Fig. 6

a Illustration of the relationship between the basic reproductive \(R_{0}\) and \(\mu _1\). b Illustrates the relationship between the basic reproductive number \(R_{0}\) and \(\tau\). c Illustrates the relationship between the basic reproductive number \(R_{0}\), \(\beta\) and \(\gamma\). d Illustrates the relationship between the basic reproductive number \(R_{0}\), \(\delta\) and \(\alpha\)