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Mathematics is not only a subject but it is also a language consisting of many different symbols and relations. Taught as a compulsory subject up the 10th class, students are then able to choose whether or not to study mathematics as a main subject. The present paper discusses mathematical modeling in mathematics education. The article provides models for high school, secondary, and higher mathematics education.

Although it is well known that vaccines against many infectious diseases confer positive economic externalities via indirect protection, analysts have typically ignored possible herd protection effects in policy analyses of vaccination programs. Despite a growing literature on the economic theory of vaccine externalities and several innovative mathematical modeling approaches, there have been almost no empirical applications. The first objective of the paper is to develop a transparent, accessible economic framework for assessing the private and social economic benefits of vaccination. We also describe how stated preference studies (for example, contingent valuation and choice modeling) can be useful sources of economic data for this analytic framework. We demonstrate socially optimal policies using a graphical approach, starting with a standard textbook depiction of Pigouvian subsidies applied to herd protection from vaccination programs. We also describe nonstandard depictions that highlight some counterintuitive implications of herd protection that we feel are not commonly understood in the applied policy literature. We illustrate the approach using economic and epidemiological data from two neighborhoods in Kolkata, India. We use recently published epidemiological data on the indirect effects of cholera vaccination in Matlab, Bangladesh (Ali et al., 2005) for fitting a simple mathematical model of how protection changes with vaccine coverage. We use new data on costs and private demand for cholera vaccines in Kolkata, India, and approximate the optimal Pigouvian subsidy. We find that if the optimal subsidy is unknown, selling vaccines at full marginal cost may, under some circumstances, be a preferable second-best option to providing them for free. (Contains 14 footnotes, 2 tables, and 8 figures.)