Bayesian Analysis of Tuberculosis Spread Scenarios in Regions of Russian Federation

Abstract

Understanding the heterogeneous spread of tuberculosis (TB), particularly multidrug-resistant (MDR) forms and the role of subclinical infection, is critical for achieving the WHO End TB strategy. This study develops a novel compartmental model that explicitly incorporates incipient and subclinical TB together with MDR forms, and links them to case detection and treatment pathways. The key innovation lies in integrating a sensitivity-based identifiability analysis with a Bayesian MCMC framework to quantify parameter uncertainty and correlations directly from regional surveillance data. Applied to five high-burden regions of the Russian Federation (2009–2020), the approach reveals strong heterogeneity in epidemic drivers: wide credible intervals for contagiousness, the rate of progression to bacterio-positive (BE+) states, and detection rates. The probabilistic forecasts up to 2025 are validated against 2021–2023 data. The region-specific differences in these correlated parameters dictate transmission dynamics, and improving detection of BE+ cases is the most effective lever for control.

1. Mathematical Modeling of the Tuberculosis Spread

Despite some success in the fight against tuberculosis (TB) in the Russian Federation and globally, the disease remains one of the main challenges to national healthcare systems and is the leading global cause of death from infectious diseases, one of the 10 main causes of death and the predominant cause of death in people infected with HIV. According to WHO, in 2025, 10.6 million people fell ill with TB for the first time, 1.3 million people died, and the number of MDR cases increased by 3% [1,2]. For successful control over the spread of socially significant and epidemically dangerous infectious diseases such as TB, predictive mathematical modeling of the epidemic process is necessary, capable of providing a reliable forecast of the probability of developing an unfavorable and favorable scenario of the epidemic situation in specific territories and early planning/implementation of necessary interventions [3].

The difficulty of developing a model of slow infections dynamics (HIV/AIDS, TB) comes when considering the heterogeneity of the population with various climatic, environmental, epidemiological, medical, biological, and socio-economic characteristics, including migration, demographic characteristics of regions, availability of medical services, the presence of high-risk groups (HIV-infected people, people with chronic diseases, addiction to psychoactive substances, etc.), and a high proportion of drug-resistant strains.

The models used for the dynamics of infectious diseases in populations have varying degrees of complexity, but three main approaches and their combinations can be distinguished [4]. The first approach is to divide the population into non-overlapping, homogeneously mixed groups with respect to infection and to introduce rules for transition from one group to another [5]. Typically, models of this type are systems of integral, integro-differential, and difference equations (SIR models), interconnected by the mass action law. In the review by Avilov and Romanyukha [5], examples of SIR models and their features in describing the dynamics of TB in a population are given. Such models are characterized by their parameters (coefficients and initial conditions), and describe an outbreak of an infectious disease if coefficients are constant. Accounting for medical and socio-economic interventions is possible in a generalized manner, while detailing the features of related processes remains difficult.

In work by Melnichenko and Romanyukha [6] it is shown that the parameter of TB infectiousness is sensitive to morbidity; therefore, demographic and socio-economic indicators were introduced as a step function into the change of the infectiousness parameter. This approach is demonstrated in predicting morbidity in the regions of the Russian Federation and confirms that the change in socio-economic conditions in the regions is one of the reasons for the decrease in TB spread. Furthermore, this effect is combined with the results of the work of the anti-TB services. In work by Kabanikhin et al. [7] for the model of TB and HIV co-infection, fundamental scenarios of disease spread were constructed with regulation of the parameter of the effectiveness of treatment in the regions of the Russian Federation. However, including socio-economic processes in an explicit form requires modification of the model (introduction of additional equations) and data. The second approach is agent-based modeling (e.g., see work by Vlad et al. [8]). Within this framework, each individual of the population is described by a set of parameters (age, immune status, time elapsed since infection, etc.), the rules of interaction of individuals, and the influence of the environment. This approach allows us to obtain a more detailed picture of the epidemic process: age stratification of morbidity, the influence of social status and the environment on the incidence and prevalence of infection, changes in immunity as a result of the spread of infection, etc. To specify models of this type, both differential equations and algorithmic descriptions are used. Such models are computationally expensive and require a detailed description specific to the region under consideration; however, the results of modeling and forecasting are not only advisory, but also practical for medical organizations (e.g., see work by Romanyukha et al. [9]).

The third approach to describing and forecasting infectious disease spread relies on statistical data and deep learning models [10,11]. However, their implementation requires complete data of the process under study, which is difficult to obtain for slowly progressing infections (for example, the effectiveness of COVID-19 modeling and forecasting is studied in the works by Krivorotko et al. [12], Krivorotko and Zyatkov [13]).

To build scenarios for the spread of TB in the regions of the Russian Federation, it is necessary to take into account the heterogeneity of its distribution [14], as well as the large percentage of multidrug-resistant forms of TB (MDR-TB) in the Siberian regions of the Russian Federation among newly diagnosed patients (see Figure 1). For example, more than 40% of newly diagnosed patients were MDR-TB-infected in the Novosibirsk region in 2024, while the average figure for the Siberian Federal District fluctuates around 30%.

The inclusion of the incipient and subclinical TB (ISbTB) compartment is crucial for modeling TB elimination, as it represents a major obstacle to ending TB alongside latent forms of TB [15]. This form is characterized by a low, oscillating bacterial load, qualifying as paucibacillary disease [16], yet remains sufficient for transmission, especially in prolonged close contact [17]. Such individuals are significant concealed community sources of infection [18], eluding diagnosis due to low bacterial burdens that cause false-negative results [19] and present a “hard to determine” category [20].

ISbTB escalates the total epidemic contagiousness; its basic reproduction number, even if low, contributes to the continuity of the overall TB epidemic process. Furthermore, these forms can lead to drug resistance [21] and relapse post-treatment [22]. Despite its proven epidemiological significance, many models omit this parameter. Our model aims to address this gap.

In recent years, the modeling of epidemics has also been developed with a stochastic approach, especially with stochastic control strategies [23,24].

The interest in mathematical modeling of TB was strongly developed after the issue of the WHO concept of ending TB by 2030 [25]. WHO TB researchers published a mathematical model for the prediction of TB incidence decline with fulfillment of the Sustainable Development Goal (SDG) subtargets [26]. In the study by Xu et al. [27] the susceptible-exposed-infectious-recovered (SEIR) model with different age groups was developed. It allowed assessment of the effect of age as a factor on TB transmission and to more accurately determine parameters for ending TB. Separate studies have considered the effects of drug-resistant cases [28], time lag [29], and age structure.

Questions of the hidden reservoirs of TB infection and the burden of overall drug resistance inside them were not significantly involved in mathematical modeling of TB infection in the current literature.

It has recently become clear that traditional notions of dividing TB infection into latent and active TB are a simplification that fails to reflect the biological process of transition from infection to disease. To more realistically characterize this process, the concept of two main intermediate groups of patients, existing between infection and active TB, has been introduced (ISbTB). Therefore, it is now believed that the immediate precursors of the development of active TB are not so much the infection state itself, but rather those stages of infection progression that directly precede active TB, i.e., ISbTB [30,31,32,33]. Recently, the first evidence has emerged of the important role of ISbTB in the spread of infection; subclinical TB has been shown to account for approximately 70% of global TB transmission [34,35]. However, the actual contribution of these conditions to the TB epidemic remains unclear, as diagnosing them in patients is difficult. It is clear that real strategies to combat TB must include the diagnosis and treatment of ISbTB to lead to effective progress towards TB elimination. Molecular biological methods currently being developed for detecting ISbTB will soon be introduced into healthcare practice. These methods will enable a more accurate understanding of the true contribution to the spread of TB and the prognosis of the epidemic [36,37,38,39,40,41]. Since this contribution depends on numerous dynamic variables influencing the spread of the epidemic (human population characteristics, drug resistance of circulating strains, access to medical care, etc.), it is appropriate to describe scenarios for the impact of iISbTB on the epidemic process using mathematical modeling. In our paper we construct the model, that, for the first time, examines the impact of ISbTB in TB epidemiology.

