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Leptospirosis Transmission Model with the Gender of Human and Season in Thailand

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Leptospirosis Transmission Model with the Gender of Human and Season in Thailand J. Basic. Appl. Sci. Res., 4(1)245-256, 2014 ISSN 2090-4304 Journal of Basic and Applied © 2014, TextRoad Publication Scientific Research www.textroad.com Leptospirosis Transmission Model with the Gender of Human and Season in Thailand Puntani Pongsumpun* Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Chalongkrung road, Ladkrabang, Bangkok 10520, Thailand. Received: November 10 2013 Accepted: December 8 2013 ABSTRACT Leptospirosis can be transmitted between people through direct and indirect ways by rat. Human can be infected by either touching the infected rats or contacting with water, soil containing urine from the infected rats through skin, eyes and nose. This disease can be found worldwide. In Thailand, this disease is found in men more than women. The season is influence to the transmission of this disease. The Leptospirosis cases are usually found in rainy season. In this paper, we study the transmission of this disease by formulating mathematical model considering gender of human and season in Thailand. The standard dynamical modeling method is used in this study. The analytical results are shown. The numerical solutions are presented to confirm our analytical results. KEYWORDS— gender, Leptospirosis, rat, season, transmission 1. INTRODUCTION The world's most common disease transmitted from animals to people, namely Leptospirosis. It can be transmitted to human by animal urine to come in contact with unhealed breaks in the skin, the eyes, or with the mucous membranes. A type of bacteria called a spirochete can cause the infection of Leptospirosis. Leptospira interrogans, Leptospira kirschneri, Leptospira noguchii, Leptospira borgpetersenii, Leptospira santarosai, Leptospira weilii and Leptospira inadai are 7 strains of Leptospirosis. Leptospirosis is found around the world, but it is usually found in the tropical countries. This disease has emerged in Thailand since 1997, as a major health concern [1]-[2]. Leptospirosis or Weil’s disease can cause high fever in human. In 1886, Weil is credited with first described leptospirosis as a unique disease process. Characteristics of Leptospirosis patients are high fever, headache, muscle aches, conjunctivitis (red eyes), diarrhea, vomiting, and kidney or liver problems (which may include jaundice), anemia and, sometimes, rash. Duration of symptoms is few days or several weeks. Some patients may death but they are rareness. For some patients, the infections may be mild or without obvious symptom [3]-[7]. Season and environmental factors are influence to the transmission of this disease. Agriculturist is usually found in Leptospirosis patients [2,8]. The vector of this disease consists of many animals such as rats, skunks, opossums, raccoons, foxes, and other vermin. This disease can be transferred though infected soil or water. The soil or water is polluted with the waste products of an infected animal. Human can be infected by either ingesting contaminated food or water or by broken skin and mucous membrane (eyes, nose, sinuses, mouth) contact with the contaminated water or soil. System of differential equations are used for describing the transmission of many diseases [9]-[11]. 25 20 15 10 population 5 Incidence rate per 100,000 rateper Incidence 0 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 year Fig.1 Incidence rate per 100,000 population of Leptospirosis in Thailand; 1997-2011 [15]. In 2007, W.Triampo and et al. studied an SIR (S =Susceptible, I = Infected, R = Recovered) model for the transmission of Thai leptospirosis cases [12]. The gender and season in Thailand are not included in their model. SIR *Corresponding Author: Puntani Pongsumpun, Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Chalongkrung road, Ladkrabang, Bangkok 10520, Thailand. Email: [email protected] 245 Pongsumpun,2014 model can be used for describing the transmission of many infectious diseases [13].In 2012[14], P.Pongsumpun studied the transmission model of Leptospirosis and separated the human population by age group. 6 1997 5 1998 1999 4 2000 2001 3 2002 2 2003 2004 1 2005 2006 0 2007 2008 2009 Incidence rate per 100,000 population 100,000rateper Incidence 2010 2011 year Fig.2 Monthly incidence rate per 100,000 population of Leptospirosis in Thailand; 1997-2011 [15]. 