronezh
Tver'
Nizhnii Novgorod
Vladimir Kostroma
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
Vologda Yaroslavl'
S/N L/N
Fig. 3. Estimated sizes of the groups TnU , TiU , S, and L for twenty-one Russian region on the basis of equal parameters of the patient detection system. Used were the data averaged over the period 1999–2001. (a) Groups of overlooked patients without and with bacterioexcretion (TnU and TiU ) per 100 thousands of adult population. (b) Groups of noninfected infected individuals (S and L) in fractions of the total number of adult population.
of the groups TnU and TiU of overlooked patients if ϕn and ϕi are time-independent. It follows from the data that these variations are relatively small (upon the average, 7% yearly). Figure 3 depicts the results of estimating the number of the overlooked patients (groups TnU and TiU ) with regard for their time dynamics and population level of contamination (groups S and L) and use of the additional heuristic estimates of the epidemic process parameters replacing the conditions for stationarity of the groups S and L. One can readily see that the estimates of the numbers of groups TnU and TiU as reduced to 100 thousand of population5 differ more than by the factor of three, and the fractions of groups S and L to the total population, that is, the level of contamination, varies from 26 to 81%. Such a scatter of the epidemiological indices in the Russian regions under consideration seems to be unlikely. 4. SUBMODEL FOR ESTIMATION OF THE PATIENT DETECTION PARAMETERS The study of the relations used to estimate the numbers of the groups S, L, TnU , and TiU by the estimates of dynamics of the two last groups and additional heuristic estimates of the epidemic process parameters demonstrated that more than 90% of the scatter in the results is due to the scatter of the yearly number of the detected patients discharging bacteria per capita, that is, the flow from TiU to TiT , which also varies by a factor greater than 2.8 on the considered twenty-one 5
By the total number of population is meant here the number of individuals older than 15 years. AUTOMATION AND REMOTE CONTROL
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MATHEMATICAL MODELING OF TUBERCULOSIS PROPAGATION I γL
δi
TnU Overlooked BE--- patients k1ϕp
γn
TiU Overlooked BE+ patients
(1 – d) d
ΔTna Actively detected BE--- patients
ϕp
ϕa
k2ϕp
ΔTnp Passively detected BE--- patients
ΔTnT detected BE--- patients
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(1 – d)
ΔTia Actively detected BE+ patients
d
ΔTip Passively detected BE+ patients
ΔTiT detected BE+ patients
Fig. 4. Block diagram of the submodel of patient detection. The flows of mortality rate and migration are not shown.
region of the European Russia. Since in our model this flow is defined as ϕi TiU , the assumption of constancy and equality of the rate ϕi of detection of the patients discharging bacteria in all regions leads to more than a triple variation in the size of group TiU and subsequent variations in the sizes of the rest of groups under consideration. This result signifies the need for studying the differences in the parameters of the patient detection system at the scale of the Russian regions and constructing a method of their numerical estimation. For that, a submodel of the patient detection process was proposed. This submodel describes the structure of the patient detection flows from the groups TnU and TiU to the groups TnT and TiT (Fig. 4). It considers only the dynamics of the groups TnU and TiU , and the morbidity flows from the groups S and L are combined in the total morbidity I. It is assumed that there exist two mechanisms of detecting the tuberculosis patients—active (mass roentgenofluorography, prophylactic medical examination, and so on) and passive (at visits to doctor with complaints on respiratory problems). The passive detection mostly allows one to detect seriously ill patients with explicit symptoms, the active detection is efficient both in the detection of the seriously ill patients and those with early stages of the disease. For the active and passive methods, the constants of the rate of detection of the persons discharging bacteria are denoted, respectively, by ϕa and ϕp . The full constant of the rate of detection of the persons discharging bacteria ϕi is expressed as ϕi = ϕa + ϕp . The corresponding constant of the rate of detection of the patients without bacterioexcretion are defined as k1 ϕa and k2 ϕp , the coefficients k1 and k2 define the relative efficiency of detecting patients at the early stages as compared with the late stages. The full constant ϕn of the rate of detection of the patients without bacterioexcretion is expressed as ϕn = k1 ϕa + k2 ϕp . The detected patients are examined for bacterioexcretion. It is believed that all detected patients without bacterioexcretion are registered as BE−, that is, the number of false positive results of the diagnostic laboratory is regarded as negligible. The detected patients with bacterioexcretion are diagnosed with the probability d by the laboratory as discharging bacteria and with the probability (1 − d) get the false negative decision.6 The patients getting the false negative conclusion of the diagnostic laboratory are registered as BE−, thus, formally enriching the flows of detection of the 6