This mathematical model is based on the comprehensive SEIS-type compartmental model for TB dynamics in the Russian Federation, incorporating MDR forms, detection processes, and treatment pathways. In the methods we combine differential equation modeling based on M.I. Perelman and G.I. Marchuk’s ideas [42] as well as Y. Yi et al.’s model [43], Sobol sensitivity analysis, Bayesian optimization, and MCMC-based forecasting. We apply the framework to several high-burden regions that are of epidemiological interest in the Russian Federation as the obstacle of ending TB in Russia. The key points of the current publication that allow achieving additional results, as compared with the other authors, are as follows:

• Parameter identification and optimization with the use of the concept and paradigm of inverse and ill-posed problems allowed us to evaluate the intensity of the epidemic process in the risk regions of Russia, that we need to identify with the aim to combat TB as the regional and national high stream.
• Sensitivity-based identifiability analysis allows us to find correlated TB parameters that are involved in the accuracy of parameter identifiability and forecast uncertainty.
• We construct the posterior distribution of sensitive epidemiological parameters of the TB compartmental mathematical model (such as contagiousness of TB contact with bacterioexcretion, the rate of TB activation, the rate of undetected TB contact with bacterioexcretion per year) that allows us to evaluate the expected TB-infected people in Russian Federation regions for three years ahead.

If we get robust estimations of epidemic force parameters, we have enough scientific potential to plan counterforce actions including efficient treatment of drug resistance and drug-sensitive forms of TB in the amounts enough to concur transmission. The results of the expected heterogeneity of regions and the assessment of key epidemiological parameters are consistent with the results of the works of A.A. Romanyukha [9] and O.A. Melnichenko [6], and also expand and clarify the uncertainty due to the inclusion of incipient and subclinical TB forms, i.e., the key ideas of this paper are

• Inclusion of detection stages based on papers by Romanyukha [6], Romanyukha et al. [9], and new compartments of MDR-TB based on the paper by Yu et al. [43];
• Inclusion of ISbTB as a compartment based on the paper by Avilov et al. [44];
• Usage of Sobol sensitivity analysis and Bayesian MCMC for uncertainty quantification;
• Region-specific posterior parameter estimation.

The main goal of this research is identify the key epidemic parameters of TB propagation that influence TB heterogeneously in the Russian Federation as justification for real strategies to combat TB that must include the diagnosis and treatment of ISbTB in order to lead to effective progress towards TB elimination.

The paper is organized as follows: In Section 2 we formulate the differential SIR-type model, give analysis of the stability of stationary points of the model, formulate the problem of parameter identification, and provide sensitivity analysis. In Section 3 the statistical data used on TB incidence, socio-economic characteristics of the regions of the Russian Federation, and their processing for use in modeling are presented. In Section 4 numerical estimates of the probability distribution of epidemiological parameters based on statistical data and SIR model data, as well as scenarios for the spread of TB in the regions of the Russian Federation, are presented.

2. SEIS Model of TB Dynamics with Multiple Drug-Resistant Forms

The mathematical model of TB spread and control used in this paper is based on the approach proposed by G.I. Marchuk and A.A. Romanyukha in 2004 [42], the paper by A.A. Romanyukha [44] concerning TB detection, and papers describing the interaction with MDR-TB [45,46]. Using the law of mass action, we formulate the SEIS model [43], characterized by a system of nine ordinary differential equations (the model diagram is shown in Figure 2):

$$\left\{\begin{array}{cc}\hfill {\displaystyle \frac{dS}{dt}}& =-\beta S(I_s+I_{\gamma })/N+{\gamma }_1(E_{\gamma }^T+I_{\gamma }^T)+{\gamma }_2(E_s^T+I_s^T)+bN-m{N}^2-\mu S;\hfill \\ \hfill {\displaystyle \frac{d{E}_s}{dt}}& =\beta S{I}_s/N-\omega E_s-{\phi }^{-}{E}_s-\beta E_s{I}_{\gamma }/N-\mu E_s;\hfill \\ \hfill {\displaystyle \frac{d{E}_s^T}{dt}}& =-{\gamma }_2{E}_s^T+{\gamma }_2{I}_s^T+{\phi }^{-}{E}_s-E_s^T(\beta I_{\gamma }/N+t_{\gamma })-\mu E_s^T;\hfill \\ \hfill {\displaystyle \frac{d{E}_{\gamma }}{dt}}& =\beta S{I}_{\gamma }/N-\omega E_{\gamma }-{\phi }^{-}{E}_{\gamma }+\beta E_s{I}_{\gamma }/N-\mu E_{\gamma };\hfill \\ \hfill {\displaystyle \frac{d{E}_{\gamma }^T}{dt}}& ={\gamma }_1{I}_{\gamma }^T-{\gamma }_1{E}_{\gamma }^T+{\phi }^{-}{E}_{\gamma }+E_s^T(\beta I_{\gamma }/N+t_{\gamma })-\mu E_{\gamma }^T;\hfill \\ \hfill {\displaystyle \frac{d{I}_s}{dt}}& =\omega E_s-{\phi }^{+}{I}_s-\beta I_s{I}_{\gamma }/N-(\alpha +\mu )I_s;\hfill \\ \hfill {\displaystyle \frac{d{I}_s^T}{dt}}& =-2{\gamma }_2{I}_s^T+{\phi }^{+}{I}_s-I_s^T(\beta I_{\gamma }/N+t_{\gamma })-(\alpha +\mu )I_s^T;\hfill \\ \hfill {\displaystyle \frac{d{I}_{\gamma }}{dt}}& =\omega E_{\gamma }-{\phi }^{+}{I}_{\gamma }+\beta I_s{I}_{\gamma }/N-(\alpha +\mu )I_{\gamma };\hfill \\ \hfill {\displaystyle \frac{d{I}_{\gamma }^T}{dt}}& =-2{\gamma }_1{I}_{\gamma }^T+{\phi }^{+}{I}_{\gamma }+I_s^T(\beta I_{\gamma }/N+t_{\gamma })-(\alpha +\mu )I_{\gamma }^T\hfill \end{array}\right.$$

(1)

with initial conditions

$$\begin{array}{c}\hfill \begin{array}{c}S\left(0\right)=S_0,E_s\left(0\right)=E_0^s,E_s^T\left(0\right)=E_{0T}^s,\hfill \\ E_{\gamma }\left(0\right)=E_0^{\gamma },E_{\gamma }^T\left(0\right)=E_{0T}^{\gamma },I_s\left(0\right)=I_0^s,\hfill \\ I_s^T\left(0\right)=I_{0T}^s,I_{\gamma }\left(0\right)=I_0^{\gamma },I_{\gamma }^T\left(0\right)=I_{0T}^{\gamma }.\hfill \end{array}\end{array}$$

(2)

The right-hand side functions are analytical in $(t,S,E_s,E_s^T,E_{\gamma },E_{\gamma }^T,I_s,I_s^T,I_{\gamma },I_{\gamma }^T)\in [0,T]\times {[0,k]}^9$. The Cauchy problem (1)–(2) has a unique solution according to the Cauchy–Kovalevskaya theorem [47].

Here $N=S\left(t\right)+E_s\left(t\right)+E_s^T\left(t\right)+E_{\gamma }\left(t\right)+E_{\gamma }^T\left(t\right)+I_s\left(t\right)+I_s^T\left(t\right)+I_{\gamma }\left(t\right)+I_{\gamma }^T\left(t\right)$ is a volume of the studied population that satisfies the logistic equation:

$${\displaystyle \frac{dN}{dt}}=(b-\mu )\left(1-{\displaystyle \frac{N}{k}}\right)N-\alpha (I_s+I_{\gamma }+I_s^T+I_{\gamma }^T).$$

Here $b-\mu >0$, $bN-m{N}^2$ refers to the influx of susceptible individuals into the population due to the birth rate or immigration; $m=(b-\mu )/k$, k is a population potential. We note that if ${N|}_{t=0}<k$, then ${\displaystyle \frac{dN}{dt}}<0$ if $N\left(t\right)\in (k-\delta ,k)$ for some $\delta >0$. Thus, $\forall t>0:N\left(t\right)<k$.

