Seasonality Impact on the Transmission Dynamics of Tuberculosis

Abstract

The statistical data of monthly pulmonary tuberculosis (TB) incidence cases from January 2004 to December 2012 show the seasonality fluctuations in Shaanxi of China. A seasonality TB epidemic model with periodic varying contact rate, reactivation rate, and disease-induced death rate is proposed to explore the impact of seasonality on the transmission dynamics of TB. Simulations show that the basic reproduction number of time-averaged autonomous systems may underestimate or overestimate infection risks in some cases, which may be up to the value of period. The basic reproduction number of the seasonality model is appropriately given, which determines the extinction and uniform persistence of TB disease. If it is less than one, then the disease-free equilibrium is globally asymptotically stable; if it is greater than one, the system at least has a positive periodic solution and the disease will persist. Moreover, numerical simulations demonstrate these theorem results.

Figure 1

The original time series of pulmonary TB cases in Shaanxi of China, January 2004 to December 2012.

Figure 2

(a) The trend component of time series for pulmonary TB cases in Shaanxi of China, January 2004 to December 2012. (b) The seasonal component of time series for pulmonary TB cases in Shaanxi of China, January 2004 to December 2012.

Figure 3

(a) Compare the original series to a series reconstructed using the component estimates of time series for pulmonary TB cases in Shaanxi of China, January 2004 to December 2012. (b) The irregular noise component of time series for pulmonary TB cases in Shaanxi of China, January 2004 to December 2012.

Figure 4

The transfer diagram for model (1).

Figure 5

For system (2), the graph of the average basic reproduction number [R _T] and the basic reproduction number R _T with respect to b ₀ which varies from 0.003 to 0.045, and Λ = 0.8, μ = 0.008, p = 0.08, g ₀ = 0.003, k ₁ = k ₂ = k ₃ = 1, σ = 0.5, T = 12, and a ₀ = 0.08.

Figure 6

For system (2), the graph of [R _T] and R _T with respect to g ₀ which varies from 0.003 to 0.007 when b ₀ = 0.015, and other parameter values are the same as those of Figure 5.

Figure 7

For system (2), the graph of [R _T] and R _T with respect to a ₀ which varies from 0.01 to 0.05 when b ₀ = 0.0165, and other parameter values are the same as those of Figure 5.

Figure 8

For system (2), the graph of [R _T] and R _T with respect to b ₀ which varies from 0.003 to 0.045 when T = 1, and other parameter values are the same as those of Figure 5.

Figure 9

For system (2), the graph of [R _T] and R _T with respect to g ₀ which varies from 0.003 to 0.007 when T = 1, and other parameter values are the same as those of Figure 6.

Figure 10

For system (2), the graph of [R _T] and R _T with respect to a ₀ which varies from 0.01 to 0.05 when T = 1, and other parameter values are the same as those of Figure 7.

Figure 11

For system (2), the graph of [R _T] and R _T with respect to b ₀ which varies from 0.003 to 0.045 when T = 5, and other parameter values are the same as those of Figure 5.

Figure 12

For system (2), the graph of [R _T] and R _T with respect to g ₀ which varies from 0.003 to 0.007 when T = 5, and other parameter values are the same as those of Figure 6.

Figure 13

For system (2), the graph of [R _T] and R _T with respect to a ₀ which varies from 0.01 to 0.05 when T = 5, and other parameter values are the same as those of Figure 7.

Figure 14

For system (2), the graph of [R _T] and R _T with respect to k ₁ which varies from 0 to 1 when b ₀ = 0.0165, and other parameter values are the same as those of Figure 5.

Figure 15

For system (2), the graph of [R _T] and R _T with respect to k ₂ which varies from 0 to 1 when b ₀ = 0.0165 and other parameter values are the same as those of Figure 5.

Figure 16

For system (2), the graph of [R _T] and R _T with respect to k ₃ which varies from 0 to 1, when b ₀ = 0.0165 and other parameter values are the same as those of Figure 5.

Figure 17

For system (2), the graph of [R _T] and R _T with respect to T which varies from 0 to 22 when b ₀ = 0.015, and other parameter values are the same as those of Figure 5.

Figure 18

For system (2), β(t), γ(t), and α(t) are listed in Example 4. Λ = 0.8, μ = 0.008, p = 0.08, a ₀ = 0.08, g ₀ = 0.003, k ₁ = k ₂ = k ₃ = 1, σ = 0.5, T = 12, and b ₀ = 0.005, and then R _T = 0.2814. These figures show that the disease will die out, which is the same as Theorem 6.

Figure 19

For system (2), b ₀ = 0.04 and other parameter values are the same as those of Figure 18; then R _T = 2.251. These figures show that the disease will be asymptotic to a periodic solution, which is the same as Theorem 7.