It deserves noting that the so-defined “quality of the diagnostic laboratory” d is not so much the probability of correct diagnosis of bacterioexcretion, but rather the measure of relation between the laboratory sensitivity and the boundary between the states of “BE+ disease” and “BE− disease.” This boundary is defined indirectly by the choice of the parameters defining the dynamics of the groups of the overlooked patients: γL , γn , δi , μnU , μiU . AUTOMATION AND REMOTE CONTROL
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BE− patients. Determination of the method of patient detection (active or passive) is regarded as faultless. For each modeled population, therefore, the given submodel generates four flows of patient detection: (1) ΔTna —patients actively detected as BE−; (2) ΔTnp —patients passively detected as BE−; (3) ΔTia —patients actively detected as BE+; (4) ΔTip —patients passively detected as BE+. The first of the submodel Eqs. (8)–(9) is a modification of Eqs. (3) and (4), the second equation formalizes the aforementioned structure of the patient detection process.
⎡
d dt
TnU TiU
=
1 ⎢ − k1 ϕa + k2 ϕp + τ + enU n +⎢ ⎣
δi ⎛ ⎜ ⎜ ⎜ ⎝
ΔTna ΔTnp ΔTia ΔTip
⎞
⎡
⎟ ⎢ ⎟ ⎢ ⎟=⎢ ⎠ ⎣
fnU (t) + I fiU (t)
⎤
γn
− ϕa + ϕp +
k1 ϕa ϕa (1 − d) k2 ϕp ϕp (1 − d) 0 ϕa d 0 ϕp d
⎤ ⎥ ⎥ ⎥ ⎦
1 + eiU τi TnU TiU
⎥
⎥ ⎦
TnU TiU
,
(8)
.
(9)
5. ADJUSTMENT OF THE SUBMODEL TO ESTIMATION OF THE PARAMETERS OF PATIENT DETECTION: RESULTS The four flows of detection appearing in the above submodel cannot be estimated within the framework of the statistics by “Form 33.” It turned out possible to get the desired data on patient detection (with segregation both by bacterioexcretion and the method of detection) from the individualized database of the detected patients created by the Research Institute of Phthisiology and Pulmonology. This base, however, stores the data on the patients detected only in nine of the twenty-one considered regions of the European Russia. That is why the following results will concern only these nine regions. When adjusting the detection submodel to the real data, it was assumed that the parameters of the overlooked patients are identical for all regions and equal to those used in the main model (see Table). The coefficients k1 and k2 were also assumed to be equal for all regions. The yearly migration inflows fnU and fiU of the patients and the constants of the migration outflow rates enU and eiU , the yearly flows of detection ΔTna , ΔTnp , ΔTia , and ΔTip , and the relative variation of the sizes of the groups of overlooked patients were estimated for each region and for each year from the statistical data. The regions were parametrized by the four variables: I—morbidity, ϕa —constant of the rate of active detection from the group TiU , ϕp —constant of the rate of passive detection from the group TiU , and d—“laboratory quality.” This approach enabled us to estimate from Eq. (9) all four regional characteristics (I, ϕa , ϕp , d) by substituting the sizes of the groups TnU and TiU obtained from Eq. (8) by means of equating the time derivative to a certain rate of variation of the sizes of these groups (see Appendix A). The calculations were carried out both for the independent data for each of the years 1997–2005 and the smoothed data averaged over three “adjacent” years with equal weight. Estimation of the domain of admissible values of the coefficients k1 and k2 is described in Appendix B. The absolute values of the resulting estimates are substantially dependent on the choice of k1 and k2 , but the relation of the estimates for different regions and, consequently, their ranking is weakly dependent AUTOMATION AND REMOTE CONTROL
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(a) ϕa, 1/year
(b) ϕp, 1/year
12
8
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9
6
6
4 3
2 0 1998 150
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0 2004 1998
2002
(c) I, 1/(year H 105)
120
2000
2002
2004
(d) d
0.90 0.75
90 0.60 60 30 1998 160
2000 (e)
TnU,
2002 person/105
0.45 2004 1998
45
100
30
70
15
40 1998
2000
2002
Bryansk reg. Orel reg. Tula reg.
