Department of Mathematics, University of Trento, via Sommarive 14, 38123 Povo (Tn), Italy.
Math Biosci. 2011 Nov;234(1):33-46. doi: 10.1016/j.mbs.2011.08.004. Epub 2011 Aug 18.
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 behaviour, 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 monomorphic equilibria are stable. This fact has been rarely observed in models of competition between pathogen strains.
我们在这里分析了 Porco 和 Blower [1] 提出的具有交叉免疫性的疫苗模型。Porco 和 Blower [1] 表明，接种疫苗可以改变竞争平衡，有利于一种在没有接种疫苗的情况下会被淘汰的菌株，并且接种疫苗还可以促进不同菌株的共存，这通常是不被期望的[2]。他们的结果主要是通过数值模拟得出的，因此，竞争平衡或共存发生变化的条件尚未完全确定。我们至少从均衡的角度对其行为进行了相当完整的描述。我们找到了疫苗接种可能导致竞争平衡发生变化的确切条件，并表明，在这些条件下，始终存在一个接种率范围，在此范围内存在共存均衡。我们还发现，在一个“双稳定性”区域中存在共存均衡（且不稳定），其中两个单态均衡是稳定的。这一事实在病原体菌株竞争的模型中很少被观察到。
