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The World Health Organization estimated that 8 million adults between 15 and 49 years old acquired syphilis globally in 2022. China CDC reported that there were 530 116 cases of syphilis in mainland China in 2023. Since syphilis is a sexually transmitted disease and age structure of the host population plays a crucial role, in this series of two papers we develop an age-structured model with four infection stages (primary, secondary, latent and tertiary) to study the transmission dynamics of syphilis. In part I (Wuet al. 2025Proc. R. Soc. A481: 20240218 (doi:10.1098/rspa.2024.0218)), we investigated the well-posedness of the model and studied stability of the steady states. In part II, first, we consider the optimal control of the age-structured model. Second, utilizing the Markov Chain Monte Carlo method, we calibrate the reported syphilis data in China by using a demographic model. Finally, we apply the relevant simulation results to numerically simulate the age-structured model. Our results indicate that (i) for the syphilis demographic model, the basic reproduction number ${\mathcal{R}}_0\approx 2.4876$with CI (95%) (1.6823, 3.1434); (ii) tertiary stage infection is more severe in the elderly population; (iii) reducing the number of secondary and latent stage syphilis individuals can effectively reduce the total number of infected populations. 

Abstract In this paper, we study a predator–prey mite model of Leslie type with generalized Holling IV functional response. The model is shown to have very rich bifurcation dynamics, including subcritical and supercritical Hopf bifurcations, degenerate Hopf bifurcation, focus‐type and cusp‐type degenerate Bogdanov–Takens bifurcations of codimension 3, originating from a nilpotent focus or cusp of codimension 3 that acts as the organizing center for the bifurcation set. Coexistence of multiple steady states, multiple limit cycles, and homoclinic cycles is also found. Interestingly, the coexistence of two limit cycles is guaranteed by investigating generalized Hopf bifurcation and degenerate homoclinic bifurcation, and we also find that two generalized Hopf bifurcation points are connected by a saddle‐node bifurcation curve of limit cycles, which indicates the existence of global regime for two limit cycles. Our work extends some results in the literature. 

In this paper, we analyse Turing instability and bifurcations in a host–parasitoid model with nonlocal effect. For a ordinary differential equation model, we provide some preliminary analysis on Hopf bifurcation. For a reaction–diffusion model with local intraspecific prey competition, we first explore the Turing instability of spatially homogeneous steady states. Next, we show that the model can undergo Hopf bifurcation and Turing–Hopf bifurcation, and find that a pair of spatially nonhomogeneous periodic solutions is stable for a(8,0)-mode Turing–Hopf bifurcationand unstable for a(3,0)-mode Turing–Hopf bifurcation. For a reaction–diffusion model with nonlocal intraspecific prey competition, we study the existence of the Hopf bifurcation, double-Hopf bifurcation, Turing bifurcation, and Turing–Hopf bifurcation successively, and find that a spatially nonhomogeneous quasi-periodic solution is unstable for a(0,1)-mode double-Hopf bifurcation. Our results indicate that the model exhibits complex pattern formations, including transient states, monostability, bistability, and tristability. Finally, numerical simulations are provided to illustrate complex dynamics and verify our theoretical results.