From the other side, if the compartment reaches 0, its derivative stays non-negative, i.e.:

$$\begin{array}{cc}\hfill {\displaystyle \frac{dS}{dt}}{|}_{S=0}& ={\gamma }_1(E_{\gamma }^T+I_{\gamma }^T)+{\gamma }_2(E_s^T+I_s^T)+N(b(k-N)+\mu N)/k;\hfill \\ \hfill {\displaystyle \frac{d{E}_s}{dt}}{|}_{E_s=0}& =\beta S{I}_s/N;\hfill \\ \hfill {\displaystyle \frac{d{E}_s^T}{dt}}{|}_{E_s^T=0}& ={\gamma }_2{I}_s^T+{\phi }^{-}{E}_s;\hfill \\ \hfill {\displaystyle \frac{d{E}_{\gamma }}{dt}}{|}_{E_{\gamma }=0}& =\beta S{I}_{\gamma }/N+\beta E_s{I}_{\gamma }/N;\hfill \\ \hfill {\displaystyle \frac{d{E}_{\gamma }^T}{dt}}{|}_{E_{\gamma }^T=0}& ={\gamma }_1{I}_{\gamma }^T+{\phi }^{-}{E}_{\gamma }+E_s^T(\beta I_{\gamma }/N+t_{\gamma });\hfill \\ \hfill {\displaystyle \frac{d{I}_s}{dt}}{|}_{I_s=0}& =\omega E_s;\hfill \\ \hfill {\displaystyle \frac{d{I}_s^T}{dt}}{|}_{I_s^T=0}& ={\phi }^{+}{I}_s;\hfill \\ \hfill {\displaystyle \frac{d{I}_{\gamma }}{dt}}{|}_{I_{\gamma }=0}& =\omega E_{\gamma }+\beta I_s{I}_{\gamma }/N;\hfill \\ \hfill {\displaystyle \frac{d{I}_{\gamma }^T}{dt}}{|}_{I_{\gamma }^T=0}& ={\phi }^{+}{I}_{\gamma }+I_s^T(\beta I_{\gamma }/N+t_{\gamma })\hfill \end{array}$$

(3)

Thus, if the compartment is non-negative at the initial time, it stays non-negative. In combination with the fact that the sum of all compartments $N<k$, then all the compartments stay less than k.

The population is divided into nine non-overlapping groups according to Figure 2 (the subscript ^T denotes the group of detected infected individuals). We note that we do not depict mortality rates in the diagram in Figure 2, so as not to overload it. The description of the variables and parameters of the model is given in

Table 1. Variables and parameters of SEIS model of TB spread.

Symbol	Description	Value	Ref.
Variables (ppl.)
S	susceptibles and carries of latent infection
$E_s$, $E_s^T$	TB-infected without bacterioexcretion (TB BE−)
$E_{\gamma }$, $E_{\gamma }^T$	MDR-TB-infected without bacterioexcretion (MDR-TB BE−)
$I_s$, $I_s^T$	TB-infected with bacterioexcretion (TB BE+)
$I_{\gamma }$, $I_{\gamma }^T$	MDR-TB-infected with bacterioexcretion (MDR-TB BE+)
Main parameters (parameters of inflow)
b	birth rate	see Section 3	[48]
$\beta$	contagiousness of TB contact with bacterioexcretion	$[0, 10]$	[42]
k	population potential (ppl.)	$4·{10}^6$
$t_{\gamma }$	fraction of non-MDR-infected turning into MDR-infected per year	$[0.001, 0.1]$	f.33 *
$\omega$	rate of TB BE− turning into BE+ per year	$[0.1, 4]$	[42]
Outflow parameters
$\mu$	speed of population outflow	see Section 3	[48]
$\alpha$	additional death rate of TB-infected with bacterioexcretion	$[0.03, 0.2]$	f.33
Treatment parameters
${\phi }^{-}$	rate of undetected TB BE− infected that is detected per year	$[0.2, 0.5]$	f.33
${\phi }^{+}$	rate of undetected TB BE+ infected that is detected per year	$[0.5, 3]$	f.33
${\gamma }_1$	rate of detected MDR-TB-infected that is treated per year	$[0.25, 0.5]$	f.33
${\gamma }_2$	rate of detected TB-infected that is treated per year	$[0.3, 2]$	f.33
Initial state
${\sigma }^{-}$	fraction of undetected TB BE− infected in the initial moment	$[0, 0.5]$
${\sigma }^{+}$	fraction of undetected TB BE+ infected in the initial moment	$[0, 0.5]$

* f.33 stands for statistical reports form 33 “Information on tuberculosis patients”.

.

This model is based on the following principles for the Russian Federation:

• TB BE− and TB BE+ forms are distinguished, and TB BE− is considered an early stage of the disease preceding the development of severe TB BE+ stages;
• We assume treated individuals return to full susceptibility S, consistent with evidence of high reinfection rates in high-incidence settings [49,50];
• Patients are divided into detected and undetected;
• the proportion of those cured from TB with bacterial excretion can transit to a form without bacterial excretion;
• the probability of transition from forms without MDR to TB-MDR is nonlinear. Infection with MDR-TB occurs both primarily through contact with an infected and in the case of incorrect treatment (with the ^T index);
• During treatment of TB BE+, infected people can transit to both latent forms S and TB BE− with equal probability.

A number of TB patients who successfully complete treatment, when the symptoms have disappeared and bacterial excretion has stably stopped, undergo recurrence. In the Siberian Federal District in 2024 the recurrence rate among those cured averaged 33% (in different regions—from 20 to 43% [49]). This means that cured patients remain susceptible to developing TB infection. TB recurrence results from immunological deficiency (dependent on risk factors such as unfavorable social, medical, and stress factors, etc.) and is caused by either relapse of an original infection or exogenous reinfection with a new or the same strain of M. tuberculosis. The source of relapse is bacteria that remain in a persistent, viable state in cured (BE−) persons for many months and years within post-TB residual lung foci, lymph nodes, or even within hematopoietic or mesenchymal stem cells [51,52,53,54]. The source of reinfection is bacteria acquired through continued contacts of cured patients with people with subclinical or active TB [35,50]. In addition, indirect evidence that cured patients remain in a TB-sensitive risk group is that Interferon-Gamma Release Assay tests after cure remain positive in the absolute majority of cases for decades and even until the end of life [55].

Within the framework of the model, all populations, regardless of status, are subject to an average mortality rate with the parameter $\mu$. A susceptible or latent infected individual from group S moves to the infected non-bacterial stage $E_s$ (or closed form of TB) after contact with a TB-infected without MDR $I_s$ governing the contagiousness parameter $\beta$ or to the TB-infected non-bacterial stage with MDR $E_{\gamma }$. Infected closed-form TB $E_s$, $E_{\gamma }$ migrate to the bacterial form with $\omega$ probability. Infected patients without MDR can migrate to infected MDR-TB with $t_{\gamma }$ probability, and also with rate $\beta$ when in contact with infected TB without bacterioexcretion. In the model, each group of TB-infected people is divided into detected (with index T) and undetected with expression coefficients ${\phi }^{-}$ for TB BE−, MDR-TB BE− and ${\phi }^{+}$ for TB BE+, MDR-TB BE+, respectively. In the model, additional mortality of infected people with bacterioexcretion $\alpha$ is observed and they move to the state S with probabilities ${\gamma }_1$ (for form with MDR) and ${\gamma }_2$.

The model principle construction was based on real infection reservoirs. This is TB with bacterioexcretion and TB without bacterioexcretion—the incipient and subclinical TB, the latter always bearing a real hazard of converting to the bacterioexcretion clinical form. That does not mean that we exclude the significance of latent TB, especially in HIV-infected individuals, but consider this of lower value and as a perspective of further studies with the foci being the ending TB.

The limits of the change in the parameters of the models that are used in the linear parts of equations were determined from statistics (TB-infected registration form No. 33). For example, the proportion of identified non-MDR TB cured in 1 year, ${\gamma }_2$, in a linear approximation is inversely proportional to the time spent in therapy. That is, the limits ${\gamma }_2\in [0.3, 2]$ correspond to $[0.5, 3.33]$ years undergoing therapy for TB+ non-MDR patients.

Restrictions on initial conditions in the regions of the Russian Federation are also assumed, namely the percentage of undetected infected people is less than 50% due to the absence of a sharp outbreak of TB:

$$\begin{array}{c}\hfill E_0^s={\sigma }^{-}{E}_{0T}^s,E_0^{\gamma }={\sigma }^{-}{E}_{0T}^{\gamma },I_0^s={\sigma }^{+}{I}_{0T}^s,I_0^{\gamma }={\sigma }^{+}{I}_{0T}^{\gamma }.\end{array}$$

2.1. ODE System Analysis

The behavior of the ODE systems is broadly studied from the perspective of asymptotic properties, and this paper is also the case. We note that for such systems it is often possible to find an analytical solution using Lie algebra [56]; however, it is effectively applicable only for smaller systems, or systems with symmetries. Thus, the model considered is too complicated for this approach, so we present only the results of stationary points stability.

