Andronov–Hopf and Neimark–Sacker bifurcations in time-delay differential equations and difference equations with applications to models for diseases and animal populations

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 12 Critical bifurcation values for endemic equilibrium of the ELM models

Version		1(dnn)		3(nnd), 7(ddd)	5(dnd)	6(ndd)
Range of \(R_{0}\)		\(1< R_{0}<2\)		\(R_{0}>1\)	\(R_{0}>1\)	\(R_{0}>2\)
γ = 2		\(1< R_{0}<1+\frac{2}{\gamma }\)				\(R_{0}>1+\frac{2}{\gamma}\)
\(R_{0}\)		1.02	1.98	1.98	1.98	2.48
ρ		2.88	−2.88	0	3	−3
η		3	3	5.88	8.88	5.88
DDE
\(\tau _{c}\)		0.3379	3.402	0.2671	0.1467	0.4165
\(\phi _{c}\)		0.8400	0.8400	5.880	8.358	5.057
\(\frac{d\mu }{d\tau } \vert _{\tau _{c}}\)		8.682	5.655e−3	9.971	38.45	2.693
Discrete
\(\tau _{c}\)	m = 22	0.3376	2.6737	0.2612	0.14486	0.3958
\(\tau _{c}\)	m = 23	0.3377	2.7012	0.2614	0.14494	0.3967
\(\omega _{c}\)	m = 22	0.0126	0.1267	0.0698	0.0545	0.0936
\(\omega _{c}\)	m = 23	0.0121	0.1213	0.0668	0.0522	0.0896
\(\frac{dr}{d\tau } \vert _{\tau _{c}}\)	m = 22	0.0058	1.499e−4	0.0060	0.0119	0.0030
\(\frac{dr}{d\tau } \vert _{\tau _{c}}\)	m = 23	0.0053	1.358e−4	0.0055	0.0109	0.0028
\(\Re [e^{-i\omega _{c}}c_{1} (\tau _{c} ) ]\)	m = 22	0.6968	0.0889	(3) −0.0086	−0.0063	−0.0240
				(7) −0.0280
	m = 23	0.6692	0.0855	(3) −0.0082	−0.0060	−0.0230
				(7) −0.0268