arXiv subjects
Publications and source records attributed to Arni S. R. Srinivasa Rao.
In point of fact the Indian tradition in mathematics is long and glorious. It dates back to earliest times, and indeed many of the Indian discoveries from 5000 years ago correspond rather naturally to modern mathematical results.
A new concept called multilevel contours is introduced through this article by the author. Theorems on contours constructed on a bundle of complex planes are stated and proved. Multilevel contours can transport information from one complex plane to another. Within a random environment, the behavior of contours and multilevel contours passing through the bundles of complex planes are studied. Further properties of contours by a removal process of the data are studied. The concept of 'islands' and 'holes' within a bundle is introduced through this article. These all constructions help to understand the dynamics of the set of points of the bundle. Further research on the topics introduced here will be followed up by the author. These include closed approximations of the multilevel contour formations and their removal processes. The ideas and results presented in this article are novel.
We state and prove a theorem on the partitioning of a randomly selected large population into stationary and non-stationary components by using a property of stationary population identity. Applications of this theorem for practical purposes is summarized at the end.
In this article, we have described the Rao distance (due to C.R. Rao) and ideas of conformal mappings on 3D objects with angle preservations. Three propositions help us to construct distances between the points within the 3D objects in \mathbb{R}^{3} and line integrals within complex planes. We highlight the application of these concepts to virtual tourism.
Understanding sustainability through modeling involves one of the complex and interdisciplinary activities where mathematics plays a key role. We provide arguments favoring the need for developing global models for measuring the status of sustainability. A global model (applicable in broader perspective) and global sustainability indices are proposed which can be used with real-world data. The solutions of the proposed Partial Differential Equations (PDEs) are blended with the weight functions of Riemann Stieltjes integrals to capture the differential importance of sustainability associated factors. The ideas, methods, and models are new and are prepared for handling multi-dimensional and multi-variate data. A practically adaptable formula for measuring the sustainability index is developed with few key variables. We provide a real-world example arising in civil engineering applications with a numerical example to demonstrate our models.
A population is considered stationary if the growth rate is zero and the age structure is constant. It thus follows that a population is considered non-stationary if either its growth rate is non-zero and/or its age structure is non-constant. We propose three properties that are related to the stationary population identity (SPI) of population biology by connecting it with stationary populations and non-stationary populations which are approaching stationarity. One of these important properties is that SPI can be applied to partition a population into stationary and non-stationary components. These properties provide deeper insights into cohort formation in real-world populations and the length of the duration for which stationary and non-stationary conditions hold. The new concepts are based on the time gap between the occurrence of stationary and non-stationary populations within the SPI framework that we refer to as Oscillatory SPI and the Amplitude of SPI. This article will appear in Bulletin of Mathematical Biology (Springer)
In usual demographic analysis, force of mortality is a function of one variable, that is, of age. In this article bi-variate and multivariate force of mortality functions are introduced for the first time to explain mortality differentials. The pattern of mortality in a population is one of the strong influencing factors in determining the life expectancies at various ages in the population. Considering univariate functions of age only to understand the human mortality data without associating with other variables could lead to incomplete analysis. The reasons behind declining forces of mortality globally could be studied using the proposed functions. Other applications of multivariate forces of mortality could be in actuarial sciences.
We derive formulas to compute mean number of worms in a newly Helminth infected population before secondary infections are started (population is closed). We have proved the two types of growth functions arise in this process as measurable functions.
Estimating the human longevity and computing of life expectancy are central to the population dynamics. These aspects were studied seriously by scientists since fifteenth century, including renowned astronomer Edmund Halley. From basic principles of population dynamics, we propose a method to compute life expectancy from incomplete data.
In this paper, we have proposed a two phase procedure (combining discrete graphs and wavelets) for constructing a true epidemic growth. In the first phase graph theory based approach was developed to update partial data available and in the second phase we used this partial data to generate a plausible complete data through wavelets. This procedure although novel and implementable, still leave some questions unanswered.
A new approach is developed to understand stability of a population and further understanding of population momentum.