For the derivation of stationary points of System (1), in this section we only assume that ${\gamma }_1={\gamma }_2=\gamma$. Let us consider the system obtained from System (1) by adding the equations and replacing:

$$\begin{array}{c}\hfill I=I_s+I_{\gamma },I^T=I_s^T+I_{\gamma }^T,E=E_s+E_{\gamma },E^T=E_s^T+E_{\gamma }^T.\end{array}$$

Then we obtain the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{dS}{dt}}\hfill & =bN-m{N}^2-\beta SI/N+\gamma I^T-\mu S;\hfill \\ {\displaystyle \frac{dE}{dt}}\hfill & =\beta SI/N-(\mu +\omega +{\phi }^{-})E;\hfill \\ {\displaystyle \frac{d{E}^T}{dt}}\hfill & ={\phi }^{-}E+\gamma I^T-\mu E^T;\hfill \\ {\displaystyle \frac{dI}{dt}}\hfill & =\omega E-(\alpha +\mu +{\phi }^{+})I;\hfill \\ {\displaystyle \frac{d{I}^T}{dt}}\hfill & ={\phi }^{+}I-(2\gamma +\alpha +\mu )I^T.\hfill \end{array}\right.\end{array}$$

(4)

Lemma 1.

System (4) has the only stationary state that is trivial.

Proof. 

By equating the right-hand sides to 0, we obtain a system for finding stationary solutions:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\beta SI/N+\mu S\hfill & =bN-m{N}^2+\gamma I^T;\hfill \\ (\mu +\omega +{\phi }^{-})E\hfill & =\beta SI/N;\hfill \\ \mu E^T+\gamma I^T\hfill & ={\phi }^{-}E;\hfill \\ (\alpha +\mu +{\phi }^{+})I\hfill & =\omega E;\hfill \\ (2\gamma +\alpha +\mu )I^T\hfill & ={\phi }^{+}I.\hfill \end{array}\right.\end{array}$$

(5)

From the second and fourth equations of System (5), we obtain:

$$S={\displaystyle \frac{(\mu +\omega +{\phi }^{-})(\alpha +\mu +{\phi }^{+})}{\beta \omega }}N.$$

Therefore, we are left with Equations (1), (3), and (5) of System (5) with a single nonlinear term $N^2$.

Note that if $N\left(0\right)<k$, then $\forall t:N\left(t\right)<k$. In the edge cases, when $N\ll k$ or $N\approx k$, we obtain a linear system for the stationary point; thus, it is unique (except for the trivial point). As soon as the solution is confined, it may tend towards one of these solutions.

We rewrite System (5) the following way:

$$\left\{\begin{array}{cc}(\mu +\omega +{\phi }^{-})E+\mu S=bN-m{N}^2\hfill & +\gamma I^T;\hfill \\ E^T=I({\phi }^{-}(\alpha +\mu +{\phi }^{+})/\omega \hfill & \hfill \\ -\gamma {\phi }^{+}/(2\gamma +\alpha +\mu ))/\mu \hfill & =I{c}_{Et};\hfill \\ E=I(\alpha +\mu +{\phi }^{+})/\omega \hfill & =I{c}_E;\hfill \\ I^T=I{\phi }^{+}/(2\gamma +\alpha +\mu )\hfill & =I{c}_{It}.\hfill \end{array}\right.$$

As soon as for stationary point $dN/dt$: $bN-m{N}^2=\alpha (I+I^T)$ = 0, and for the first equation, we get:

$$\begin{array}{cc}\hfill (\mu +\omega +{\phi }^{-})I{c}_E+\mu S& =\alpha I(1+c_{It})+\gamma I{c}_{It}.\end{array}$$

Thus,

$$\begin{array}{cc}\hfill I& =-{\displaystyle \frac{\mu S}{(\mu +\omega +{\phi }^{-})c_E-\alpha (1+c_{It})-\gamma c_{It}}}=c_{I}S.\end{array}$$

Then, as soon as $dN/dt=0$, we get:

$$\begin{array}{c}\hfill (b-\mu )\left(1-{\displaystyle \frac{N}{k}}\right)N-\alpha (I+I^T)=0\end{array}$$

Thus,

$$\begin{array}{c}\hfill N=k\left(1-{\displaystyle \frac{\alpha c_S{c}_I(1+c_{It})}{b-\mu }}\right).\end{array}$$

(6)

That means that there is a unique stationary point, except for the trivial and disease-free point. However, by the substitution of coefficients $c_S,c_I,c_E,c_{It}$ into (6), we obtain $N=0$ and this point coincides with the trivial point (checked with Wolfram Mathematica, the code is available in GiHub repository). □

Our model operates in absolute values and reflects the logistic curve of the population growth. Most human populations in the regions of the Russian Federation show slow population dynamics and in these dynamics the nontrivial stationary state of TB prevalence can exist. However, we can apply the model to the risk groups where fast growth and diminution can occur.

2.1.1. Stationary Points’ Stability

The matrix of the linearized System (5):

$$\left(\begin{array}{ccccc}b-2mN-{\displaystyle \frac{\beta I(N-S)}{N^2}}-\mu & b-2mN& b-2mN& b-2mN-{\displaystyle \frac{\beta S(N-I)}{N^2}}& b-2mN+\gamma \\ {\displaystyle \frac{\beta I(N-S)}{N^2}}& -\mu -\omega -{\phi }^{-}& 0& {\displaystyle \frac{\beta S(N-I)}{N^2}}& 0\\ 0& {\phi }^{-}& -\mu & 0& \gamma \\ 0& \omega & 0& -\alpha -\mu -{\phi }^{+}& 0\\ 0& 0& 0& {\phi }^{+}& -2\gamma -\alpha -\mu \end{array}\right)$$

(7)

There are two stationary points $N=0$ and $S=N=k$.

Then from the form of (7) with $I=0$, point $S=N=k$ is stable, when

$$\mu <band(\mu +\omega +{\phi }^{-})(\alpha +\mu +{\phi }^{+})<\beta \omega$$

(8)

and point $N=0$ is stable when $b-\mu <0$, or $\mu >b$.

In our examples we will consider cases $b<\mu$ and $b\approx \mu$, which depict the situation in the Russian Federation [57]. In that case the point $N=0$ is globally stable for admissible area $0<N<k$, as soon as $\forall t:{\displaystyle \frac{dN}{dt}}<0$ if $N<k$, and ${\displaystyle \frac{dN}{dt}}\to 0$ only when $N\to 0$ or k. The stability analysis of stationary points is not informative, since the difference $\mu -b$ is quite small. Thus the convergence to the point $N=0$ is slow, so we are interested in short-term dynamics (compared to the convergence to a stationary point).

The 10-year window may seem too large for short-term forecasting; however, that is not the case for TB and the current difference between birth and mortality rates. TB has an infectivity rate of around 2–4 and recovery rate around 0.1 (for annual time scale). It indicates that the TB infection characteristic duration time is around 10 years. The difference between the birth and mortality rates is too small (0.003), which implies that the solution will reach the point with $N=0$ in more than 4000 years if the rates stay constant.

2.1.2. Basic Reproduction Number ${\mathcal{R}}_0$

To derive the basic reproduction number ${\mathcal{R}}_0$, we consider two states of people in System (1): infected without MDR $x_1={(E_s,E_s^T,I_s,I_s^T)}^{\mathrm{T}}$ and with MDR-TB $x_2={(E_{\gamma },E_{\gamma }^T,I_{\gamma },I_{\gamma }^T)}^{\mathrm{T}}$ at the disease free equilibrium (DFE) point ${\mathcal{E}}_0=(k,0,0,0,0,0,0,0,0)$, i.e., $S_0=N=k$. Following the approach from Article [58], we derive the basic reproduction number for non-MDR TB ${\mathcal{R}}_1$ and MDR-TB ${\mathcal{R}}_2$.

