In this paper, the dynamics of a ratio-dependent predator–prey model with strong Allee effect and Holling IV functional response is investigated by using dynamical analysis. The model is shown to have complex dynamical behaviors including subcritical or supercritical Hopf bifurcation, saddle-node bifurcation, Bogdanov–Takens bifurcation of codimension-2, a nilpotent focus or cusp of codimension-2. The codimension-2 Bogdanov–Takens bifurcation point acts as an organizing center for the whole bifurcation set. The coexistence of stable and unstable positive equilibria, homoclinic cycle is also found. Our analysis shows that the ratio-dependent model may collapse suddenly due to certain parameter variation, i.e. the numbers of predator and prey will decrease sharply to zeroes after undergoing a short time of sustained oscillations with small amplitudes. Of particular interest is that the coalescence of saddle-node bifurcation point and Hopf bifurcation point may indicate the occurrence of relaxation oscillations and the critical state of extinction of predator and prey. Numerical simulations and phase portraits including one-parameter bifurcation curve and two-parameter bifurcation curves are given to illustrate the theoretical results.
<abstract><p>This paper is devoted to investigating the impact of vaccination on mitigating COVID-19 outbreaks. In this work, we propose a compartmental epidemic ordinary differential equation model, which extends the previous so-called SEIRD model <sup>[<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>,<xref ref-type="bibr" rid="b3">3</xref>,<xref ref-type="bibr" rid="b4">4</xref>]</sup> by incorporating the birth and death of the population, disease-induced mortality and waning immunity, and adding a vaccinated compartment to account for vaccination. Firstly, we perform a mathematical analysis for this model in a special case where the disease transmission is homogeneous and vaccination program is periodic in time. In particular, we define the basic reproduction number $ \mathcal{R}_0 $ for this system and establish a threshold type of result on the global dynamics in terms of $ \mathcal{R}_0 $. Secondly, we fit our model into multiple COVID-19 waves in four locations including Hong Kong, Singapore, Japan, and South Korea and then forecast the trend of COVID-19 by the end of 2022. Finally, we study the effects of vaccination again the ongoing pandemic by numerically computing the basic reproduction number $ \mathcal{R}_0 $ under different vaccination programs. Our findings indicate that the fourth dose among the high-risk group is likely needed by the end of the year.</p></abstract>
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