(f)
60
130
2002
2000
0 2004 1998
T iU,
2000
Yaroslavl' reg. Belgorod reg. Voronezh reg.
2004
person/105
2002
2004
Lipetsk reg. Tambov reg. Saratov reg.
Fig. 5. Results of adjustment of the detection submodel for k1 = 0.08 and k2 = 0.004. (a) r.c. of active detection of persons discharging bacteria ϕa , 1/year; (b) r.c. of passive detection of persons discharging bacteria ϕp , 1/year; (c) total yearly morbidity I per 100 thousands of the modeled population; (d) “laboratory quality” d; (e) number of the overlooked patients without bacterioexcretion TnU per 100 thousands of the modeled population; (f) number of the overlooked patients with bacterioexcretion TiU per 100 thousands of the modeled population.
on the choice of k1 and k2 . For illustration sake, these results are shown for fixed values k1 = 0.08 and k2 = 0.004 of these variables taken from the domain providing estimates of the rest of the parameters that are the most likely in the order of values. Figure 5 shows the parameter estimates obtained from the data smoothed over the period 1998– 2004 (the data for 1997–2005 were used for smoothing). The results obtained when using the nonsmoothed data do not differ appreciably from those based on the smoothed data. 6. CONCLUSIONS The present work aims mostly at getting a comprehensive and, possibly, objective picture of tuberculosis morbidity and estimating the efficiency of the antituberculosis service. Importantly, we mean analysis of the actual data acquired annually in all Russian regions. The distinction of these studies lies in the impossibility of carrying out direct experiment and verification of predictions. For example, the number of the overlooked patients can be estimated on the basis of plausible AUTOMATION AND REMOTE CONTROL
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assumptions and data for the detected patients, but experimental verification of this estimate is impossible. Even if a single town or settlement is thoroughly examined, the data obtained will correspond only to this town or settlement and not to the entire region. This fact is rooted in the substantial nonuniformity of the epidemiological indices, including morbidity: morbidity detected in similar inhabited localities may differ by the factor of 2 to 3. Together with the study of the previously published mathematical models of tuberculosis spread, as well as the attempt to adjust the model of [14] to the real data with identical assumed parameters of the patient detection system for all regions of the European Russia, this fact demonstrated the need for independent estimation of the efficiency of patient detection for each of the regions under consideration. However, as follows from the analysis of “Form 33,” the amount of statistical data providing information about the overlooked patients, that is, detection flows, is smaller than the number of parameters describing the tuberculosis situation and efficiency of patient detection. All this gave rise to the need for more detailed data (individualized database) and more detailed patient detection mathematical model adapted to the data available. The results of submodel adjustment suggested the following: • Relative time stability of the results and their consistency with the qualitative expert estimates corroborates correctness of the assumptions made. • Assumption about the significant differences between the European Russian regions in the efficiency and strategy of patient detection, as well as about the relative stability of the tuberculosis situations in these regions was confirmed. • Model