We rewrite System (1) as follows:

$$\dot{x_i}=(f_i-v_i)x_i,i=1,2,$$

where $f_i$ characterize infection in the group due to external flows, $v_i$ – recovery, mortality in the group. Then we have:

$$f_1=\left(\begin{array}{c}\beta S{I}_s/N\\ 0\\ 0\\ 0\end{array}\right),v_1=\left(\begin{array}{c}{\displaystyle \frac{\beta E_s{I}_{\gamma }}{N}}+\omega E_s+{\phi }^{-}{E}_s+\mu E_s\\ {\displaystyle \frac{\beta E_s^T{I}_{\gamma }}{N}}+{\gamma }_2{E}_s^T-{\gamma }_2{I}_s^T-{\phi }^{-}{E}_s+t_{\gamma }{E}_s^T+\mu E_s^T\\ {\displaystyle \frac{\beta I_s{I}_{\gamma }}{N}}-\omega E_s+{\phi }^{+}{I}_s+(\alpha +\mu )I_s\\ {\displaystyle \frac{\beta I_s^T{I}_{\gamma }}{N}}+2{\gamma }_2{I}_s^T-{\phi }^{+}{I}_s+t_{\gamma }{I}_s^T+(\alpha +\mu )I_s^T\end{array}\right).$$

$$f_2=\left(\begin{array}{c}{\displaystyle \frac{\beta S{I}_{\gamma }}{N}}+{\displaystyle \frac{\beta E_s{I}_{\gamma }}{N}}\\ {\displaystyle \frac{\beta E_s^T{I}_{\gamma }}{N}}\\ {\displaystyle \frac{\beta I_s{I}_{\gamma }}{N}}\\ {\displaystyle \frac{\beta I_s^T{I}_{\gamma }}{N}}\end{array}\right),v_2=\left(\begin{array}{c}\omega E_{\gamma }+{\phi }^{-}{E}_{\gamma }+\mu E_{\gamma }\\ {\gamma }_1{E}_{\gamma }^T-{\gamma }_1{I}_{\gamma }^T-{\phi }^{-}{E}_{\gamma }-t_{\gamma }{E}_s^T+\mu E_{\gamma }^T\\ -\omega E_{\gamma }+{\phi }^{+}{I}_{\gamma }+(\alpha +\mu )I_{\gamma }\\ 2{\gamma }_1{I}_{\gamma }^T-{\phi }^{+}{I}_{\gamma }-t_{\gamma }{I}_s^T+(\alpha +\mu )I_{\gamma }^T\end{array}\right).$$

For the linearized system $\dot{x_i}=(F_i-V_i)x_i$, $i=1,2$, we calculate the Jacobian matrices

$$F_i={\displaystyle \frac{\partial f_k}{\partial x_i^j}}\left({\mathcal{E}}_0\right),V_i={\displaystyle \frac{\partial v_k}{\partial x_i^j}}\left({\mathcal{E}}_0\right),j,k=1,\dots ,4,$$

after which we find the eigenvalues of the matrices product: $\lambda \left(F_i{V}_i^{-1}\right)$, $i=1,2$. The matrices $F_i$ and $V_i$ are presented in Appendix A.1. Thus, we get

$${\mathcal{R}}_1={\mathcal{R}}_2={\displaystyle \frac{\beta \omega }{(\omega +{\phi }^{-}+\mu )({\phi }^{+}+\alpha +\mu )}}.$$

As soon as ${\mathcal{R}}_0=max\{{\mathcal{R}}_1,{\mathcal{R}}_2\}$, the basic reproduction number in the model (1) has the form

$$\begin{array}{c}\hfill {\mathcal{R}}_0={\displaystyle \frac{\beta \omega }{(\omega +{\phi }^{-}+\mu )({\phi }^{+}+\alpha +\mu )}}.\end{array}$$

(9)

The basic reproduction number ${\mathcal{R}}_0$ is based on infectious rates (contagiousness of TB, rate of undetected TB, rate of TB BE− turning into BE+). The difference between the MDR and non-MDR strains in the model (1) is the rate of treatment, which is not reflected in the basic reproduction number (since it describes the rate of infection in a non-immune population). The identical ${\mathcal{R}}_0$ suggests that, in the absence of treatment, MDR and non-MDR spread similarly. This highlights the role of treatment in controlling MDR.

Lemma 2.

The stationary point without infection ${\mathcal{E}}_0$ is locally asymptotically stable when basic reproduction number (9) ${\mathcal{R}}_0<1$.

The ${\mathcal{R}}_0<1$ in Lemma 2 means that $\beta \omega <(\omega +{\phi }^{-}+\mu )({\phi }^{+}+\alpha +\mu )$. It coincides with the part of (8) that represents stability with respect to infection outbreak.

The exclusion of latent forms of TB increases the basic reproduction number ${\mathcal{R}}_0$; namely, due to undetected infectiousness and the transition to an active form (it is possible that such a transition may not occur during the simulation), the ${\mathcal{R}}_0$ will decrease (see details in Appendix A.1).

2.2. Inverse Problem

We assume that additional information about the solution in fixed points in time is given:

$$\begin{array}{c}\hfill E_s^T\left(t_j\right)+E_{\gamma }^T\left(t_j\right)+{\epsilon }_j=E^j,I_s^T\left(t_j\right)+{\epsilon }_j=I_s^j,I_{\gamma }^T\left(t_j\right)+{\epsilon }_j=I_{\gamma }^j,j=1,\dots ,M.\end{array}$$

(10)

Then the inverse problem (1), (2), (10) is to identify the parameters of the SEIS model $q=(\beta ,{\gamma }_1,{\gamma }_2,\omega ,t_{\gamma },\alpha ,{\phi }^{-},{\phi }^{+},{\sigma }^{-},{\sigma }^{+})\in {\mathbb{R}}_{+}^{10}$ with the help of additional information $y_j^{data}=(E^j,I_s^j,I_{\gamma }^j)$ (10).

We assume that observations $y_t^{data}=(E,I_s,I_{\gamma })$ are subject to independent additive Gaussian errors:

$$y_t^{data}=f_t\left(q\right)+{\epsilon }_t,{\epsilon }_t\sim \mathcal{N}(0,\Sigma ).$$

Here $f_t\left(q\right)=(E_s^T(t;q)+E_{\gamma }^T(t;q),I_s^T(t;q),I_{\gamma }^T(t;q))$, $\Sigma =\mathrm{diag}({\sigma }_1^2,{\sigma }_2^2,{\sigma }_3^2)$, and ${\sigma }_i$ are estimated jointly with q.

The inverse problem (1), (2), (10) could be reduced to the problem of the minimization of the misfit function:

$$\begin{array}{c}\hfill J(q,\sigma )={\displaystyle \sum _{n=1}^3}{\displaystyle \sum _{j=1}^M}{\displaystyle \frac{|y_n(t_j;q)-y_{n,j}^{data}{|}^2}{{\sigma }_n^2}}.\end{array}$$

(11)

The model (1) is theoretically identifiable based on algebra analysis [59].

Due to the instability and non-uniqueness of the solution to the inverse problem, the following section presents a sensitivity analysis of the model, which will allow us to estimate the degree of sensitivity of the parameters q to the measured states of the system (10) [60].

2.3. Sensitivity-Based Identifiability Analysis for SEIS Model

The sensitivity analysis of the model (1) was carried out using the Sobol method [61], with a uniform distribution of parameters, the variation limits of which are given in Table 1 (the literature and statistical evaluated parameters). This method quantifies how the variance in model outputs (predictions of TB diagnoses) can be apportioned to individual parameters and their interactions across the entire a priori feasible parameter space. The initial conditions (2) approximate the epidemiological situation of the Novosibirsk region in 2009:

$$\begin{array}{c}N=2263383,E_s=300,E_{\gamma }=500,E_s^T=1100,E_{\gamma }^T=1549,\\ I_s=100,I_{\gamma }=400,I_s^T=438,I_{\gamma }^T=1330.\end{array}$$

Thus, $S=N-E_s-E_{\gamma }-E_s^T-E_{\gamma }^T-I_s-I_{\gamma }-I_s^T-I_{\gamma }^T=2257666$ people. The modeling time is 10 years.

For model outputs that we consider in the inverse problems $y_1=E_s^T(t;q)+E_{\gamma }^T(t;q)$, $y_2=I_s^T(t;q)$, $y_3=I_{\gamma }^T(t;q)$ (10), the Sobol decomposition expresses the total variance $V\left(y_n\right)$, $n=1,2,3$:

$$\begin{array}{c}\hfill V^{\left(n\right)}=V\left(y_n\right)=\sum _i{V}_i^{\left(n\right)}+\sum _{i<j}{V}_{ij}^{\left(n\right)}+\dots +V_{1\dots m}^{\left(n\right)},n=1,2,3.\end{array}$$

Here $V_i^{\left(n\right)}=V\left[\mathrm{E}\left(y_n\right|q_i)\right]$, $i=1,\dots ,10$, is the first-order (main) effect variance, measuring the contribution of parameter $q_i$ alone, $V_{ij}^{\left(n\right)}$ is the second-order interaction variance between $q_i$ and $q_j$, and so on for higher-order interactions.

