Analysis of a vaccine model with cross-immunity: When can two competing infectious strains coexist?

Abstract

We analyse here the Vaccine Model with Cross-Immunity proposed by Porco and Blower [1]. Porco and Blower[1] show that vaccination can shift the competitive balance in favour of a strain that, without vaccination, would be out-competed and that vaccination can also promote coexistence of different strains, something that normally is not expected [2]. Their results have been mainly obtained through numerical simulations, so that the conditions under which a shift in competitive balance or coexistence occurs have not been fully established.We give a rather complete description of its behavior, at least in terms of equilibria. We find the exact conditions under which vaccination may lead to a shift in competitive balance and show that, under these conditions, there always exist a range of vaccination rates under which a coexistence equilibrium exists. We also find that a coexistence equilibrium exists (and is unstable) in a 'bi-stability' region, where both 1 *Manuscript monomorphic equilibria are stable. This fact has been rarely observed in models of competition between pathogen strains.

Reevaluation of Epidemiological Data Demonstrates That It Is Consistent With Cross-Immunity Among Human Papillomavirus Types

“…These interpretations have affected both vaccination policy and modeling [10][11][12]. Many of the models of effectiveness and cost-effectiveness of HPV vaccination have not addressed the possibility of cross-immunity, with a few exceptions [37,38]. Our results suggest that type replacement may reduce vaccine effectiveness.…”

On the formulation of epidemic models (an appraisal of Kermack and McKendrick)

“…Often when epidemic models are formulated as delay equations, the aim is to derive conditions for the (in)stability of the endemic steady state and to find out, by way of the Hopf bifurcation theorem, when one should expect to find persistent oscillations [ The trigger for the approach sketched in the present paper is somewhat different: our ultimate goal is to find conditions for strain replacement as a result of mass vaccination [17,19,27,38,44]. In this context, the need arises to incorporate (i) a realistic form of demographic turnover, (ii) the waning of immunity and (iii) cross-immunity.…”

Network epidemiology and plant trade networks

“…Similarly, a temporal study of the spatial association in England and Wales (2003–06) between P. ramorum reports in the trade and in the wild found that infected premises (garden centres and nurseries) were significantly associated with infected sites (woodlands and historic gardens) at short distances (<1 km) over the studied period, thus suggesting a role of the trade in spreading the epidemic (Xu et al 2009). Given that various Phytophthora species are found in plant nurseries, and that some of these species have hybridized to produce new emerging pathogens, plant epidemiology too may benefit from models of the interactions between different pathogen strains spreading in spatial networks (Alexander and Kobes 2011; Dorigatti and Pugliese 2011; Karrer and Newman 2011; Weng et al 2012). …”