Understanding animal movements and modelling the routes they travel can be essential in studies of pathogen transmission dynamics. Pathogen biology is also of crucial importance, defining the manner in which infectious agents are transmitted. In this article we investigate animal movement with relevance to pathogen transmission by physical rather than airborne contact, using the domestic chicken and its protozoan parasite Eimeria as an example. We have obtained a configuration for the maximum possible distance that a chicken can walk through straight and non-overlapping paths (defined in this paper) on square grid graphs. We have obtained preliminary results for such walks which can be practically adopted and tested as a foundation to improve understanding of non-airborne pathogen transmission. Linking individual non-overlapping walks within a grid-delineated area can be used to support modeling of the frequently repetitive, overlapping walks characteristic of the domestic chicken, providing a framework to model faecal deposition and subsequent parasite dissemination by faecal/host contact.We also pose an open problem on multiple walks on finite grid graphs. These results grew from biological insights and have potential applications. Keywords: Spread of bird diseases, Eimeria, Maximum walks, longest paths, NP-Complete. MSC: 92A17, 68Q17
Carey's Equality pertaining to stationary models is well known. In this paper, we have stated and proved a fundamental theorem related to the formation of this Equality. This theorem will provide an in-depth understanding of the role of each captive subject, and their corresponding follow-up duration in a stationary population. We have demonstrated a numerical example of a captive cohort and the survival pattern of medfly populations. These results can be adopted to understand age-structure and aging process in stationary and non-stationary population population models. Key words: Captive cohort, life expectancy, symmetric patterns.
A mid-point theorem is proved in an elementary way for the U type shape of functions that arise out of exponential quadratic functions. These results are inspired from epidemic patterns and growth over a time period. Key words: natural numbers mapping, mean value theorem.
With the launch of second line anti-retroviral therapy for HIV infected individuals, there has been an increased expectation on surviving period of people with HIV. We consider previously well-known models in HIV epidemiology where the parameter for incubation period is used as one of the important components to explain the dynamics of the variables. Such models are extended here to explain the dynamics with respect to a given therapy that prolongs life of an HIV infected individual. A deconvolution method is demonstrated for estimation of parameters in the situations when no-therapy and multiple therapies are given to the infected population. The models and deconvolution method are extended in order to study the impact of therapy in age-structured populations. A generalization for a situation when n-types of therapies are available is given. Models are demonstrated using hypothetical data and sensitivity of the parameters are also computed.
In conducting preliminary analysis during an epidemic, data on reported disease cases offer key information in guiding the direction to the in-depth analysis. Models for growth and transmission dynamics are heavily dependent on preliminary analysis results. When a particular disease case is reported more than once or alternatively is never reported or detected in the population, then in such a situation, there is a possibility of existence of multiple reporting or under reporting in the population. In this work, a theoretical approach for studying reporting error in epidemiology is explored. The upper bound for the error that arises due to multiple reporting is higher than that which arises due to under reporting. Numerical examples are provided to support the arguments. This article mainly treats reporting error as deterministic and one can explore a stochastic model for the same.
Mathematical models could be helpful in assisting the Indian Government's new initiative of issuing biometric cards to its citizens. In this note, we look into the role of mathematical models in estimating the missing, non-enumerated population numbers, estimating annual numbers of cards required by age, gender and regions in India. The linkage between National Population Register and biometric cards is also highlighted. See technical Appendices. There are other scientific issues, namely, electronic, data storage management, identity verification etc, which we do not address in this paper.
The paper describes a method to understand time required to vaccinate against viruses in total as well as subpopulations. As a demonstration, a model based estimate for time required to vaccinate H1N1 in India, given its administrative difficulties is provided. We have proved novel theorems for the time functions defined in the paper. Such results are useful in planning for future epidemics. The number of days required to vaccinate entire high risk population in three subpopulations (villages, tehsils and towns) are noted to be 84, 89 and 88 respectively. There exists state wise disparities in the health infrastructure and capacities to deliver vaccines and hence national estimates need to be re-evaluated based on individual performances in the states.