Mathematics

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Shiv Nadar University

In this work, we studied various properties of arithmetic function ˜ϕ, where ˜ϕ(n) = |{m ∈ N|1 ≤ m ≤ n, (m, n) = 1, m is not a prime}|. © Indian Mathematical Society, 2019.

(SoNS) School of Natural Sciences

Mathematics Student

Number of non-primes in the set of units modulon

Overwhelming antibiotic use poses a serious challenge today to the public-health policymakers worldwide. Many empirical studies pointed out this ever-increasing antibiotic consumption as primary driver of the community-acquired antibiotic drug-resistance, especially in the middle- and lower-income countries. The association is well documented across spatio-temporal gradients in many parts of the world, but there is rarely any study that emphasizes the mechanism of the association, which is important for combating drug-resistance. Formulating a mathematical model of emergence and transmission of drug-resistance, we in this paper, present how amalgamating three components: socio-economic growth, population ecology of infectious disease, and antibiotic misuse can instinctively incite proliferation of resistance in the society. We show that combined impact of economy, infections, and self-medication yield synergistic interactions through feedbacks on each other, presenting the emergence of drug-resistance as a self-reinforcing cycle in the population. Analysis of our model not only determines the threshold of antibiotic use beyond which the emergence of resistance may occur, but also characterizes how fast it develops depending on economic growth, and lack of education and awareness of the population. Our model illustrates that proper and timely government aid in population health can break the self-reinforcing process and reduce the burden of drug-resistance in the community. © 2019, The Author(s).

Fulltext

Scientific Reports

Antibiotic drug-resistance as a complex system driven by socio-economic growth and antibiotic misuse

Background: Diabetes mellitus (DM) is burgeoning as a global chronic health condition. Some studies suggest that tuberculosis (TB) can even cause diabetes in those not previously known to be diabetic, which as a corollary can add to the already heavy global DM burden. The World Health Organization (WHO) recommends screening for DM at the start of TB treatment; however, it remains to be elucidated which patients with TB-induced hyperglycaemia are at risk for developing DM and who would benefit from a more regular follow-up. This systematic review will aim to firstly synthesise literature on the irreversibility of TB-induced hyperglycaemia in individuals with previously undiagnosed type 2 diabetes mellitus and secondly to synthesise literature on risk factors for progression from TB-induced hyperglycaemia to overt DM in previously undiagnosed. Methods: We will search for relevant studies in electronic databases such as PubMed, EMBASE, PROQUEST, and SCOPUS. Furthermore, references will be hand searched to identify other studies. A flow diagram will be drawn to identify the studies retrieved from each database. We will review all publications that include studies containing data on impaired glucose metabolism upon TB diagnosis, and the quality of all eligible studies will be assessed using the Newcastle-Ottawa Scale. We will further conduct a meta-analysis to pool estimates on the risk of progression of persistent hyperglycaemia to overt DM within this population group, as well as the risk factors for this progression. We will use a random effect model to assess heterogeneity, will carry out sensitivity analysis to explore the influence of a single study on the overall estimate, and will report our findings from our systematic review and meta-analysis according to PRISMA guidelines. Egger's test will be performed to explore the presence of selective reporting bias. If data allow, we will perform a subgroup/meta-regression analysis. Summary effects will be reported using odds ratio, hazard ratio, and relative risk ratios. Furthermore, any clinical, epidemiological, and public health research gaps we identify will be described in a research proposal. © 2019 The Author(s).

World University root node

Systematic Reviews

The epidemiology of intransient TB-induced hyperglycaemia in previously undiagnosed diabetes mellitus 2 individuals: A protocol for a systematic review and meta-analysis

Wildlife habitat corridors are components of landscapes, which facilitate the movement of organisms and processes between areas of intact habitat, and thus provide landscape corridor as well as serve as an ideal component to study and understand physiological ecology. Corridors are thus regions within a given landscape that generally comprise native vegetation, and connect otherwise fragmented, disconnected, non-contiguous wildlife habitat patches in the landscape. The purpose of designing corridors as a conservation strategy is primarily to counter, and to the extent possible, mitigate the impacts of habitat fragmentation and loss on the biodiversity of the landscape, as well as support continuance of land use for essential local and global economic activities in the region of reference. In this paper, we use game theory and graph theory to model and design a wildlife corridor in the Central India – Eastern Ghats landscape complex, with tiger (Panthera tigris tigris) as the focal species. We construct a graph using the habitat patches supporting wild tiger populations in the landscape complex as vertices and the possible paths between these vertices as edges. A cost matrix is constructed to indicate the cost incurred by the tiger for passage between the habitat patches in the landscape (based on Shelford's Law of Tolerance), by modelling a two-person Prisoner's Dilemma game. A minimum spanning tree is then obtained by employing Kruskal's algorithm, which would suggest a feasible tiger corridor network for the tiger population within the landscape complex. Additionally, analysis of the graph is done using various centrality measures, in order to identify and focus on potentially important habitat patches, and their potential community structure. Correlation analysis is performed on the centrality indices to draw out interesting trends in the network. © 2019 Elsevier B.V.

