### Introduction

### Materials and methods

*S*), the exposed class(

*E*), (or high-risk latent, that is, recently exposed but not yet infectious), the active-TB infectious class (

*I*), and the low risk latent class or treated infected class (

*L*).

*Λ*and

*μ*, respectively.

*βc*is the transmission rate for the susceptible, exposed and low risk latent individuals with an infectious individual per contact per unit of time. It is assumed that it is less likely to get infected for the low risk latent individuals with a reduction constant, 0≤

*σ*≤1(5).

*k*is the rate at which an individual leaves the latent class by becoming infectious.

*d*is the per-capita disease-induced death rate and

*γ*is the per-capita treatment rate. Individuals who do not progress from the class

*E*to the class

*I*are moving to the low risk latent class at the rate

*α*, (e.g., taking the TB medications before occuring active TB or called “case finding effort”). The term

*pβcE*(

*I*/

*N*) models the exogenous reinfection rates with

*p*representing the level of reinfection. A value of 0<

*p*<1 implies that reinfection is less likely than a new infection. Most developed countries have low incidence TB rate, therefore, the exogeneous reinfection can be ignored. As mentioned earlier, this is not the case of Korea, since Korea has a higher proportion of exposed or low risk latent individuals like developing countries. Hence, it is better to take account of exogeneous reinfection in the TB transmission model. Also, we assume a constant per-capita removal rate to focus exclusively on the role of exogenous reinfection then, the basic reproductive number

*R*

_{0}for (1) can be obtained as follows:

*βc*/(

*μ*+

*γ*+

*d*), that is, by the average population of infected people from one infectious individual during his or her infectious period and

*k*/(

*μ*+

*k*+

*α*), the rate of the population, which survives the exposed period and go to the infectious class successfully. Therefore

*R*

_{0}gives the number of secondary infectious cases produced by an infectious individual for a period of his or her infectious period in a population of susceptible individuals.

### Results

*μ*(

*t*) and

*b*(

*t*) is time-dependent parameter for every year taken from [13]. The other parameters for

*σ*,

*k*,

*d*are defined as constant taken from [12]. The rest of parameters are estimated through the least-square method:

*γ*(

*t*) is an estimate of the treatment rate;

*βc*(

*t*) is an estimate of the transmission rate;

*α*(

*t*) is an estimate of the case finding rate from

*E*to

*I*;

*p*(

*t*) is an estimate of the level of reinfection.

*γ*(

*t*),

*βc*(

*t*),

*α*(

*t*) and

*p*(

*t*) for two distinct time windows, which are listed in Table 1. Figure 1 displays the actual data of active-TB incidence and its best fit or the model outputs from 1970 to 2011 yr. The active-TB incidence (shown as∘) represents the number of new infectious individuals per each year compared with the model outputs, dashed and solid curves under two different sets of parameter values estimated in Table 1. The results demonstrate that our fitted TB model agrees well with the reported TB incidence data.

*S*(

*t*)/

*N*(

*t*),

*E*(

*t*)/

*N*(

*t*),

*I*(

*t*)/

*N*(

*t*),

*and*

*L*(

*t*)/

*N*(

*t*). The fraction of susceptible individuals is increased from aroud 0.36 to over 0.5 in a steady manner during the simulated time. The fraction of exposed individuals and the fraction of infectious individuals have been decreased significantly from 1970 until 2000 but they have been slightly increased after 2001.The fraction of latent individuals stays around 0.4 throughout the entire period of time. When we take a closer look at the fraction of active TB infected individuals, the number of infectious individuals is expected to grow slowly since 2001.

*γ*), the case holding effort(

*σ*) and the case finding effort(

*α*) in the model equations (1) as noted in [14]. Numerical simulations are performed using three different values for each of these effort parameters. The effort

*γ*denotes the effort of treatment of active TB infected individuals in

*I*while

*σ*denotes the effort of preventing low risk latent individuals from going back to the high risk or exposed class by forcing them to finish their treatment. Lastly,

*α*denotes the effort of identifying exposed individuals so that they can be treated at their early stage and move to the low risk class. Here only the sensitivity analysis for the case finding effort is shown since this parameter is the most significant impact on the reduction of active TB cases.

*S*(

*t*)/

*N*(

*t*),

*E*(

*t*)/

*N*(

*t*),

*I*(

*t*)/

*N*(

*t*),

*and*

*L*(

*t*)/

*N*(

*t*) using three different values of

*α*to illustrate the impact of the case finding on the transient dynamics of TB after 2011 up to 2030. The corresponding results are plotted in red dotted curve, blue solid curve and black dashed curve using

*α*=0.0252 and

*α*=0.0352 (the baseline value from the estimation as given in Table 1) and

*α*=0.0752, respectively. As shown in Figure 3, the proportion of active-TB incidences

*I*(

*t*)/

*N*(

*t*) will be as twice as the one in 2011 year when

*α*=0.0352 (or even more so with

*α*=0.0252). When we put more effort on this case finding rate as

*α*=0.0752, the proportion of active-TB incidences

*I*(

*t*)/

*N*(

*t*) will be almost half around 2030 year. The results of sensitivity analysis for the other two cases, the treatment efforts and the case holding effort, are similar as the ones of the case finding effortof course, not as significant as the case finding case but some impact to a certain extent.