Figure 1

Solution trajectory. Numerical simulation of the model system (1)–(3) showing solution trajectory, projected onto the \((x, y)\)-plane, tending toward a limit cycle surrounding the endemic equilibrium point. Here, \(x ( 0 ) = 0.3\), \(y ( 0 ) = 0.4\), \(z ( 0 ) = 0.5\), \(a_{1} = 2.407407\), \(a_{2} = 0.023077\), \(a_{3} = 0.00196\), \(a_{4} = 0.0007\), \(\omega _{1} = 0.7\), \(\omega _{2} = 0.7\), \(\omega _{3} = 0.7K_{s} = 0.7\), \(K_{\gamma } = 4.3\), \(K_{R} = 5.0\), \(\varepsilon _{r} = 0.9\), \(\gamma = 4.0\), \(\mu = 0.01\), \(z^{*} = 5.0\), and \(\psi _{R} = 1.2\)