Le III (Case 0 , = three.0, = two.five). There is certainly now evidence that rates of secondary tuberculosis in high endemic communities (for example semiclosed communities), in individuals with LTB orand currently treated for principal disease, are actually higher than in folks presenting with primary infection [21, 22]. Taking this into consideration we consider now the following numerical values for the parameters: = 0.00014, = 3.0, = 2.five. In this case the fundamental reproduction quantity takes the value 0 = 0.9653059690. Moreover we have 0 = 0.0001450317354, = 0.0001066568066, = 0.0001225687204. (31)Figure eight: Numerical simulation for 0 = 0.9653059690, = 3.0, and = two.five. The method can evolve to two d-Bicuculline distinctive equilibria = 0 (red lines) or = 285 (dark green lines) according to different initial circumstances.+ +1600 1200 2000 3000 800 4000 5000 2 400 6000 1 7000 8000Figure 9: Numerical simulation for 0 = 0.9653059690, = three.0, and = 2.5. Phase space representation on the method with many equilibrium points.For these parameter we have that the condition 0 is fulfilled as well as the method has the possibility of a number of equilibria. In fact, we have 
within this case the following stationary points = (, , , , ): 1 = (9009, 0, 0, 0, 0) , two = (8507, 182, 9, 5, 2166) , 3 = (3221, 1406, 285, 103, 1566) . (32)1 is often a steady disease-free equilibrium point (steady node), 3 is really a stable endemic equilibrium (stable focus), and two is an unstable equilibrium point (saddle point). Figure 8 shows the convergence to = 0 or to = 285 in accordance with with different initial conditions. In Figure 9 is shown yet another representation (phase space) from the evolution from the system toward 1 or to three according to unique initial situations. The representation is really a threedimensional phase space in which the horizontal axes are12 susceptible and recovered individuals, even though the vertical axis will be the prevalence + + . For the previously numerical values, the system experiences a backward bifurcation [37] at the worth = 0.0001261648723 with 0 . For , the method possesses two stable equilibrium points and one unstable (see Figure 4). Instance IV (Case 0 , = three.0, = two.5). Take into consideration now a a lot more intense scenario with = two.5, = 3.0, and = 0.7 (the other parameters kept the same values offered in Table 4). Within this case the situation 0 is fulfilled. This instance is shown so that you can illustrate additional complex and rich dynamics that may well admit technique (1), which can be mathematically probable and could in principle be a model case for an extreme hypothetical situation inside a semiclosed higher burden neighborhood. For these parameters we’ve got 0 = 0.0001679568390, = 0.0001729256777, = 0.0001489092005, which clearly satisfy the condition 0 . Consequently, as was explained in the earlier section, the program has the possibility of a number of equilibria. The truth is, for the bifurcation worth 1 = 0.0001673533706 from the disease transmission rate, which satisfies the PubMed ID:http://www.ncbi.nlm.nih.gov/pubmed/21337810 situation 1 0 , the method acquires two positive equilibria, apart from the disease-free equilibrium. When = 0 seem 3 good equilibrium points plus the disease-free equillibrium becomes unstable. For 2 = 0.0001688612368 with 0 2 the program admits a exclusive and stable endemic equilibrium (see Figure ten). We take now the worth = 0.0001675, which satisfies the condition 1 0 . With these numerical values the fundamental reproduction quantity is 0 = 0.9972800211 1, and for that reason, the diseasefree equilibrium is steady. We’ve in th.