Mathematics

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Immunization programs have often been impeded by vaccine scares, as evidenced by the measles-mumps-rubella (MMR) autism vaccine scare in Britain. A "free rider" effect may be partly responsible: vaccine-generated herd immunity can reduce disease incidence to such low levels that real or imagined vaccine risks appear large in comparison, causing individuals to cease vaccinating. This implies a feedback loop between disease prevalence and strategic individual vaccinating behavior. Here, we analyze a model based on evolutionary game theory that captures this feedback in the context of vaccine scares, and that also includes social learning. Vaccine risk perception evolves over time according to an exogenously imposed curve. We test the model against vaccine coverage data and disease incidence data from two vaccine scares in England & Wales: the whole cell pertussis vaccine scare and the MMR vaccine scare. The model fits vaccine coverage data from both vaccine scares relatively well. Moreover, the model can explain the vaccine coverage data more parsimoniously than most competing models without social learning and/or feedback (hence, adding social learning and feedback to a vaccine scare model improves model fit with little or no parsimony penalty). Under some circumstances, the model can predict future vaccine coverage and disease incidence-up to 10 years in advance in the case of pertussis-including specific qualitative features of the dynamics, such as future incidence peaks and undulations in vaccine coverage due to the population's response to changing disease incidence. Vaccine scares could become more common as eradication goals are approached for more vaccine-preventable diseases. Such models could help us predict how vaccine scares might unfold and assist mitigation efforts. © 2012 Bauch, Bhattacharyya.

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(SoNS) School of Natural Sciences

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PLoS Computational Biology

Evolutionary game theory and social learning can determine how vaccine scares unfold

The traditional frameworks for fuzzy linear optimization problems are inspired by the max-min model proposed by Zimmermann using the Bellman-Zadeh extension principle. This paper attempts to view fuzzy multiobjective linear programming problem (FMOLPP) from a perspective of preference modeling. The fuzzy constraints are viewed as negative preferences for rejecting what is unacceptable while the objective functions are viewed as positive preferences for depicting satisfaction to what is desired. This bipolar view enable us to handle fuzzy constraints and objectives separately and help to combine them in distinct ways. The optimal solution of FMOLPP is the one which maximizes the disjunctive combination of the weighted positive preferences provided it satisfy the negative preferences combined in conjunctive way. © 2012 Springer-Verlag.

Communications in Computer and Information Science

Fuzzy multiobjective linear programming: A bipolar view

Regular and chaotic oscillations in a modified discrete two dimensional coupled predator-prey model, proposed recently, is re-investigated by observing the bifurcation diagram, calculating Lyapunov exponents and correlation dimensions for various orbits. The map evolves from one cycle to three cycles followed by period doubling and then to a chaotic regime and show bistability as its parameter λ varies, 0 < ≤ 1.211. Initial population size of the species and the value their coupling coefficients play crucial role for subsequent evolutionary phenomena of the system. Regular and chaotic evolutions depend completely on the coefficient ,λ, and on the initial prey and predator population. Lyapunov exponents and correlation dimensions provide true measure of chaos and also, to identify the chaotic orbits. Some recently discovered indicators, FLI, SALI and DLI have also been used to understand the nature of orbits.Mathematical analysis have been carried out to find stable and unstable fixed points and to obtain numerical value of the parameter λ for sensitive state of the system to initial conditions i.e. emergence of chaos. Bistability condition have been discussed for variation of λ values. Graphical representation of Lyapunov exponents and correlation dimensions provide better understanding of the nature of orbits which are further justified by the use of FLI, SALI and DLI.

Institute for Innovations & Inventions with Mathematics and IT

Applied Mathematical Sciences

Chaotic evaluations in a modified coupledLogistic type predator-prey model

Linear Algebra and Its Applications

Norm inequalities related to the matrix geometric mean

In non-mandatory vaccination policies, individual choice can be a major driver of vaccine uptake. Choice thereby influences whether public health targets can be achieved. Individual vaccinating decisions can be influenced by perceptions of vaccine risks or infection risks. There is also the potential for non-vaccinators to strategically 'free-ride' on herd immunity provided by vaccinators. This strategic interaction between individuals generates a social dilemma - a conflict between self-interest and what is best for the group as a whole. Game theory and related mathematical approaches that couple mechanistic models of vaccinating decisions with mechanistic models of disease spread can capture this social dilemma and address relevant questions. The past decade has seen significant growth in the theoretical literature developing and analyzing such models. Here, we argue that using these models to address specific public health challenges will require more work that integrates information from empirical studies into the development and validation of such models, as well as more collaboration between mathematical modelers, psychologists, economists and public health experts. © 2012 Landes Bioscience.

