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Theory and Modern Applications

Table 11 Critical bifurcation values for endemic equilibrium of the HIV 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

1(dnn)

3(nnd), 7(ddd)

5(dnd)

6(ndd)

Range of \(R_{0}\)

\(1 < R_{0} < 3\)

\(R_{0} > 1\)

\(R_{0} > 1\)

\(R_{0} > 3\)

\(R_{0}\)

1.1

2.9

2.9

2.9

3.5

ρ

0.00405

−0.00405

0

0.0045

−0.0045

η

0.0045

0.0045

0.0086

0.0131

0.0068

DDE

\(\tau _{c}\)

229.94

1371.68

183.72

99.492

457.26

\(\phi _{c}\)

1.962e−3

1.962e−3

8.550e−3

0.01225

5.031e−3

\(\frac{d\mu }{d\tau } \vert _{\tau _{c}}\)

1.848e−5

7.663e−8

2.108e−5

8.381e−5

1.729e−6

Discrete

\(\tau _{c}\)

m = 22

229.58

1183.78

179.60

98.25

426.73

m = 23

229.60

1191.27

179.78

98.31

428.01

\(\omega _{c}\)

m = 22

0.02005

0.1194

0.06981

0.05417

0.10219

m = 23

0.01920

0.1143

0.06684

0.05187

0.09785

\(\frac{dr}{d\tau } \vert _{\tau _{c}}\)

m = 22

8.466e−6

5.153e−7

8.693e−6

1.763e−5

2.401e−6

m = 23

7.760e−6

4.690e−7

7.959e−6

1.615e−5

2.193e−6

\(\Re [e^{-i\omega _{c}}c_{1} (\tau _{c} ) ]\)

m = 22

0.3454

0.6627

(3) −0.0371

−0.0222

−0.5919

   

(7) −0.2868

  

m = 23

0.3317

0.6377

(3) −0.0355

−0.0212

−0.5667

   

(7) −0.2745