Determination of optimal vaccination strategies using an orbital stability threshold from periodically driven systems

Onyango, Nelson Owuor; Müller, Johannes

doi:10.1007/s00285-013-0648-8

Cited by 14 publications

(21 citation statements)

References 29 publications

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“…This result goes along with Agur et al (1993). See also the discussion in Onyango and Müller (2014). Corollary 1.…”

Section: (T) For Simplicity We Writesupporting

confidence: 57%

“…We are naturally led to consider temporary immunity, as discussed in (Onyango and Müller, 2014): in the case of diseases with life-long immunity and lifespan much longer than the period (µ ∼ α 1/T ) the vaccination is not affected by periodicity. For instance, measles has periodic transmission rate but, since it confers permanent immunity, periodic vaccination is not used.…”

Section: Discussionmentioning

confidence: 99%

“…Now, we show the alternative in Lemma 1. Let R 0 = βS 0 γ+µ , as defined in Onyango and Müller (2014) and in a more general setting in Rebelo et al (2012); Wang and Zhao (2008). We show that the only relevant assumption in Rebelo et al (2012, theorem 2) is the value of R 0 .…”

Section: A Proof Of Lemmamentioning

confidence: 99%

“…We then construct the optimal vaccination strategy as the limit of the preventive strategies for which the vaccination effort is minimized. In (Onyango and Müller, 2014), optimality for time dependent vaccination profiles is defined based on the effective reproductive number.…”

Section: Optimal Vaccinationmentioning

confidence: 99%

“…Typically, the objective is to define an optimal vaccination strategy by minimizing combinations of the vaccination effort/cost and of the effective reproduction number R 0 (i.e., the number of secondary infections generated by a primary case) (Müller and Hadeler, 1996;Castillo-Chavez and Feng, 1998;Laguzet and Turinici, 2015a). This can give rise to particularly interesting problems when we consider non-homogeneous models for which vaccination strategy depends on age (Müller and Hadeler, 1996;Castillo-Chavez and Feng, 1998;Tartof et al, 2013), risk-groups (Scott et al, 2015;Long and Owens, 2011) or when it is time-dependent (Onyango and Müller, 2014;d'Onofrio, 2002;Browne et al, 2015;Houy, 2016). The current work is concerned with timedependent epidemic models with vaccination when both the transmission rate and the vaccination are assumed to be periodic.…”

Section: Introductionmentioning

confidence: 99%

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Optimal vaccination strategies and rational behaviour in seasonal epidemics

et al. 2016

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We consider a SIRS model with time dependent transmission rate. We assume time dependent vaccination which confers the same immunity as natural infection. We study two types of vaccination strategies: i) optimal vaccination, in the sense that it minimizes the effort of vaccination in the set of vaccination strategies for which, for any sufficiently small perturbation of the disease free state, the number of infectious individuals is monotonically decreasing; ii) Nash-equilibria strategies where all individuals simultaneously minimize the joint risk of vaccination versus the risk of the disease. The former case corresponds to an optimal solution for mandatory vaccinations, while the second corresponds to the equilibrium to be expected if vaccination is fully voluntary. We are able to show the existence of both optimal and Nash strategies in a general setting. In general, these strategies will not be functions but Radon measures. For specific forms of the transmission rate, we provide explicit formulas for the optimal and the Nash vaccination strategies.

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“…This result goes along with Agur et al (1993). See also the discussion in Onyango and Müller (2014). Corollary 1.…”

Section: (T) For Simplicity We Writesupporting

confidence: 57%

Section: Discussionmentioning

confidence: 99%

Section: A Proof Of Lemmamentioning

confidence: 99%

Section: Optimal Vaccinationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal vaccination strategies and rational behaviour in seasonal epidemics

et al. 2016

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From regional pulse vaccination to global disease eradication: insights from a mathematical model of poliomyelitis

2014

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Mass-vaccination campaigns are an important strategy in the global fight against poliomyelitis and measles. The large-scale logistics required for these mass immunisation campaigns magnifies the need for research into the effectiveness and optimal deployment of pulse vaccination. In order to better understand this control strategy, we propose a mathematical model accounting for the disease dynamics in connected regions, incorporating seasonality, environmental reservoirs and independent periodic pulse vaccination schedules in each region. The effective reproduction number, Re, is defined and proved to be a global threshold for persistence of the disease. Analytical and numerical calculations show the importance of synchronising the pulse vaccinations in connected regions and the timing of the pulses with respect to the pathogen circulation seasonality. Our results indicate that it may be crucial for mass-vaccination programs, such as national immunisation days, to be synchronised across different regions. In addition, simulations show that a migration imbalance can increase Re and alter how pulse vaccination should be optimally distributed among the patches, similar to results found with constant-rate vaccination.Furthermore, contrary to the case of constant-rate vaccination, the fraction of environmental transmission affects the value of Re when pulse vaccination is present.against polio and measles is mass immunisation, which may be regarded as a pulse vaccination [9]. The complex logistics required for these mass-immunisation campaigns magnifies the need for research into the effectiveness and optimal deployment of pulse vaccination [43].Pulse vaccination has been investigated in several mathematical models, often in disease models with seasonal transmission. Many diseases show seasonal patterns in circulation; thus inclusion of seasonality may be crucial. Agur et al. (1993) argued for pulse vaccination using a model of seasonal measles transmission, conjecturing that the pulses may antagonise the periodic disease dynamics and achieve control at a reduced cost of vaccination [1]. Shulgin et al. (1998) investigated the local stability of the disease-free periodic solution in a seasonally forced population model with three groups: susceptible (S), infected (I) and recovered (R).They considered pulse vaccination and explicitly found the threshold pulsing period [31]. Recently, Onyango and Müller considered optimal periodic vaccination strategies in the seasonally forced SIR model and found that a well-timed pulse is optimal, but its effectiveness is often close to that of constant-rate vaccination [28].In addition to seasonality, spatial structure has been recognised as an important factor for disease dynamics and control [38]. Heterogeneity in the population movement, along with the patchy distribution of populations, suggests the use of metapopulation models describing disease transmission in patches or spatially structured populations or regions. Mobility can be incorporated and tracked in these models in vari...

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Optimal time-profiles of public health intervention to shape voluntary vaccination for childhood diseases

2018

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In order to seek the optimal time-profiles of public health systems (PHS) Intervention to favor vaccine propensity, we apply optimal control (OC) to a SIR model with voluntary vaccination and PHS intervention. We focus on short-term horizons, and on both continuous control strategies resulting from the forward-backward sweep deterministic algorithm, and piecewise-constant strategies (which are closer to the PHS way of working) investigated by the simulated annealing (SA) stochastic algorithm. For childhood diseases, where disease costs are much larger than vaccination costs, the OC solution sets at its maximum for most of the policy horizon, meaning that the PHS cannot further improve perceptions about the net benefit of immunization. Thus, the subsequent dynamics of vaccine uptake stems entirely from the declining perceived risk of infection (due to declining prevalence) which is communicated by direct contacts among parents, and unavoidably yields a future decline in vaccine uptake. We find that for relatively low communication costs, the piecewise control is close to the continuous control. For large communication costs the SA algorithm converges towards a non-monotone OC that can have oscillations.

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Determination of optimal vaccination strategies using an orbital stability threshold from periodically driven systems

Cited by 14 publications

References 29 publications

Optimal vaccination strategies and rational behaviour in seasonal epidemics

Optimal vaccination strategies and rational behaviour in seasonal epidemics

From regional pulse vaccination to global disease eradication: insights from a mathematical model of poliomyelitis

Optimal time-profiles of public health intervention to shape voluntary vaccination for childhood diseases

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