Ecological Informatics

A graph theoretic approach for modelling tiger corridor network in Central India-Eastern Ghats landscape complex, India

Ecosystems are constantly under threat by invasive species which alter existing ecological networks as they spread. We use Geographic Information System methods coupled with complex network analysis to model the movement and spread of Lantana camara in Rajaji Tiger Reserve, India, as understanding spatial aspects of the spread mechanism are essential for better resource management in the region. Lantana mapping was carried out by field observations along multiple transects and plots, and the data generated was used as input for maximum entropy modelling to identify land patches in the study area that are favourable for lantana growth. The patch information so obtained was integrated with a raster map generated by identifying different topographical features in the study area which are favourable for lantana growth. The integrated data was analysed with a complex network perspective, where relatively dense and large potential lantana distribution patches were considered as vertices, connected by relatively sparse and thin potential lantana continuities, identified as edges. The network centrality analysis reveal key patches in the study area that play specialized roles in the spread of lantana in a large region. Hubs in the lantana network are primarily identified as dry seasonal river beds. The lantana network is found to exhibit small-world architecture with a well formed community structure. We infer that the above properties of the lantana network are major contributors in regulating the movement and spread of the plant through the entire region of study. © 2019 Elsevier B.V.

Ecological Complexity

A network theoretic study of potential invasion and spread of Lantana camara in a part of Rajaji Tiger Reserve, India

In the Supplementary Information file originally published with this Article, the labelling of equations, figures and references in the text was omitted. These errors have been corrected in the Supplementary Information that now accompanies the Article. © 2019, The Author(s).

Author Correction: Antibiotic drug-resistance as a complex system driven by socio-economic growth and antibiotic misuse (Scientific Reports, (2019), 9, 1, (9788), 10.1038/s41598-019-46078-y)

In this paper, we explore the problem of establishing a network among the stocks of a market at high frequency level and give an application to program trading. Our work uses high frequency data from the National Stock Exchange, India, for the year 2014. To begin, we analyse the spectrum of the correlation matrix to establish the presence of linear relations amongst the stock returns. A comparison of correlations with pairwise mutual information shows the further existence of non-linear relations which are not captured by correlation. We also see that the non-linear relations are more pronounced at the high frequency level in comparison to the daily returns used in earlier work. We provide two applications of this approach. First, we construct minimal spanning trees for the stock network based on mutual information and study their topology. The year 2014 saw the conduct of general elections in India and the data allows us to explore their impact on aspects of the network, such as the scale-free property and sectorial clusters. Second, having established the presence of non-linear relations, we would like to be able to exploit them. Previous authors have suggested that peripheral stocks in the network would make good proxies for the Markowitz portfolio but with a much smaller number of stocks. We show that peripheral stocks selected using mutual information perform significantly better than ones selected using correlation. © 2019 Sharma, Habib. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

PLoS ONE

Mutual information based stock networks and portfolio selection for intraday traders using high frequency data: An Indian market case study

The propagation of rumours about rare but severe adverse vaccination or infection events through social networks can strongly impact vaccination uptake. Here we model a coupled behaviour-disease system where individual risk perception regarding vaccines and infection are shaped by their personal experiences and the experiences of others. Information about vaccines and infection either propagates through the network or becomes available through globally available sources. Dynamics are studied on a range of network types. Individuals choose to vaccinate according to their personal perception of risk and information about infection prevalence. We study events ranging from common and mild, to severe and rare. We find that vaccine and infection adverse events have asymmetric impacts. Vaccine (but not infection) adverse events may significantly prolong the tail of an outbreak. Similarly, introducing a small risk of a vaccine adverse event may cause a steep decline in vaccine coverage, especially on scale-free networks. Global dissemination of information about infection prevalence boosts vaccine coverage more than local dissemination. Taken together, these findings highlight the dangers associated with vaccine rumour propagation through scale-free networks such as those exhibited by online social media, as well as the benefits of disseminating public health information through mass media. © 2019, The Author(s).

The impact of rare but severe vaccine adverse events on behaviour-disease dynamics: a network model

Social distancing' during an infectious disease outbreak can play a major role in controlling the spread of the disease. Individuals' interests in isolating themselves however, are constrained by the inherent costs in it. Rational decision making requires comparing the cost of social distancing with the cost of being infected such as availability of the vaccines, drugs, and treatment facilities that may depend on the current epidemic burden in the community. To understanding these, we develop a differential population game model of social distancing integrating with a simple SIR model describing the disease process. Using several type of cost functions of infections, we compute the Nash equilibrium strategies under variable efficiencies of social distancing. We also derive the closed form of the analytical solution of the utility functions under a special case. Depending on the efficiency of social distancing and the functional dependence of the cost of infections, we have shown that individuals behave very rationally to isolate themselves. This information may be useful in designing the public health policies during an epidemic outbreak. © 2019 Oxford University Press. All rights reserved.

IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)

Game dynamic model of social distancing while cost of infection varies with epidemic burden

In this paper, we introduce and study some strongly almost convergent double sequence spaces by Riesz mean associated with four-dimensional bounded regular matrix and a Musielak-Orlicz function over n-normed spaces. We make an effort to study some topological and algebraic properties of these sequence spaces. We also study some inclusion relations between the spaces. Finally, we establish some relation between weighted lacunary statistical sequence spaces and Riesz lacunary almost statistical convergent sequence spaces over n-normed spaces. © 2018 World Scientific Publishing Company.