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For many infectious diseases, including malaria and COVID-19, the host may experience more than one episode of infection, where reinfection occurs due to waning immunity. In this paper, we propose a new age-structured epidemic model to investigate the dynamics of such diseases with multiple infections. The model is based on a system of partial differential equations that describes the interplay between completely susceptible individuals, temporarily immune individuals, and infected individuals at different stages. The model incorporates both time and age-dependent variables and parameters. We derive the basic reproduction number and conduct rigorous analyses on the equilibrium solutions and their stability properties. Specifically, we study the global asymptotic stability of the disease-free equilibrium and obtain the explicit conditions for the occurrence of a backward bifurcation. Our findings could provide useful insights into the effects of disease prevention and intervention strategies such as vaccination campaigns. 

Abstract Butterflies are a diverse and charismatic insect group that are thought to have evolved with plants and dispersed throughout the world in response to key geological events. However, these hypotheses have not been extensively tested because a comprehensive phylogenetic framework and datasets for butterfly larval hosts and global distributions are lacking. We sequenced 391 genes from nearly 2,300 butterfly species, sampled from 90 countries and 28 specimen collections, to reconstruct a new phylogenomic tree of butterflies representing 92% of all genera. Our phylogeny has strong support for nearly all nodes and demonstrates that at least 36 butterfly tribes require reclassification. Divergence time analyses imply an origin ~100 million years ago for butterflies and indicate that all but one family were present before the K/Pg extinction event. We aggregated larval host datasets and global distribution records and found that butterflies are likely to have first fed on Fabaceae and originated in what is now the Americas. Soon after the Cretaceous Thermal Maximum, butterflies crossed Beringia and diversified in the Palaeotropics. Our results also reveal that most butterfly species are specialists that feed on only one larval host plant family. However, generalist butterflies that consume two or more plant families usually feed on closely related plants.