Optimal Control of Multiple Transmission of Water-Borne Diseases

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 421419, 16 pages doi:10.1155/2012/421419 Research Article Optimal Control of Multiple Transmission of Water-Borne Diseases G. Devipriya and K. Kalaivani P. G. Department of Mathematics, Women’s Christian College, Chennai 600006, India Correspondence should be addressed to G. Devipriya, Received 29 March 2012; Accepted 25 May 2012 Academic Editor: B. N. Mandal Copyright q 2012 G. Devipriya and K. Kalaivani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A controlled SIWR model was considered which was an extension of the simple SIR model by adjoining a compartment W that tracks the pathogen concentration in the water. New infections arise both through exposure to contaminated water as well as by the classical SIR person-person transmission pathway. The controls represent an immune boosting and pathogen suppressing drugs. The objective function is based on a combination of minimizing the number of infected individuals and the cost of the drugs dose. The optimal control is obtained by solving the optimality system which was composed of four nonlinear ODEs with initial conditions and four nonlinear adjoint ODEs with transversality conditions. The results were analysed and interpreted numerically using MATLAB. 1. Introduction Most of the major problems that humanity face in the twenty-first century are related to water quantity and/or quality issues. These problems are going to be more aggravated in future by climate change, resulting in higher water temperatures, melting of glaciers, and an intensification of the water cycle 1, with potentially more floods and droughts 2. With respect to human health, the most direct and most severe impact is the lack of improved sanitation, and related to it is the lack of safe drinking water, which currently affects more than one-third of the global population. Additional threats include, for example, exposure to pathogens or to chemical toxicants via the food chain, for instance, the result of irrigating plants with contaminated water and of bioaccumulation of toxic chemicals by aquatic organisms, including seafood and fish or during recreation like swimming in polluted surface water. 2 International Journal of Mathematics and Mathematical Sciences Water-borne diseases are infectious diseases caused by pathogenic microorganisms that most commonly are transmitted in contaminated fresh water, whether in bathing, washing, drinking, or in the preparation of food. Though these diseases spread either directly or indirectly through flies or filth, water is the chief medium for spread of these diseases, and hence, they are termed as waterborne diseases. More than one-third of Earth’s accessible renewable freshwater is consumptively used for agricultural, industrial, and domestic purposes 3. As most of these activities lead to water contamination with diverse synthetic and geogenic natural chemicals, it comes as no surprise that chemical pollution of natural water has become a major public concern in almost all parts of the world. The main acute disease risk associated with drinking water in developing and transition countries is due to pathogens, which include viruses, bacteria, and protozoa, which spread via the oral-fecal route 4. These diseases are more prevalent in areas with poor sanitary conditions. Campylobacter jejuni, Microsporidia, Yersinia enterocolitica, Cyclospora, Caliciviruses, and environmental bacteria like Mycobacterium sp., Aeromonas sp., Legionella pneumophila, and multidrug-resistant Pseudomonas aeruginosa have been associated with waterborne illnesses. These pathogens travel through water sources and interfuse directly through persons handling food and water. The main distribution of many water-borne pathogens varies substantially from one country to another. Some pathogens such as Vibrio cholerae, Hepatitis E virus, and Schistosomiasis are restricted to certain tropical countries; others, such as Cryptosporidiosis and Campylobacteriosis, are probably widespread. Rotavirus infections predominate in the winter months and account for approximately 140 million cases/year with 600,000–800,000 deaths/year 5. Recent evidence on Entamoebahistolytica in children Dhaka, Bangladesh, the causative agent of Amoebiasis Amoebic dysentery, shows that infection occurred in 80 percentage of children over a four-year period with a reinfection rate of 53 percentage 6. According to WHO records of infectious disease outbreaks in 132 countries from 1998 to 2001, outbreaks of waterborne diseases are at the top of the list, with cholera as the most frequent disease, followed by acute diarrhea and typhoid fever 7. Recent literature related to this work has been discussed below. The literature on economic epidemiology is varied and growing, and there are several good surveys, such as Gersovitz and Hammer 8 and Klein et al. 9. The earliest contribution, by Sanders 10, considered the treatment in different versions of the SIS model from a planner’s perspective. Goldman and Lightwood 11 studied treatment in the controlled SIS model but considered different cost structures, that is, on the assumption that the medical authorities operate without an explicit budget constraint. Rowthorn 12 extended the analysis of the controlled SIS model, by considering how different kinds of budget constraints affect the optimal solution. Arnone and Walling 13 presented the information on pathogen sources, health effects of waterborne pathogens, relevant water quality legislation, and an evaluation of pathogen indicators. Joh et al. 14 predicted that in the case of waterborne diseases, suppressing the pathogen density in aquatic reservoirs may be more effective than minimizing the number of infected individuals. Shannon et al. 15 have developed improved disinfection, decontamination, reuse, and desalination methods to work in concert to improve health, safeguard the environment, and reduce water scarcity, not just in the industrialized world, but in the developing world, where less chemical and energy intensive technologies are greatly needed. Batterman et al. 16 studied the historical practices and different disciplinary approaches to water-related infectious disease and proposed an interdisciplinary public-health-oriented systems approach to research and intervention design. Finally, they International Journal of Mathematics and Mathematical Sciences 3 illustrated using a case study that focuses on diseases associated with water and sanitation management practices in developing countries where the disease burden is the most severe. Schwarzenbach et al. 17 discussed the main groups of aquatic contaminants and their effects on human health and app (...truncated)