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SIS criss-cross model of tuberculosis in heterogeneous population

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  • In this paper we propose a model of tuberculosis (TB) transmission in a heterogeneous population consisting of two different subpopulations, like homeless and non-homeless people. We use the criss-cross model to describe the illness dynamics. This criss-cross model is based on the simple SIS model with constant inflow into both subpopulations and bilinear transmission function. We find conditions for the existence and local stability of stationary states (disease-free and endemic) and fit the model to epidemic data from Warmian-Masurian Province of Poland. Basic reproduction number $ \mathcal{R}_0 $ is considered as a threshold parameter for the general model. Applying local center manifold theory we show that when $ \mathcal{R}_0 = 1 $ a supercritical bifurcation occurs, and with $ \mathcal{R}_0 $ increasing above this threshold the disease-free stationary state loses stability and locally asymptotically stable endemic stationary state appears. Our analysis confirms the hypothesis that homeless individuals may be a specific reservoir of the pathogen and the disease may be transmitted from this subpopulation to the general population.

    Mathematics Subject Classification: Primary: 92B05; Secondary: 34C11, 34D20, 34K60, 92C60.


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  • Figure 1.  Nullclines $C - xy + y - \mu x = 0$, $x - k = 0$ and phase portraits for system (2), for $\frac{C}{\mu} <k$ (A) and $\frac{C}{\mu}>k$ (B)

    Figure 2.  Graphic representation of solutions of system (10) when (A) $C_1 >\mu_1 k_1$ and $C_2 >\mu_2 k_2$, (B) $C_1 <\mu_1 k_1$ and $C_2 >\mu_2 k_2$, (C) $C_1 > \mu_1 k_1$ and $C_2 <\mu_2 k_2$, (D) $C_1 = \mu_1 k_1$ and $C_2 = \mu_2 k_2$. Red curves represent the graph of the function $y_2(y_1)$, blue ones represent the graph of $y_1(y_2)$. Dotted lines bound the region defined by $\left(\xi_1,\tfrac{C_1}{\mu_1\kappa_1}\right) \times \left(\xi_2,\tfrac{C_2}{\mu_2\kappa_2}\right)$

    Figure 3.  Graphic representation of solutions of system (10) for $C_i <\mu_i k_i$, $i = 1,\ 2$, if condition (11) holds (A) and does not hold (B). Red curves represent the graph of the function $y_2(y_1)$, blue ones represent the graph of $y_1(y_2)$. Red line is the tangent line to the function $y_2(y_1)$ at zero, blue one is the tangent line to $y_1(y_2)$ at zero. Dotted lines bound the region defined by $\left(\xi_1,\tfrac{C_1}{\mu_1\kappa_1}\right) \times \left(\xi_2,\tfrac{C_2}{\mu_2\kappa_2}\right)$

    Figure 4.  Tuberculosis in the Warmian-Masurian province over the years 2001-2016 (number of infected non-homeless individuals). Comparison between the actual data and the model

    Figure 5.  Phase portraits for system (5) in the phase planes $(S_1,I_1)$ (A) and $(S_2,I_2)$ (B) with fitted values of parameters summarized in Table 1

    Figure 6.  Bifurcation diagram for system (5). The solid line depicts the graph of $\big(\mu_1(\gamma_1 + \alpha_1 + \mu_1)-C_1\beta_{11}\big)\big(\mu_2(\gamma_2 + \alpha_2 + \mu_2)-C_2\beta_{22}\big)- \beta_{12}\beta_{21}C_1C_2 = 0$. The dotted black lines depict $C_1 = \tfrac{\mu_1(\gamma_1 + \alpha_1 + \mu_1)}{\beta_{1}}$ and $C_2 = \tfrac{\mu_2(\gamma_2 + \alpha_2 + \mu_2)}{\beta_{2}}$. The red point depicts the point $(C_1,C_2)$ for the values of $C_1$ and $C_2$ taken from Table 1

    Figure 7.  Bifurcation diagram for system (6) with $\beta_{1},\ \beta_{2}$ tending to zero. The solid black lines depict $C_1 = \mu_1 k_1$ and $C_2 = \mu_2 k_2$

    Table 1.  Parameters for the model described by system (5)

    Name Definition Value
    $\alpha_1$, $\alpha_2$ Disease-related death rates $0.09$
    $\gamma_1$, $\gamma_2$ Recovery coefficients $0.9$
    $\mu_1$, $\mu_2$ Natural death rate $0.009$
    $C_1$ Constant inflow of humans into the subpopulation of the non-homeless $11 000$
    $C_2$ Constant inflow of humans into the subpopulation of the homeless $60$
    $\beta_{11}$ Transmission coefficient $0.57566\cdot 10^{-6}$ (estimated)
    $\beta_{12}$ Transmission coefficient $11.276\cdot 10^{-6}$ (estimated)
    $\beta_{21}$ Transmission coefficient $3.7679\cdot 10^{-6}$ (estimated)
    $\beta_{22}$ Transmission coefficient $98.249\cdot 10^{-6}$ (estimated)
     | Show Table
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