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

Darlai, Rachadawan; Moore, Elvin J.; Koonprasert, Sanoe

doi:10.1186/s13662-020-02646-5

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 4 Andronov–Hopf bifurcation conditions and values for \(\tau _{c}\) in HIV and ELM delay models

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

Version	Disease-free
	HIV: \(R_{0} = \frac{\varepsilon }{\delta }\)		ELM: Λ = β − r, \(R_{0} = \frac{\beta }{r}\)
	Bifurcation	\(\tau _{c}\)	Bifurcation	\(\tau _{c}\)
dnn	\(R_{0} < 1\)	\(\frac{1}{\phi _{c}}\cos ^{-1} (\frac{\varepsilon }{\delta } ) \)	\(R_{0}<1\)	\(\frac{1}{\phi _{c}}\cos ^{-1} (\frac{\beta }{r} )\)
dnn	ε<δ	\(\phi _{c}= \sqrt{\delta ^{2}-\varepsilon ^{2}}\)	β<r	\(\phi _{c} = \sqrt{r^{2}+\beta ^{2}}\)
ndn	No	–	No	–
nnd	No	–	No	–
ddn	\(R_{0} < 1\)	\(\frac{\pi }{2\phi _{c}} \)	\(R_{0}<1\)	\(\frac{\pi }{2\phi _{c}}\)
ddn	ε<δ	δ − ε	β<r	\(\phi _{c} = r-\beta \)
dnd	\(R_{0} < 1\)	\(\frac{1}{\phi _{c}}\cos ^{-1} (\frac{\varepsilon }{\delta } ) \)	\(R_{0}<1\)	\(\frac{1}{\phi _{c}}\cos ^{-1} (\frac{\beta }{r} )\)
dnd	ε>δ	\(\phi _{c} = \sqrt{\varepsilon ^{2}-\delta ^{2}}\)	β<r	\(\phi _{c} = \sqrt{r^{2}+\beta ^{2}}\)
ndd	No	–	No	–
ddd	\(R_{0} < 1\)	\(\frac{\pi }{2\phi _{c}} \)	\(R_{0}<1\)	\(\frac{\pi }{2\phi _{c}}\)
ddd	ε<δ	\(\phi _{c} = \delta -\varepsilon \)	β<r	\(\phi _{c} = r-\beta \)

Version	Endemic
	HIV: \(R_{0} = \frac{\varepsilon }{\delta }\)		ELM: Λ = β − r, \(R_{0} = \frac{\beta }{r}\)
	Bifurcation	\(\tau _{c}\)	Bifurcation	\(\tau _{c}\)
dnn	\(1 < R_{0} <3\)	\(\frac{1}{\phi _{c} }\cos ^{-1} (\frac{2\delta -\varepsilon }{\delta } ) \)	\(1 < R_{0} < 1+\frac{2}{\gamma }\)	\(\frac{1}{\phi _{c}}\cos ^{-1} (\frac{r-\gamma \varLambda }{r} )\)
dnn	δ<ε<3δ	\(\phi _{c}= \sqrt{4\delta \varepsilon - \varepsilon ^{2}-3\delta ^{2}}\)	0<γΛ<2r	\(\phi _{c} = \sqrt{2r\gamma \varLambda +\gamma ^{2}\varLambda ^{2}}\)
ndn	No	–	No	–
nnd	\(R_{0} > 1\)	\(\frac{\pi }{2\phi _{c}} \)	\(R_{0} > 1\)	\(\frac{\pi }{2\phi _{c}}\)
nnd	ε>δ	\(\phi _{c}= \varepsilon -\delta \)	Λ>0	\(\phi _{c} = \gamma \varLambda \)
ddn	No	–	No	–
dnd	\(R_{0} > 1\)	\(\frac{1}{\phi _{c}}\cos ^{-1} (\frac{\delta }{\varepsilon } ) \)	\(R_{0} > 1\)	\(\frac{1}{\phi _{c}}\cos ^{-1} (\frac{r}{r+\gamma \varLambda } )\)
dnd	ε>δ	\(\phi _{c} = \sqrt{\varepsilon ^{2}-\delta ^{2}}\)	Λ>0	\(\phi _{c} = \sqrt{\gamma ^{2}\varLambda ^{2}+2r\gamma \varLambda }\)
ndd	\(R_{0} > 3\)	\(\frac{1}{\phi _{c}}\cos ^{-1} (\frac{\delta }{\varepsilon -2\delta } ) \)	\(R_{0} > 1 +\frac{2}{\gamma } \)	\(\frac{1}{\phi _{c}}\cos ^{-1} (\frac{r}{\gamma \varLambda -r} )\)
ndd	ε>3δ	\(\phi _{c}= \sqrt{\varepsilon ^{2}+3\delta ^{2}-4\delta \varepsilon } \)	γΛ>2r	\(\phi _{c} = \sqrt{\gamma ^{2}\varLambda ^{2}-2r\gamma \varLambda }\)
ddd	\(R_{0}>1\)	\(\frac{\pi }{2\phi _{c}} \)	\(R_{0} > 1\)	\(\frac{\pi }{2\phi _{c}}\)
ddd	ε>δ	\(\phi _{c} = \varepsilon -\delta \)	Λ>0	\(\phi _{c} = \gamma \varLambda \)

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