We compute the total-order Sobol index

$$S_{T_i}^{\left(n\right)}={\displaystyle \frac{V^{\left(n\right)}-V\left[\mathrm{E}\left(y_n\right|q_{\sim i})\right]}{V^{\left(n\right)}}},n=1,2,3,$$

and the first-order and second-order Sobol indexes

$$S_{1_i}^{\left(n\right)}={\displaystyle \frac{V_i^{\left(n\right)}}{V^{\left(n\right)}}},S_{ij}^{\left(n\right)}={\displaystyle \frac{V_{ij}^{\left(n\right)}}{V^{\left(n\right)}}},n=1,2,3,$$

where $q_{\sim i}$ denotes all parameters except $q_i$, $1=1,\dots ,10$. $S_{T_i}^{\left(n\right)}$ measures the total contribution of $q_i$, including all its interactions with other parameters. A parameter with very low total-order indices $S_{T_i}^{\left(n\right)}$ (e.g., $<0.01$) has a negligible influence on the measurements $y_n$, $n=1,2,3$. Its value cannot be inferred from the data. If $S_{T_i}^{\left(n\right)}>0.5$ for some $q_i$ to data $y_n$, then the total contribution of $q_i$, including all its interactions with other parameters, is significant. However, if $S_{T_i}^{\left(n\right)}>0.5$ and $S_{1_i}^{\left(n\right)}<0.5$, then parameter $q_i$ may still be identifiable, but its estimate will be strongly correlated with others.

We use Sensitivity Analysis Library in Python 3.14 (SALib package) and set $N=1024·(2·8+2)$ = 18,432 Sobol samples in the range of parameter changes relative to their initial value (Table 1) by 20%. In Figure 3 (right) the total-order Sobol coefficients are given for the model coefficients $\tilde{q}=(\beta ,{\gamma }_1,{\gamma }_2,\omega ,t_{\gamma },\alpha ,{\phi }^{-},{\phi }^{+})\in {\mathbb{R}}_{+}^8$ (without initial conditions), including the first indices, the second (correlation), etc., with 95% confidence intervals. The most sensitive to measurements of detected infected is the TB contagiousness parameter $\beta$. The TB detection parameter BE+ ${\phi }^{+}$ is sensitive to infected TB BE+ without MDR $I_s^T\left(t_j\right)$. The proportion of infected TB converting to the BE+ form $\omega$ is sensitive to infected TB-MDR BE+ with and without MDR. The additional mortality rate $\alpha$ due to TB is sensitive to MDR-TB BE+ measurements.

Note that the first Sobol index for $\beta$ coefficient $S_1^{\left(1\right)}$ for TB BE− measurement is more than 0.5 (this parameter is sensitive to the $y_1$), while others are less than 0.5, which means the parameter’s influence is primarily through interactions (see Figure 3 left). It may still be identifiable, but its estimate will be strongly correlated with others. The second Sobol indexes $S_{\beta \omega }^{\left(2\right)}$, $S_{\beta \omega }^{\left(3\right)}$, $S_{\beta {\phi }^{+}}^{\left(2\right)}$ are more than 0.2 and $S_{\beta j}^{\left(2\right)}$ close to 0.1–0.2 for all parameters j (see correlation matrices in Appendix A.2).

As we note that Sobol indices show strong parameter correlations, the formal practical identifiability analysis is performed, i.e., we construct the correlation matrix (see

Table 2. Correlation matrix ${\rho }_{ij}$ for the Novosibirsk region posterior.

	$\mathit{\beta }$	${\mathit{\gamma }}_1$	${\mathit{\gamma }}_2$	$\mathit{\omega }$	${\mathit{\phi }}^{-}$	${\mathit{\phi }}^{+}$	${\mathit{t}}_{\mathit{\gamma }}$	$\mathit{\alpha }$
$\beta$	1	−0.609	−0.739	0.309	−0.886	0.984	0.881	0.869
${\gamma }_1$	−0.609	1	0.947	−0.481	0.875	−0.506	−0.669	−0.849
${\gamma }_2$	−0.739	0.947	1	−0.559	0.926	−0.639	−0.855	−0.935
$\omega$	0.309	−0.481	−0.559	1	−0.294	0.145	0.644	0.666
${\phi }^{-}$	−0.886	0.875	0.926	−0.294	1	−0.843	−0.825	−0.901
${\phi }^{+}$	0.984	−0.506	−0.639	0.145	−0.843	1	0.800	0.771
$t_{\gamma }$	0.881	−0.669	−0.855	0.644	−0.825	0.800	1	0.943
$\alpha$	0.869	−0.849	−0.935	0.666	−0.901	0.771	0.943	1

for Novosibirsk region) based on the Fisher information matrix

$${\rho }_{ij}={\displaystyle \frac{FI{M}_{ij}^{-1}}{\sqrt{FI{M}_{ii}^{-1}FI{M}_{jj}^{-1}}}},$$

where

$$FIM={\left({\displaystyle \frac{\partial f_t}{\partial q}}\right)}^T{\Sigma }^{-1}{\displaystyle \frac{\partial f_t}{\partial q}}$$

and ${\displaystyle \frac{\partial f_t}{\partial q}}$ is a sensitivity matrix.

Table 2 shows that the parameters with $|{\rho }_{ij}|>0.95$ are difficult to distinguish. For example, ${\rho }_{\beta {\phi }^{+}}>0.98$ indicates that the data cannot independently inform the parameters $\beta$ and ${\phi }^{+}$, only their product is well-identified. This is shown during the sensitivity Sobol analysis as well.

3. Data Analysis

The birth b and mortality $\mu$ rates are presented in

Table 3. Number of deaths in Russian Federation regions per 1000 residents.

Region	2009	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019	2020	2021	2022
Zabaikalsky district	13.8	13.8	13.3	13.1	12.5	12.5	12.9	12.3	11.7	12.3	12.4	13.7	15.8	13.8
Tyva Republic	12	11.6	11	11.2	10.9	10.9	10.3	9.8	8.7	8.8	8.3	9.4	9	8.6
Novosibirsk region	14	13.9	13.6	13.6	13.4	13.3	13.1	13	12.9	12.9	12.7	15.3	17	13.7
Irkutsk region	14.3	14.4	14	13.9	13.6	13.7	13.6	13.4	12.9	13	13.2	15	17.7	14.1
Altai district	14.7	15	14.6	14.6	14.2	14.2	14.1	14.1	14	14.3	14	16.5	19.1	15.8

and

Table 4. Number of births in Russian Federation regions per 1000 residents.

Region	2009	2010	2011	2012	2013	2014	2015	2016	2017	2018	2019	2020	2021	2022
Zabaikalsky district	16.1	15.9	15.6	16.3	16.1	16.2	15.7	14.9	13.7	13	12.2	12.2	11.9	11.2
Tyva Republic	26.9	26.8	27.4	26.6	26	25.2	23.7	23.1	21.8	20.1	18.4	20	19.7	17.7
Novosibirsk region	12.9	13.2	13.1	13.9	14.1	14	14.2	13.7	12.3	11.7	10.7	10.3	10.1	9.6
Irkutsk region	15.6	15.2	15.3	15.9	15.7	15.2	15.3	14.7	13.4	12.8	11.8	11.3	11	10.4
Altai district	12.7	12.7	12.8	13.8	13.6	13.4	12.9	12.4	11.2	10.3	9.4	9	8.7	8.2

for five regions of the Russian Federation.

To identify parameters of the model for regions of the Russian Federation, we used the following data:

• Let $p_{\gamma }=I_{\gamma }^T/(I_{\gamma }^T+I_s^T)$; then from [49], we have data of the fraction and absolute number of patients with MDR among the patients with bacillary forms of TB, which represent $p_{\gamma }$ and $I_{\gamma }^T$ respectively.
• From data of f.33 (Information about patients with TB from the Order of Rosstat dated 31.12.2010 № 483), we have cohorts that represent $I_s^T+I_{\gamma }^T+E_s^T+E_{\gamma }^T=C_{EI}$. Then we may obtain the following:
  • $I_{\gamma }^T$ is given in data;
  • $I_s^T=I_{\gamma }^T(1/p_{\gamma }-1)$;
  • $E_{\gamma }^T+E_s^T=C_{ET}-I_{\gamma }^T-I_s^T$.