12000 10000 8000 6000 Male Female Case numbers 4000 2000 0 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 year Fig.3 Incidence rate per 100,000 population of Leptospirosis in Thailand classified by gender; 1997-2011 [15]. The data of Leptospirosis patients during 1997 and 2011 in Thailand are shown in fig.1, fig.2 and fig.3 [15]. We can see that Leptospirosis patients are found in Thailand every year. The distribution of this disease is different in each month. From fig.2, we can see that this disease is usually found in rainy season. Male patients are more than female patients as shown in fig.3. Thus, season and gender are influence to the transmission of this disease. In this paper, we formulated the model for Leptospirosis. Season and gender are included into our model. Standard dynamical modeling method is used for analyzing our model. 2. TRANSMISSION MODEL In this study, we formulate the transmission model of Leptospirosis considering the gender of human and season in Thailand. We suppose that there are the different transmission rates between male and female. Transmission rates are assumed to be different in each season. Transmission diagram of our model is described as shown in fig.4: 246 J. Basic. Appl. Sci. Res., 4(1)245-256, 2014 4a) 4b) 4c) Fig.4 Transmission diagram of our model 4a) For male human population 4b) For female human population 4c) For rat population The variables and parameters are defined as follows: Nh is the total human population, h is the birth rate of human population, m is the fraction of male human, Sm is the number of susceptible male human, I is the number of infectious male human in summer season, ms I is the number of infectious male human in rainy season, mr I is the number of infectious male human in winter season, mw Rm is the number of recovered male human, is the transmission rate of Leptospirosis from rat to male human in summer season, sm 247 Pongsumpun,2014 is the transmission rate of Leptospirosis from rat to male human in rainy season, rm is the transmission rate of Leptospirosis from rat to male human in winter season, wm is the death rate of human population, d is the recovery rate of human population, S f is the number of susceptible female human, 1 m is the fraction of female human, I is the number of infectious female human in summer season, fs I is the number of infectious female human in rainy season, fr I is the number of infectious female human in winter season, fw Rf is the number of recovered female human, is the transmission rate of Leptospirosis from rat to female human in summer season, s f is the transmission rate of Leptospirosis from rat to female human in rainy season, rf is the transmission rate of Leptospirosis from rat to female human in winter season, w f NR is the number of rat population, SR is the number of susceptible rat population, IR is the number of infectious rat population, R is the birth rate of rat population, dR is the death rate of rat population, R is the transmission rate of Leptospirosis between rat. The differential equations can be described as follows: ' dSm Sm t h mN h s r w S m I R dS m dt m m m (1) dI I' t ms S I d I msdt s m m R m s (2) dI ' mr Im t r S m I R d I m rdt m r (3) dI I' t mw S I d I mwdt w m m R m w (4) ' dRm Rm t I m I m I m dR m (5) dt s r w ' dS f S t 1 m N S I dS fdt h h sf r f w f f R f (6) dI I' t fs S I d I fsdt s f f R f s (7) dI I' t fr S I d I frdt r f f R f r (8) dI I' t fw S I d I fwdt w f f R f w (9) ' dRf R t I I I dR f fs f r f w f (10) dt dS S' t R N S I d S RRRRRRRRdt (11) 248 J. Basic. Appl. Sci. Res., 4(1)245-256, 2014 ' I R (t) RS R I R d R I R (12) where NSIIIR m m ms m r m w m NSIIIR f f fs f r f w f N N N h m f NSI (13) RRR We suppose that each population group has constant size; thus the rate of change for each group is equivalent to zero. Then we have N mN , N (1 m)N , d, d . m h f h h R R We introduce the new variables: SRIImIm m SRIII sm,,,, i s i r i w r m , sf,,,, i fs i fr i f w r f , m ms m r m w m f fs f r f w f NNNNNm m m m m NNNNNf f f f f SIRR sRR, i . Then the reduce equations become NNRR s' t d1 s i N s (14) m m sm r m w m R R m i' t s i N d i (15) ms s m m R R m s i' t s i N d i (16) mr r m m R R m r i' t s i N d i (17) mw w m m R R m w s' t d1 s i N s (18) f f sf r f w f R R f i' t s i N d i (19) fs s f f R R f s i' t s i N d i (20) fr r f f R R f r i' t s i N d i (21) fw w f f R R f w ' iRRRRRR t 1 i N d i (22) where 1 =s + i + i + i + r m ms m r m w m 1 =s + i + i + i + r f fs f r f w f 3.
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