of patient detection taking into account the distinctions of the actual patient detection system and the existing dependences between the patient status (estimation with variable accuracy) and the method of patient detection (estimated with high accuracy) was constructed. The specified estimates of the parameters of patient detection and total morbidity enable one to make more justified estimates of the levels of overlooked morbidity and contamination by the tuberculosis infectious agent. However, the justified estimation of the contamination level requires additional analysis of the process of infection transmission. It deserves special noting that systematic estimation of the performance of the patient detection system in all Russian regions requires some modification of “Form 33” by introducing two additional parameters: fractions of the actively detected patients among the newly detected BE− and BE+ patients. On the whole, the mathematical models of tuberculosis spread and treatment together with the methods of their adjustment to the actual data enable estimation of the current tuberculosis situation beyond the “visibility zones” of the antituberculosis institutions, forecasting of the situation in the near future, as well as solution of the problem of optimal allocation of the funds between different tasks challenging the antituberculosis institutions. APPENDIX A We derive here the formulas for estimation of I, ϕa , ϕp , and d on the basis of Eqs. (8), (9). It is assumed that the values of the detection flows ΔTna , ΔTnp , ΔTia , and ΔTip , parameters k1 and k2 , patient migration inflows fnU and fiU , constants of the rates of patient migration outflows enU and eiU , relative rates ξnU and ξiU of gain in the sizes of groups TnU and TiU , respectively, as well as all rate constants related with the natural dynamics of the patient groups (τn , τi , γn , and δi ) are known. We introduce for convenience the variable r = ΔTia /ΔTip . Since ΔTia and ΔTip are assumed to be known, r can be calculated with ease. Consideration of the third and fourth lines of Eq. (9) AUTOMATION AND REMOTE CONTROL
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ϕa = rϕp .
(A.1)
readily leads to
By summing the first and third, as well as the second and fourth lines of Eq. (9) and applying (A.1), one can obtain the linear equation comprising only one of the estimated variables ϕp :
ΔTna + ΔTia ΔTnp + ΔTip
k1 r r k2 1
= ϕp
TnU TiU
.
(A.2)
The expression for the sizes of the groups TnU and TiU appearing in the right-hand side of (A.2) is deduced from Eq. (8). The time derivatives of the group sizes are replaced by the products of the sizes themselves by the relative increment rates ξnU and ξiU :
ξnU TnU ξiU TiU
=
fnU + I fiU
⎡
1 ⎢ − k1 ϕa + k2 ϕp + τ + enU n +⎢ ⎣ δi
Obvious transformation with regard for (A.1) provides
fnU + I fiU
= −Amod
⎤
γn
− ϕa + ϕp +
TnU TiU
1 + eiU τi
⎥
⎥ ⎦
TnU TiU
.
,
(A.3)
where Amod is the matrix from Eq. (8) modified with allowance for the dynamics of the groups of overlooked patients: ⎡
⎢ − (k1 r + k2 )ϕp +
Amod = ⎢ ⎣
1 + enU + ξnU τn
⎤
γn
⎥
⎥
⎦. 1 − (r + 1)ϕp + + eiU + ξiU τi
δi
(A.4)
From Eq. (A.3) we obtain an expression for the sizes of the groups of overlooked patients:
TnU TiU
=
−A−1 mod
fnU + I fiU
.
(A.5)
Substitution of (A.5) in (A.2) and obvious transformations provide
Amod
k1 r k2
r 1
−1
ΔTna + ΔTia ΔTnp + ΔTip
= −ϕp
fnU + I fiU
.
(A.6)
By using the well-known formula for the inverse 2 × 2 matrix, one can readily demonstrate that
Amod
1 −r −k2 k1 r
ΔTna + ΔTia ΔTnp + ΔTip
= −ϕp r(k1 − k2 )
fnU + I fiU
.
