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Joshua Moore
Joshua Moore

Mathematical Analysis : Functions, Limits, Seri... __FULL__



In mathematics, a series is, roughly speaking, the operation of adding infinitely many quantities, one after the other, to a given starting quantity.[1] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance.




Mathematical Analysis : Functions, Limits, Seri...



Audience: This text is appropriate for any undergraduate mathematics course in real analysis ormathematical analysis, or for a preparatory class for beginning mathematics graduate studentswho will later advance to courses in measure theory and functional analysis.Knowledge of this material will also be of benefit to graduate students in economics, signal and image processing,fluid and structural mechanics, etc.


Limits in maths are defined as the values that a function approaches the output for the given input values. Limits play a vital role in calculus and mathematical analysis and are used to define integrals, derivatives, and continuity. It is used in the analysis process, and it always concerns the behavior of the function at a particular point. The limit of a sequence is further generalized in the concept of the limit of a topological net and related to the limit and direct limit in the theory category. Generally, the integrals are classified into two types namely, definite and indefinite integrals. For definite integrals, the upper limit and lower limits are defined properly. Whereas indefinite integrals are expressed without limits, and it will have an arbitrary constant while integrating the function. Let us discuss the definition and representation of limits of the function, with properties and examples in detail.


H. Pishro-Nik is a professor in the Department of Electrical andComputer Engineering at the University of Massachusetts Amherst. Hereceived his B.S. degree from Sharif University of Technology, and M.Sc. andPh.D. degrees from Georgia Institute of Technology, all in Electrical andComputer Engineering. His research interests include Information Theory,Error Control Coding, and mathematical analysis of wireless networks.


Over and above the simple scenario described in the basic models, many other factors such as, host factors, demographic heterogeneity, geographic distribution of populations, rules of social interactions, climate and environmental influences, and the ecology of the area play important roles in the development of malaria in space and time. Age-specific host immunity, parasite diversity, DDT and drug resistance dynamics, vector population dynamics, effect of global warming are also interacting factors and variables that influence disease dynamics at different scales. There has not been A MODEL that has been able to incorporate all factors and variables because of the overwhelming complexity of the system. Also, a model's utility may not always lie in its mathematical analysis or incorporating finer details. The ability to base it on relevant details and ask specific questions that can be tested, are the hallmark of useful models. Along with fitting the past data and predicting the future, it should also be able to point to areas where data needs to be generated in order to increase our conceptual grasp. Such improvements in modelling generally occur in multiple steps, one leading to the other, as more information become available. The next section elaborates on some representative next-generation mathematical models that evolved from the above-mentioned basic models, and includes the increasing complexities of host-vector-parasite interactions. Specifically, the factors considered here are - (i) Age and immunity, (ii) Host-Pathogen variability and resistant Strains, (iii) Environmental factors, (iv) Social and economic factors, and (v) Migration and visitation.


Immunity can be described as a continuum of different levels of protection rather than a single class, as anti-malarial immunity develops slowly among people exposed to continuous and intense malaria transmission. Yang [67] divided the immune class (R h ) in human population into immune (R h1 ), partially immune (R h2 ) and non-immune but with immunologic memory (R h3 ), with each class having differential immunity (see Additional file 1 Table S3). The mathematical analysis of Yang model shows that the effects of these three types of immune responses lead to delay in the reappearance of the individuals, who already had experienced malaria, to the susceptible population. Hence the community under high threat of malaria (high R 0 ) shows low prevalence of individuals with asexual blood-stage infection and without infectious gametocytes, whereas, the same community is relatively free of severe infection due to the increase in immunity by re-infection.


A model is a mathematical abstraction of reality. The level of abstraction depends on the questions asked and the scale at which the underlying causative processes are studied. For example, in inter-host transmission of an infection, most molecular events in host-pathogen interactions (e.g., types of immune cells involved, parasite development inside the host, signaling pathways) are not considered. Many of these processes are condensed into a single parameter in the immune function or inoculation rate. In intra-host models, on the other hand, how the titers of the infective agent or related molecules change in an individual is studied, because that is what decides the diseased state of the host individual. In epidemiological models, intra-host processes between host-parasite-vector are neglected, but the host and vector population are subdivided in terms of the infection/diseased state (i.e., Susceptible, Exposed, Infected, and Immune/Recovered). These models aim to match their results with the available epidemiological data, where incidence of the infectious disease (and death due to it) determines the status of an epidemic in the population. Here, from the public health point of view, one is more interested in knowing if the infection will die out, or persist in a population through the important parameter R 0 . Yet, as more molecular studies are coming to the fore and both detection of infection and mode of infection propagation (genes, proteins, pathways, immune interactions) are elucidated, the epidemiological models also would need to consider these processes for inclusion as parameters or subclasses. This is clearly visible in the later models, where both the infected and recovered classes are divided into subclasses (such as, asymptomatic), which have different time scales and/or transmission modes. In this review, efforts have been taken to group the epidemiological models of malaria in terms of the complexity of infection processes included in its description, which makes them more realistic. The age-specific distribution of infection due to differential immunity across age is one such case. The assumption is that more realistic models would enhance the understanding of the infection transmission process at the population level, which, in turn, may help in better prediction of intervention strategies. The specific models discussed here are only indicative and not exhaustive. Pure mathematical analysis of the models, even though not so popular among the biologists, is important. They allow clear understanding of the logic of the system behaviour in terms of the relationship among the parameters and variables, which are representative to real biological processes. It will be useful to develop connections between mathematical analysis and their real world implications, since such analyses may help us to understand hitherto unknown scenario, such as the effect of temperature, seasonal forcing, excessive rainfall, correlation between different variables and parameter changes. Among the innumerable statistical models based on malaria incidence data, only a few approaches have been described here. The results of these models are mostly data specific and applicable primarily to the particular data set studied. They are highly useful for prediction in that specific context, but may not work in other places/scenarios. Models that incorporate the essentials of host-parasite-vector interaction, proper clinical population subdivisions for disease transmission and also describe multiple data sets from different ecological regions, promise to be an ideal combination of both approaches. It is now clear that the role of indirect factors such as, social structure, economic status, play an overwhelming role in the transmission and persistence of malaria in a region. It also underlies the failure of several control measures where local heterogeneity was not considered. It is the need of the hour to include factors such as the role of heterogeneity in host population due to social status, local differences in ecology due to poverty, differential effects of disease transmission in populations residing in habitats of different temperature, in the mathematical models. Such a description has the possibility of yielding understanding of malaria transmission for populations with societal differences and climate change. Mathematical models have the ability to address several multiplicative, feedback and nonlinear effects that enhance or suppress the effects of factors such as, exposure, immunity, spatiotemporal heterogeneities, control measures and environment, in order to capture key linkages to the complex transmission dynamics. They can also include stochasticity in different variables and parameters to simulate realistic scenario. This comparative analysis of different mathematical models of malaria would contribute to consolidate our understanding about the evolution of these models, and may also help in developing new models by incorporating features discussed above to improve predictions and deciding realistic control measures. 041b061a72


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