The purpose of this paper is to analyze the mechanism for the interplay of deterministic and stochastic models for contagious diseases. Deterministic models for contagious diseases are prone to predict global stability. Small natural birth and death rates in comparison to disease parameters like the contact rate and the removal rate ensures that the globally stable endemic equilibrium corresponds to a tiny average proportion of infected individuals. Asymptotic equilibrium levels corresponding to low numbers of individuals invalidate the deterministic results.

Diffusion effects force probability mass functions of the stochastic model to possess similar stability properties as the deterministic model. Particular simulations of the stochastic model predict, however, oscillatory patterns. Small and isolated populations show longer periods, more violent oscillations, and larger probabilities of extinction.

We prove that evolution maximizes the infectiousness of the disease as measured by the ability to increase the proportion of infected individuals. This holds provided the stochastic oscillations are moderate enough to keep the proportion of susceptible individuals near a deterministic equilibrium.

We close our paper with a discussion of the herd-immunity concept and stress its close relation to vaccination-programs.

The present case study explores the assessment practice of two examiners from different academic backgrounds on an undergraduate thesis work in mathematics education. Reflection notes and feedback from two instances of a thesis-writing process are interrogated using a framework based on a semiotics perspective to meaning-making. It is shown that the examiners utilize aspects that are ‘immediate and non-contested’ to successively make accessible that which is ‘withdrawn yet to be revealed.’ In this process, varied interventional approaches are observed to support desirable aspects of teacher professional knowledge including critical-analytical disposition. The study also highlights knowledge integration for teaching, scientific disposition, collaboration, and knowledge transformation as some of the desirable aspects of teacher professional knowledge. The findings indicate that knowledge integration and transformation remain an issue of concern for pre-service teachers. This raises the question of how desirable traits for a teacher-as-a researcher can be promoted within the context of thesis work.

Linjär algebra innehåller ett rikt spektrum av metoder, och en förståelse av dessa krävs inom nästan alla samhällsområden där datorer, beräkningar, bildbehandling eller liknande används. Även om verkligheten många gånger är icke-linjär, kräver icke-linjär analys ofta en mycket god förståelse av de linjära specialfallen. Få delområden i matematiken kan på en lika elementär nivå klargöra skillnaderna mellan effektiva och mindre effektiva algoritmer och åskådliggöra matematikens potential för den fortsatta samhällsutvecklingen.

Denna fjärde upplaga har kompletterats med flera nya avsnitt, såsom allmänna likformighets- och kongruensavbildningar, linjära differentialekvationer, kvadratiska former, singulärvärdeuppdelning och nya övningsuppgifter. Målet är att framställningen ska utgå från grunderna samtidigt som den når så långt som möjligt genom att tydliggöra strukturen i ämnet.

Boken riktar sig till studenter som behöver förstå egenvärden och egenvektorer redan under sin första kurs i linjär algebra. Ambitionerna sträcker sig dock betydligt längre än så och boken kan med fördel användas i såväl grundkurser i linjär algebra, vilka börjar med vektorgeometri, som i fortsättningskurser i ämnet där vektorrum och komplexa matriser behandlas. 

Despite that outbreaks had been observed for hundreds of years for many populations, it took until the 1920s before the first mechanisms that did not involve human interference were suggested. Just a few mechanisms were included in the first models and the question whether the inclusion of other, very plausible, mechanisms could alter the predictions remained. In this chapter, we follow the development of models that have been proposed to explain oscillatory population dynamics from the early models suggested by Lotka (1925) and Volterra (1926) until global dynamical questions that are still open for models incorporating explicit resource dynamics, like the chemostat, cf Kuang (1989).

In this short introduction, the section “Mathematics, Science, and DynamicalSystems” of the Handbook of the Mathematics of the Arts and Sciences issummarized.

The ability of mixotrophs to combine phototrophy and phagotrophy is now well recognized and found to have important implications for ecosystem dynamics. In this paper, we examine the dynamical consequences of the invasion of mixotrophs in a system that is a limiting case of the chemostat. The model is a hybrid of a competition model describing the competition between autotroph and mixotroph populations for a limiting resource, and a predator--prey-type model describing the interaction between autotroph and herbivore populations. Our results show that mixotrophs are able to invade in both autotrophic environments and environments described by interactions between autotrophs and herbivores. The interaction between autotrophs and herbivores might be in equilibrium or cycle. We find that invading mixotrophs have the ability to both stabilize and destabilize autotroph-herbivore dynamics depending on the competitive ability of mixotrophs. The invasion of mixotrophs can also result in multiple attractors.

The idea of a dynamical system is predicting the future of a given system with respect to some initial conditions. If the dynamical system is formulated as a differential equation, then there is usually a direct relation between the dynamical system and the processes involved. Today, we can easily say that dynamical systems can predict a huge number of phenomena, including chaos. The real question is therefore, not whether complicated phenomena may occur, but whether restrictions on the possible dynamics exist.

In this chapter, we commence with major theorems that are frequently used for justifying phase space analysis. We continue with simple examples that either possess limit cycles and classes of differential equations that never possess limit cycles. We end up with the ideas behind two major theorems that put bounds for the number of limit cycles from above: Sansone's (1949) theorem and Zhang's (1986) theorem. Both theorems apply to systems that have a clear mechanistic interpretation. We outline the major arguments behind the quite precise estimates used in these theorems and describe their differences. Our objective is not to formulate these theorems in their most general form, but we give references to recent extensions.

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