For demonstration of the numerical results we choose five regions of the Russian Federation with an unfavorable situation in terms of the prevalence of TB: Novosibirsk and Irkutsk regions, Tyva Republic, and Altai and Zabaikalsky districts.

4. Model Scenarios of TB Spread

As a result of solving the inverse problem (1), (2), (10), the vector of SEIS model parameters q is determined for each region, the substitution of which into the Models (1)–(2) leads to a comparison of the model and real data on identified infected people.

Bayesian Approach

To obtain the distribution of the model parameters and to construct a forecast considering the uncertainty, we use the Bayesian MCMC (Markov Chain Monte Carlo) approach, in which the posterior distribution for the parameters $q\in {\mathbb{R}}_{+}^{10}$ has the form:

$$p(q\mid y)\propto p(y\mid q)·{\displaystyle \prod _{i=1}^{10}}p\left(q_i\right).$$

Here $p\left(q_i\right)$ is the prior distributions of the SEIS model parameters; $p(y\mid q)$ is the likelihood of the data.

The likelihood functional is defined by the logarithmic density function of the normal distribution with the mean being the difference of the real data and solution of the direct problem for the given current parameters (11). The variance of likelihood functional is determined during the MCMC algorithm (see Appendix A.3).

The results of the inverse problem solution are in

Table 5. Restored parameters of SEIS model of TB spread (1) in regions of Russian Federation.

		Novosibirsk		Tyva		Altai		Irkutsk		Zabaikalsky
Symbol	A Priori Interval	Mean	95% CI	Mean	95% CI	Mean	95% CI	Mean	95% CI	Mean	95% CI
$R_0$	-	0.973	[0.594; 1.598]	0.974	[0.683; 1.400]	0.942	[0.557; 1.595]	0.876	[0.518; 1.502]	0.926	[0.540; 1.593]
$\beta$	$[0.03, 0.2]$	2.676	[2.369; 2.928]	2.972	[2.718; 3.171]	3.566	[3.199; 3.909]	2.927	[2.628; 3.228]	2.128	[1.876; 2.375]
$t_{\gamma }$	$[0, 10]$	0.057	[0.046; 0.068]	0.022	[0.018; 0.026]	0.019	[0.015; 0.023]	0.049	[0.039; 0.058]	0.068	[0.056; 0.081]
$\omega$	$[0.25, 0.5]$	3.164	[2.727; 3.588]	3.725	[3.272; 3.983]	1.368	[1.143; 1.597]	1.605	[1.348; 1.871]	0.844	[0.725; 0.971]
$\alpha$	$[0.3, 2]$	0.065	[0.053; 0.077]	0.038	[0.031; 0.045]	0.150	[0.120; 0.181]	0.066	[0.054; 0.079]	0.125	[0.101; 0.149]
${\phi }^{-}$	$[0, 0.5]$	0.326	[0.265; 0.391]	0.267	[0.212; 0.321]	0.488	[0.422; 0.556]	0.295	[0.243; 0.350]	0.275	[0.235; 0.318]
${\phi }^{+}$	$[0, 0.5]$	2.407	[2.123; 2.634]	2.787	[2.536; 2.957]	2.610	[2.349; 2.836]	2.724	[2.442; 2.968]	1.577	[1.376; 1.772]
${\gamma }_1$	$[0.1, 4]$	0.373	[0.326; 0.419]	0.290	[0.238; 0.342]	0.328	[0.273; 0.390]	0.307	[0.258; 0.356]	0.286	[0.245; 0.326]
${\gamma }_2$	$[0.2, 0.5]$	0.519	[0.455; 0.585]	0.870	[0.741; 1.012]	0.674	[0.552; 0.816]	0.275	[0.227; 0.323]	0.292	[0.249; 0.338]
${\sigma }^{-}$	$[0.5, 3]$	0.278	[0.230; 0.322]	0.186	[0.152; 0.218]	0.482	[0.447; 0.498]	0.254	[0.213; 0.295]	0.281	[0.234; 0.326]
${\sigma }^{+}$	$[0.001, 0.1]$	0.354	[0.298; 0.411]	0.381	[0.321; 0.440]	0.476	[0.429; 0.499]	0.376	[0.319; 0.433]	0.471	[0.420; 0.497]

. Parameters with a total Sobol index less than 0.01 for all outputs were considered practically non-identifiable and assigned informative Normal priors centered at Optuna values with width 10% of the prior range (see results for mean values in Appendix A.4). The prior distribution $p\left(q_i\right)$ for sensitive parameters $\beta ,{\phi }^{+},\omega$ is proposed as uniform (see Table 1).

We apply the Bayesian approach to find the posteriority distribution of the parameters q with 20,000 draws of four Markov chains (i.e., 80,000 samples for each parameter), mean (column “Mean”), and 95% credible interval (CI). Differences in the parameters of pathogen contagiousness $\beta$, the proportion of infected TB converting to the BE+ form $\omega$, and the rate of detection ${\phi }^{+}$ of TB-infected individuals with bacterial excretion indicate heterogeneity in the prevalence of TB in the Russian Federation regions.

The Pareto front (see Figure A3, Appendix A.5) for three measured statistics (10) shows the uniqueness of the given solution of the inverse problem (TB parameter identification); then the CI values depend on the numerical accuracy of the correlated parameters.

Figure 4 demonstrates the mean and CI of model parameters as noted in Table 5. Note that all parameters $\beta ,\omega$, and ${\phi }^{+}$ with a wide CI are sensitive to the measurement with their interaction with each other (see Section 2.3), i.e., the total-order Sobol indexes are greater than 0.5, but the first-order indexes are sufficiently small. Scenarios of the number who are TB BE+ infectious may be unreliable because $I_s^T$ and $I_{\gamma }^T$ depend on the combination of all these parameters. Small errors in identifying any one of them lead to unrealistic forecasts. Therefore, when analyzing forecast scenarios for TB BE+ groups, it is appropriate to use interval estimates rather than point estimates, as was done in this study. While the model is structurally identifiable, strong parameter correlations (seen in Sobol indices) affect practical identifiability.

In Figure 5, Figure 6 and Figure 7 the probability distribution of TB-infected based on a Bayesian approach is demonstrated for Novosibirsk region and Tyva Republic of the Russian Federation (Figure 5), Altai district and Irkutsk region (Figure 6), as well as Zabaikalsky district (Figure 7). The approximation for Novosibirsk region is more accurate because of small 95% credible intervals that contain the real data (dashed dot line). We choose statistical standards for all five regions: Mean Absolute Percentage Error (MAPE) and Root Mean Square Error (RMSE):

$$MAP{E}_n={\displaystyle \frac{1}{M}}{\displaystyle \sum _{j=1}^M}\left|{\displaystyle \frac{y_n\left(t_j\right)-y_{n,j}^{data}}{y_{n,j}}}\right|,RMS{E}_n=\sqrt{{\displaystyle \frac{1}{M}}{\displaystyle \sum _{j=1}^M}{(y_n\left(t_j\right)-y_{n,j}^{data})}^2},n=1,2,3.$$

MAPE describes the percentage deviation of the simulated data from the actual data and was approximately 25–28% (see

Table 6. Quantitative forecast accuracy metrics (MAPE and RMSE) for measured data.

	${\mathit{E}}_{\mathit{s}}^{\mathit{T}}+{\mathit{E}}_{\mathit{\gamma }}^{\mathit{T}}$	${\mathit{I}}_{\mathit{s}}^{\mathit{T}}$	${\mathit{I}}_{\mathit{\gamma }}^{\mathit{T}}$
MAPE	0.2572	0.285	0.2476
RMSE	$2.7·{10}^{-4}$	$5·{10}^{-5}$	$8.7·{10}^{-5}$

). This value is due to the low base effect: the number of TB patients is small compared to the region’s population. For example, in Novosibirsk region, the number of TB patients is approximately 1000 out of a population of 2.7 million, or 37 out of every 100,000 people. Therefore, we additionally present the RMSE metric, which shows the average deviation of the modeled results from the actual data in absolute terms (number of people). For example, for data $I_s^T$ RMSE = $5·{10}^{-5}$ means that the error is 5 per 100,000 people. The single situation when the mean prediction is worse is the modeling results for $I_s^T$, which is compensated by adjustments in other groups. The $I_s^T$ forecasts are less accurate only for the Zabaikalsky district, and the data is poorly fit for the Tyva Republic. The Zabaikalsky district data has two “falls” in data that cannot be described with the chosen model, as it is able to describe only one peak of infection with a descent without an additional “fall”. The Tyva Republic data has an unusual feature of $I_s^T$ being significantly lower than $I_{\gamma }^T$. This may be due to a simplification of the model by ignoring latent TB infections, which may actually account for some of the detected cases. The basic reproduction number ${\mathcal{R}}_0$, calculated using Formula (9), for all considered regions is close to 0.9. The most important terms in ${\mathcal{R}}_0$ are the contagiousness of TB contact with bacterioexcretion $\beta$ and the rate of TB BE− turning into BE+ per year $\omega$. The larger these parameters, the larger ${\mathcal{R}}_0$. The most unfavorable situation in terms of TB incidence among the regions examined is unfolding in the Tyva Republic due to the overestimated value of the $\omega$ parameter and its confidence intervals, which is confirmed by social data on population overcrowding in the region.

