LAUSR

Viral diseases of honey bees are important economically and ecologically and have been widely modelled. The models reflect the fact that, in contrast to the typical case for vertebrates, invertebrates cannot acquire immunity to a viral disease, so they are of SIS or (more often) SI type. Very often, these diseases may be transmitted vertically as well as horizontally, by vectors as well as directly, and through the environment, although models do not generally reflect all these transmission mechanisms. Here, we shall consider an important additional complication the consequences of which have yet to be fully explored in a model, namely that both infected honey bees and their vectors may best be described using more than one infection class. For honey bees, we consider three infection classes. Covert infections occur when bees have the virus under control, such that they do not display symptoms of the disease, and are minimally or not at all affected by it. Acutely overtly infected bees often exhibit severe symptoms and have a greatly curtailed lifespan. Chronically overtly infected bees typically have milder symptoms and a moderately shortened lifespan. For the vector, we consider just two infection classes which are covert infected and overt infected as has been observed in deformed-wing virus (DWV) vectored by varroa mites. Using this structure, we explore the impact of spontaneous transition of both mites and bees from a covertly to an overtly infected state, which is also a novel element in modelling viral diseases of honey bees made possible by including the different infected classes. The dynamics of these diseases are unsurprisingly rather different from the dynamics of a standard SI or SIS disease. In this paper, we highlight how our compartmental structure for infection in honey bees and their vectors impact the disease dynamics observed, concentrating in particular on DWV vectored by varroa mites. If there is no spontaneous transition, then a basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0 exists. We derive a condition for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0>1$$\end{document}R0>1 that reflects the complexities of the system, with components for vertical and for direct and vector-mediated horizontal transmission, using the directed graph of the next-generation matrix of the system. Such a condition has never previously been derived for a honey-bee–mite–virus system. When spontaneous transitions do occur, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0 no longer exists, but we introduce a modification of the analysis that allows us to determine whether (i) the disease remains largely covert or (ii) a substantial outbreak of overt disease occurs.