Human Vaccines and Immunotherapeutics

Mathematical models of the interplay between individual vaccinating decisions and disease dynamics: A need for closer integration of models and data

The equivalence between the interval-valued fuzzy set (IVFS) and the intuitionistic fuzzy set (IFS) is exploited to study linear programming problems involving interval uncertainty modeled using IFS. The non-membership of IFS is constructed with three different viewpoints viz., optimistic, pessimistic, and mixed. These constructions along with their indeterminacy factors result in S-shaped membership functions in the fuzzy counterparts of the intuitionistic fuzzy linear programming models. The solution methodology of Yang et al. [45], and its subsequent generalization by Lin and Chen [33] are used to compute the optimal solutions of the three fuzzy linear programming models. © 2011 Elsevier B.V. All rights reserved.

Fuzzy Sets and Systems

Fuzzy linear programming under interval uncertainty based on IFS representation

During the 2009 H1N1 pandemic, many individuals did not seek vaccination immediately but rather decided to "wait and see" until further information was available on vaccination costs. This behaviour implies two sources of strategic interaction: as more individuals become vaccinated, both the perceived vaccination cost and the probability that susceptible individuals become infected decline. Here we analyze the outcome of these two strategic interactions by combining game theory with a disease transmission model during an outbreak of a novel influenza strain. The model exhibits a "wait and see" Nash equilibrium strategy, with vaccine delayers relying on herd immunity and vaccine safety information generated by early vaccinators. This strategic behaviour causes the timing of the epidemic peak to be strongly conserved across a broad range of plausible transmission rates, in contrast to models without such adaptive behaviour. The model exhibits not only feedback mechanisms but also a feed-forward mechanism: a high initial perceived vaccination cost perpetuates high perceived vaccine costs (and lower vaccine coverage) throughout the remainder of the outbreak. This suggests that any effect of risk communication at the start of a pandemic outbreak will be amplified compared to the same amount of risk communication effort distributed throughout the outbreak. © 2011 Elsevier Ltd.

Vaccine

"Wait and see" vaccinating behaviour during a pandemic: A game theoretic analysis

This paper presents an approach based on value and ambiguity indexes defined in [1] to solve linear programming problems with data as triangular intuitionistic fuzzy numbers. © 2011. The authors-Published by Atlantis Press.

Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology, EUSFLAT 2011 and French Days on Fuzzy Logic and Applications, LFA 2011

Linear programming with triangular intuitionistic fuzzy number

We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences, then a general bounded Bessel sequence in an arbitrary Hilbert space can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and prove that for these spaces bounded Bessel sequences of normalized kernel functions are finite unions of Riesz basic sequences.

Indiana University Mathematics Journal

The feichtinger conjecture and reproducing kernel hilbert spaces

In insect ecosystem, the dynamics of prey and predator is regulated by complex interactions between them. Insect pests are spatially aggregated in patches forming a spatial pattern in the environment. An efficient predator dynamically changes its strategies and time for its random search movements to concentrate on higher resource patches based on the benefit of assessment. This food-gathering activity of both prey and predator plays a major role in stabilizing the system by influencing the per unit food consumption. Extending Holling time-budget argument by migration, here we formulate a two patch prey-predator model and show that how several foraging parameters such as handling time, dispersal rate can have important consequences in stability of prey-predator system. Specifically, the ratio between timings that a predator remains mobile in searching and handling their food, is the most important one and simulation on this suggests that the stabilizing effect continues to operate when the dispersal process is modeled more realistically. Thus we conclude that the migration submodel is an important constituent of a spatial predator-prey system. These results are shown to have important implications for possible biological control. © 2010 Korean Society for Computational and Applied Mathematics.