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Theory and Modern Applications

Table 6 Table of \(b_{0}(Y_{n},Y_{n})\) and \(c_{0}(Y_{n},Y_{n},Y_{n})\) formulas for HIV and ELM 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

\(b_{0}(Y_{n},Y_{n})\)

HIV

ELM Λ = β − r, \(\nu =\frac{\beta }{K} (\frac{\varLambda }{\beta })^{\frac{\gamma -1}{\gamma }}\)

dnn

\(-2\varepsilon h_{m} \tau y_{n} y_{n}\)

\(-(\gamma ^{2}+\gamma )\nu h_{m} \tau y_{n} y_{n}\)

nnd, dnd

\(-\varepsilon h_{m} \tau (y_{n} y_{n-m}+y_{n-m}y_{n})\)

\(\begin{array}[t]{l}{-}\nu h_{m} \tau [(\gamma ^{2}-\gamma ) y_{n-m}y_{n-m} \\ \quad {}+\gamma (y_{n}y_{n-m}+y_{n-m}y_{n})]\end{array}\)

ndd, ddd

\(-2\varepsilon h_{m} \tau y_{n-m} y_{n-m}\)

\(-\nu h_{m} \tau (\gamma ^{2}+\gamma )y_{n-m}y_{n-m}\)

\(c_{0}(Y_{n},Y_{n},Y_{n})\)

ELM Λ = β − r, \(\xi =\frac{\beta }{K^{2}}(\frac{\varLambda }{\beta })^{\frac{\gamma -2}{\gamma }}\)

dnn

\(-(\gamma ^{3}-\gamma )\xi h_{m} \tau y_{n}y_{n}y_{n} \)

nnd, dnd

\(\begin{array}[t]{l}{-}\xi h_{m} \tau [(\gamma ^{3}-3\gamma ^{2}+2\gamma )y_{n-m} y_{n-m}y_{n-m} \\ \quad {}+(\gamma ^{2}-\gamma )(y_{n}y_{n-m}y_{n-m}+y_{n-m}y_{n}y_{n-m}+y_{n}y_{n-m}y_{n-m})] \end{array}\)

ndd, ddd

\(-\xi h_{m} \tau (\gamma ^{3}-\gamma )y_{n-m}y_{n-m}y_{n-m} \)