We also point out that the data points of 2019 are not used in the inverse problem because of a change in the rules of statistical records of TB in Russia [62], which made figures from 2019 unrepresentative. However, that does not affect the epidemiological process, and thus the following years are not affected significantly (see Appendix A.6).

In 2018–early 2019, a surge in TB detection occurred in the Russian Federation due to the emergence of regulatory official orders that significantly expand the coverage of prophylactic medical examination/observation service and the introduction of modern diagnostic and preventive treatment methods [63,64,65]. We suggest that from 2020 onwards, the impact of these official acts was significantly reduced due to the emergence of the COVID-19 pandemic and, as a result, a decrease in TB detection [66].

A short-term forecast for 3 years ahead was constructed for TB-infected with bacterioexcretion BE+. The credible interval and forecast accuracy are higher for TB BE+ without MDR. To improve the accuracy of the MDR-TB BE+ forecast, it is necessary to use a priori information on treatment, immunology, etc. The model cannot capture sudden spikes; forecasts assume stable epidemiological conditions.

5. Discussion

The literature on mathematical modeling of the early stages of untreated pulmonary TB, i.e., its incipient and subclinical forms, is extremely scarce. We found modeling of the early stages of TB in the works of Avilov et al. [44,67], where two models were used, one of which assumes the infectiousness of the initial stages of the disease, and the second considers all cases to be initially non-infectious, but some of them subsequently become infectious. The authors do not give preference to either model, although they consider the first model more plausible. Our model is based on the latest ideas about the very high contagiousness of the early stages of the TB process [35,50], and combines both models, assuming the infectiousness of TB cases in the early stages and the possible gradual development of some cases to an infectious state. At the same time, like Avilov et al. [5], we defined the development of infectiousness as the transition of patients from the BE− to the BE+ state, which allows us to predict the hidden incidence and prevalence of TB within the framework of the model and to better understand the real epidemic situation in different regions.

Coming through recent ideas in TB forecasting based on statistical methods and statistically based machine learning [3], in our work we also relied on long-term statistical data of departmental reporting for the territories of the Siberian and Far Eastern Federal Districts, being the most unfavorable districts in the Russian Federation. However, we used the generalized SEIS-type compartment model with the law of mass action, and machine learning was added not only to the statistical analysis but also integrated directly into the mathematical model.

We consider the closest work to our research to be Study [68]. This prototype model comprises five differential equations, while our model has nine differential equations. The authors used common methodology to derive the formula of ${\mathcal{R}}_0$ and calculate the sensitivity of ${\mathcal{R}}_0$. However, the methodology of comparing the real and the model data is completely different in our works. Our ${\mathcal{R}}_0$ values (from 0.876 to 0.974) are within the range of estimates available in the literature (from 0.24 to 4.3) [69]. It should be noted that the lack of direct analogs—calculated for individual regions of the Russian Federation—prevents a direct quantitative comparison, highlighting the novelty of our study.

The very recent work by Ochieng [70] presented the sensitivity analysis that has shown that a major positive influence on $R_0$ is induced by the per-capita TB transmission rate, rate of bacteria spread to the environment by infectious individuals, and the aerosol-transmission coefficient. The negative influence on $R_0$ is induced by the mycobacterium environmental death rate and the transition rate from the low-risk latent TB to high-risk latent TB. Even though, in terms of clinical epidemiology, the terminology and methodology in our works are quite different, we achieve very similar results.

An analysis of the model sensitivity in our model shows that the TB contagiousness parameter $\beta$, the proportion of infected TB converting to the BE+ form $\omega$, and the TB detection parameter BE+ ${\phi }^{+}$ are globally sensitive to measurements of TB BE+ $I_s^T$, and the first two parameters $\beta$ and $\omega$ are sensitive to MDR-TB BE+ and TB BE−. The sensitivity is also based on correlations.

Here we fully correspond to [43], where the simulation results proved that the key to better prevent and control the spread and development of TB is to improve the detection rate, and the conversion rate from TB to MDR-TB.

On the Bayesian approach we calculated the credible intervals (CIs) of the posterior distribution of ten model parameters based on additional information of the number of identified patients with TB BE−, TB BE+ without MDR, and MDR-TB BE+ from 2009 to 2021. The less sensitive parameters give small CIs for five considered regions that describe the TB propagation and detection in RF regions as being in common. More sensitive and correlated parameters have huge CIs that prove the heterogeneity of TB propagation in RF regions as well as open ways for its control. The Pareto front (see Figure A3) for three measured statistics shows the uniqueness of the given solution of the inverse problem (TB parameter identification), and then the CI values depend on the numerical accuracy of the correlated parameters.

Based on identified parameters and their CIs, we construct the forecasting scenarios of detected patients with TB BE−, TB BE+ without MDR, and MDR-TB BE+ from 2022 to 2026 years for five regions of the Russian Federation, with the approach being close to the results [71].

6. Conclusions

The aim of our study was to learn the new patterns and laws of TB spreading worldwide based on the example of the Russian Federation, and to evaluate combat strategies. Over the past decade and a half, the TB epidemic in the Russian Federation, including the Siberian and Far Eastern Federal Districts, has been steadily improving, meaning that TB is gradually halting its spread, and thus, the ${\mathcal{R}}_0<1$. Despite this improvement, given the significant number of people living with HIV infection, who are highly susceptible to TB, we estimate the ${\mathcal{R}}_0$ value to be 0.9. This improvement is likely due to anti-epidemic measures taken by the TB service, as well as a reduction in socio-economic and other risk factors.

In the compartment model with revealed and unrevealed cases, incipient and subclinical TB (BE−) and clinical TB (BE+), drug-sensitive and drug-resistant TB, we obtained the formula for the basic reproduction number ${\mathcal{R}}_0$. The formula comprises the TB contagiousness parameter $\beta$, the rate of TB BE− converting into BE+ $\omega$, and the rate of undetected TB BE+ ${\phi }^{+}$. Using the sensitivity analysis, we discovered the leading pattern, determining the prevalence of TB clinical forms with bacterial excretion, which are the most dangerous for humans from the aspect of lethality and disease transmission. The pattern is based on the same parameters as ${\mathcal{R}}_0$ determinacy, which are TB contagiousness parameter $\beta$, the rate of TB BE− converting into BE+ $\omega$, and the rate of undetected TB BE+ clinical forms ${\phi }^{+}$. By solving the inverse problem, we proved the variability of these crucial parameters in high-incidence regions of the Russian Federation. This heterogeneity is due to differences in medical and social systems. The TB contagiousness parameter $\beta$ depends on the intensity of contact tracing. The rate of TB BE− converting into BE+ $\omega$ is influenced by healthcare activity—the coverage of the population with X-ray film examination. The detection (reveal) indicator of BE+ clinical forms is based on the same parameter (coverage of the population with X-ray films) and also the application of sensitive methods of TB diagnostics, including molecular methods. Intensification of these activities will help to fulfill the strategy of ending TB. By addressing the prior gaps: the absence of the incipient and subclinical disease compartment, case detection rate, consideration of transitions between compartments, use of the Bayesian approach for the detection of scenarios, we came closer to understanding the TB epidemic and the measures for counteraction.

In the future, to plan detailed parameters of possible interventions and their application in specific regions, additional processes should be included, for example, by introducing quantitative trajectories of socio-economic, medical–social, molecular–epidemiological, and other indicators connected to the territories. The most complex and convenient model for use in practical healthcare will have to include an assessment of the financial and economic efficiency of possible interventions. We plan to consider a fractional-order extension of the mathematical model to capture memory effects in TB progression and solve the uncertainly of forecasting [72,73].