Is case the following stationary points = (, , , , ): 0 = (5148, 0, 0, 0, 0) , 1 = (3372, 1041, 122, 60, 482) , two = (2828, 1283, 190, 88, 651) . 0 will be the stable disease-free equillibrium point (steady node), 1 is definitely an unstable equilibrium point (saddle point), and 2 is usually a steady endemic equilibrium (steady focus). Figure 11 shows the convergence to = 0 or to = 190 as outlined by the initial situation. In Figure 12 is shown one more representation (phase space) on the evolution of the method toward 0 or to two according to the initial circumstances. Let us take now the value = 0.0001683, which satisfies the situation 0 two . In this case, the fundamental reproduction quantity has the worth 0 = 1.002043150. We still have that the situation 0 is fulfilled (34) (33)Computational and Mathematical Strategies in Medicine1 00.0.0.0.Figure 10: Bifurcation diagram (answer of polynomial (20) versus ) for the condition 0 . The method experiences a number of bifurcations at 1 , 0 , and two .300 200 one hundred 0Figure 11: Numerical simulation for 0 = 0.9972800211, = 3.0, and = 2.five. The system can evolve to two diverse NAMI-A chemical information equilibria = 0 or = 190 based on the initial situation.and the method within this case has 4 equilibrium points = (, , , , ): 0 = (5148, 0, 0, 0, 0) , 1 = (5042, 76, five, 3, 20) , 2 = (3971, 734, 69, 36, 298) , 3 = (2491, 1413, 246, 109, 750) . (35)Computational and Mathematical Methods in Medicine2000 1500 1000 500 0 0 0 2000 200 400 2000 00 400 3000 3000 0 0 5000 4000 400 4000 00 1 600 800 2 2000 1500 1000 500 three 0 2000 200 two 2000 400 40 1000 1200 1400 3000 300 3000+ ++ +4000 40 4000 0 00 1800 1000 1200Figure 12: Numerical simulation for 0 = 0.9972800211, = three.0, and = 2.five. Phase space representation with the system with multiple equilibrium points.Figure 13: Numerical simulation for 0 = 1.002043150, = 3.0, and = two.5. The system can evolve to two diverse equilibria 1 (stable node) or 3 (stable focus) based on the initial condition. 0 and 2 are unstable equilibria.0 would be the unstable disease-free equillibrium point (saddle point ), 1 is really a stable endemic equilibrium point (node), two is definitely an unstable equilibrium (saddle point), and 3 can be a steady endemic equilibrium point (focus). Figure 13 shows the phase space representation of this case. For additional numerical analysis, we set each of the parameters inside the list in accordance with the numerical values given in Table 4, leaving free the parameters , , and associated to the principal transmission rate and reinfection prices with the illness. We are going to discover the parametric space of method (1) and relate it for the indicators of your coefficients of the polynomial (20). In Figure 14, we take into consideration values of such that 0 1. We are able to observe from this figure that because the primary transmission price of your illness increases, and with it the fundamental reproduction number 0 , the method below biological plausible condition, represented within the figure by the square (, ) [0, 1] [0, 1], evolves such PubMed ID:http://www.ncbi.nlm.nih.gov/pubmed/21337810 that initially (for lower values of ) coefficients and are both positive, then remains constructive and becomes negative and finally both coefficients grow to be unfavorable. This alter inside the coefficients signs because the transmission price increases agrees using the outcomes summarized in Table two when the condition 0 is fulfilled. Next, in an effort to discover an additional mathematical possibilities we’ll modify some numerical values for the parameters in the list within a far more intense manner, taking a hypothetical regime with = { = 0.03885, = 0.015.