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Abstract Recent experimental evidence suggests that spatial heterogeneity plays an important role in within‐host infections caused by different viruses including hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV). To examine the spatial effects of viral infections, in this paper we study the asymptotic spreading in a within‐host viral infection model, which describes the spatial expansion speeds of viruses and infected cells within an infected host. We first establish the boundedness of solutions to the Cauchy problem via local ‐estimates and dual arguments. Then the spreading speed is estimated when the basic reproduction number of the corresponding kinetic system is larger than one. More precisely, the upper bounds of the spreading speed are given by constructing suitable upper solutions while the lower bounds of the spreading speed are obtained by introducing an auxiliary equation with nonlocal delay. When the basic reproduction number of the corresponding kinetic system is less than or equal to one, the virus dies out uniformly. Finally, we present some numerical simulations to illustrate our theoretical findings and discuss the biological relevance of these results. 

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. 

Antimicrobial de-escalation refers to reducing the spectrum of antibiotics used in treating bacterial infections. This strategy is widely recommended in many antimicrobial stewardship programs and is believed to reduce patients’ exposure to broad-spectrum antibiotics and prevent resistance. However, the ecological benefits of de-escalation have not been universally observed in clinical studies. This paper conducts computer simulations to assess the ecological effects of de-escalation on the resistance prevalence ofPseudomonas aeruginosa—a frequent pathogen causing nosocomial infections. Synthetic data produced by the models are then used to estimate the sample size and study period needed to observe the predicted effects in clinical trials. Our results show that de-escalation can reduce colonization and infections caused by bacterial strains resistant to the empiric antibiotic, limit the use of broad-spectrum antibiotics, and avoid inappropriate empiric therapies. Further, we show that de-escalation could reduce the overall super-infection incidence, and this benefit becomes more evident under good compliance with hand hygiene protocols among health care workers. Finally, we find that any clinical study aiming to observe the essential effects of de-escalation should involve at least ten arms and last for four years—a size never attained in prior studies. This study explains the controversial findings of de-escalation in previous clinical studies and illustrates how mathematical models can inform outcome expectations and guide the design of clinical studies. 

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.