(A.7)
It is easy to see that only two unknowns ϕp and I appear in Eq. (A.7), the second line containing only ϕp . Consequently, the expression for ϕp can be derived from the second line of (A.7), and then the expression for I can be established by substituting the result into the first line: ϕp =
r(δi + k1 εiU )ΔT p − (δi + k2 εiU )ΔT a , r(k1 − k2 )fiU + (r + 1)(k2 ΔT a − k1 rΔT p)
where the following notation is introduced: εiU = ΔT a = (ΔTna + ΔTia ), I=
1 + eiU + ξiU , ΔT p = (ΔTnp + ΔTip ) and τi
((k1 r + k2 )ϕp + εnU )(rΔT p − ΔT a ) + γn (k1 rΔT p − k2 ΔT a ) − fnD , ϕp r(k2 − k1 )
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AVILOV, ROMANYUKHA
1 + enU + ξnU , and for ϕp expression (A.8) is used. τn The estimate of ϕa follows from (A.1) and (A.8). At the same time, one can easily determine formulas for the sizes of groups of the overlooked patients TnU and TiU : since the expressions for ϕp and I are known (Eqs. (A.8) and (A.9)), Eqs. (A.5) and (A.4) define them uniquely. To derive the expression for the parameter d, one must use the fourth line of Eq. (9) and the established formulas for ϕp and TiU : here, εnU stands for εnU =
d=
ΔTip . ϕp TiU
(A.10)
APPENDIX B We describe here estimation of the domain of the admissible values of the parameters k1 and k2 which reflect the relative efficiency of detecting the patients at the early stages of tuberculosis (TnU ) as compared with the patients at its last stages (TiU ). It seems impossible to get their direct estimates without detailed statistical data about the strategy of patient detection. The present authors do not have at their disposal such statistical data. Nevertheless, the data available and the expert estimates allow one to estimate the domain of admissible values of k1 and k2 . As follows from the physical sense of the coefficients under consideration, they can assume only the nonnegative values and k1 > k2 because for the patients at the early ROT stages the active detection is knowingly more efficient than the passive one. In the case of no false negative results of the laboratory testing for bacterioexcretion (d = 1), a ΔT n
p ΔT
=
n
k1 ϕa k2 ϕp
(B.1)
would follow from Eq. (9). However, availability of the false negative laboratory results leads to an admixture to the flow of patients detected as BE− of some amount of the bacillary patients among which the fraction of the actively detected patients is lower than among the bacillary patients. Whence, a ΔT n
p ΔT n
>
ΔTna . ΔTnp
(B.2)
Additionally, one can readily establish from the third and fourth lines of Eq. (9) that ϕa ΔTia . p = ΔTi ϕp
(B.3)
ΔTna ΔTip k1 > . k2 ΔTnp ΔTia
(B.4)
It follows form (B.1)–(B.3) that
The right-hand side of (B.4) may be determined from the statistical data for each of the nine regions of the European Russia to which the patient detection submodel is adjusted. The greatest value observed in the right-hand side of (B.4) is equal to 4.92. Since the coefficients k1 and k2 are assumed to be the same for all regions, the condition k1 5 k2
(B.5)
was used as one of the boundaries of the domain of admissible values of k1 and k2 . AUTOMATION AND REMOTE CONTROL
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The above estimates of the model parameters [14] also allow one to carry out rough estimation of k1 and k2 in the order of magnitude. These results lead to the estimates k1 , k2 ∼ 0.05. With regard for (B.5), the triangle on the plane (k1 , k2 ) defined by 0 k1 0.1,
