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Strain replacement occurs when after a vaccination campaign one (or more) strains decline in prevalence while another strain (or strains) rise in prevalence. Differential effectiveness of the vaccine is the widely accepted and the most important mechanism which leads to this replacement effect. Recent theoretical studies have suggested that strain replacement may occur even if the vaccine is perfect, that is, the vaccine is completely effective with respect to all strains present. It has already been shown that perfect vaccination, along with a trade-off mechanism, such as co-infection or super-infection, lead to strain replacement. In this paper, we examine the hypothesis that strain replacement with perfect vaccination occurs only with trade-off mechanisms which allow a strain with a lower reproduction number to eliminate a strain with a higher reproduction number in the absence of vaccination. We test this hypothesis on a two-strain model with vertical transmission. We first show that vertical transmission as a trade-off mechanism can lead to dominance of a strain with suboptimal reproduction number. Based on the hypothesis we expect, and we show, that strain replacement occurs with vertical transmission.

We consider an SEIR epidemic model for an infectious disease that spreads in the human host population through both horizontal and vertical transmission. A periodically varying contact rate is introduced to simulate recurrent outbreaks. We use the optimal control theory to assess the disease control. Optimal vaccination strategies to minimize both the disease burden and the intervention costs are analyzed. We derive the optimality system and solve it numerically. The theoretical findings are then used to simulate a vaccination campaign for rubella under several scenarios, by using epidemiological parameters obtained by real data.

A mathematical model for Banana Xanthomonas Wilt (BXW) spread by insect vector is presented. The model incorporates inflorescence infection and vertical transmission from the mother corm to attached suckers, but not tool-based transmission by humans. Expressions for the basic reproduction number $R_0$ are obtained and it is verified that disease persists, at a unique endemic level, when $R_0>1$. From sensitivity analysis, inflorescence infection rate and roguing rate were the parameters with most influence on disease persistence and equilibrium level. Vertical transmission parameters had less effect on persistence threshold values. Parameters were approximately estimated from field data. The model indicates that single stem removal is a feasible approach to eradication if spread is mainly via inflorescence infection. This requires continuous surveillance and debudding such that a 50% reduction in inflorescence infection and 2–3 weeks interval of surveillance would eventually lead to full recovery of banana plantations and hence improved production.

A delayed SEIR epidemic model with vertical transmission and non-monotonic incidence is formulated. The equilibria and the threshold of the model have been determined on the bases of the basic reproduction number. The local stability of disease-free equilibrium and endemic equilibrium is established by analyzing the corresponding characteristic equations. By comparison arguments, it is proved that, if R₀ < 1, the disease-free equilibrium is globally asymptotically stable. Whereas, the disease-free equilibrium is unstable if R₀ > 1. Moreover, we show that the disease is permanent if the basic reproduction number is greater than one. Furthermore, the sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium when R₀ > 1.