SIS criss-cross model of tuberculosis in heterogeneous population

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 $C_2 = \mu_2 k_2$