0 k2 k1 /5
(B.6)
was used as the resulting domain of the admissible values. REFERENCES 1. Waaler, H.T., Geser, A., and Andersen, S., The Use of Mathematical Models in the Study of the Epidemiology of Tuberculosis, Am. J. Publ. Health, 1962, vol. 52, pp. 1002–1013. 2. Dye, C., Garnett, G.P., Sleeman, K., and Williams, B.G., Prospects for Worldwide Tuberculosis Control under the WHO DOTS Strategy, The Lancet , 1998, vol. 362, pp. 1886–1891. 3. Murray, C.J.L. and Salomon, J.A., Modeling the Impact of Global Tuberculosis Control Strategies, Proc. Natl. Acad. Sci. USA, 1998, vol. 95, pp. 13881–13886. 4. Murray, C.J.L. and Salomon, J.A., Using Mathematical Models to Evaluate Global Tuberculosis Control Strategies, in Harvard Center for Population and Development Studies, Cambridge, 1998. 5. Waaler, H.T., A Dynamic Model for the Epidemiology of Tuberculosis, Am. Rev. Respirat. Disease, 1968, vol. 98, pp. 591–600. 6. Waaler, H.T. and Piot, M.A., The Use of an Epidemiological Model for Estimation the Effectiveness of Tuberculosis Control Measures. Sensitivity of the Effectiveness of Tuberculosis Control Measures to the Coverage of the Population, Bull. World Health Organizat., 1969, vol. 41, pp. 75–93. 7. Vynnycky, E. and Fine, P.E.M., The Natural History of Tuberculosis: The Implications of AgeDependent Risks of Disease and the Role of Reinfection, Epidemiol. Infect., 1997, vol. 119, pp. 183–201. 8. Vynnycky, E. and Fine, P.E.M., Lifetime Risks, Incubation Period, and Serial Interval of Tuberculosis, Am. J. Epidem., 2000, vol. 152, no. 3, pp. 247–263. ˇ 9. Sutherland, I., Svandov´ a, E., and Radhakrishna, S., The Development of Clinical Tuberculosis Following Infection with Tubercle Bacilli, Tubercle., 1982, vol. 63, pp. 255–268. 10. National Tuberculosis Institute, Bangalore. Tuberculosis in a Rural Population of South India: A FiveYear Epidemiological Study, Bull. World Health Organizat., 1974, vol. 51, pp. 473–488. 11. Avilov, K.K., Modeling of Epidemic Tuberculosis in Russia. Topical Issues of Phthisiology, Pulmonology, and Thoracic Surgery, in Sb. tez. Vseros. konf. studentov and molodykh uchenykh, posv. Vsemirnomu dnyu bor’by s tuberkulozom (Abstracts Papers the All-Russian Conf. Stud. Young. Res. on Tuberculosis Control), Moscow, 2006, pp. 5–7. 12. Avilov, K.K., Adjustment of the Model of Tuberculosis Spread to the Actual Data. Analysis of Error Sources, in Sb. tez. 1-i mezhdunar. konf. “Matematicheskaya biologiya i bioinformatika” (Abstratcs 1st Int. Conf. “Mathematical Biology and Bioinformatics”), 2006, pp. 140–141. 13. Avilov, K.K., Mathematical Modeling of the Process of Tuberculosis Spread. Adjustment to the Actual Data, in Sb. tez. yubileinoi nauch.-prakt. konf. posv. 45-letiyu kafedry tuberkuleza SGMU (Abstracts Jubilee Conf. 45th Anniv. SGMU Tubelculosis Chair), 2006. 14. Perelman, M.I., Marchuk, G.I., Borisov, S.E., et al., Tuberculosis Epidemiology in Russia: The Mathematical Model and Data Analysis, Russ. J. Numer. Anal. Math. Modeling, 2004, vol. 19, no. 4, pp. 305–314. 15. Regiony Rossii. Osnovnye kharakteristiki sub”ektov Rossiiskoi Federatsii. 2004. Stat. sb. (Russian Regions. Main Characteristics of the Russian Federation Subjects, Statist. Trans.), Moscow: Rosstat, 2004.
This paper was recommended for publication by V.N. Novosel’tsev, a member of the Editorial Board AUTOMATION AND REMOTE CONTROL
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