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].

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:

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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

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 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)

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' 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 sm,,,, i s i r i  w r  m , sf,,,, 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. ANALYTICAL SOLUTIONS

We set the right hand side of eqs.(14)-(22) to zero, then the steady states are given as follows: i) The disease free steady state: E1  (1,0,0,0,1,0,0,0,0) * * * * * * * * * ii) The disease endemic steady state: E2  (sm ,ims ,imr ,imw , s f ,i f s ,i fr ,i fw ,iR ) * h R0 where sm  , h R0  ( rm   sm   wm )N R (R0 1)

  N (R  1) * sm h R 0 i  , ms (   )((     )N (R  1)   R ) h rm sm wm R 0 h 0   N (R 1) * rm h R 0 i  , mr (   )((     )N (R 1)   R ) h rm sm wm R 0 h 0   N (R  1) * wm h R 0 i  , mw (   )((     )N (R 1)   R ) h rm sm wm R 0 h 0 * h R0 s f  , h R0  ( rf   s f   wf )N R (R0 1)

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  N (R 1) * s f h R 0

i fs  , (  h )(( rf   s f   wf )N R (R0  1)  h R0 )   N (R  1) * rf h R 0

i fr  , (  h )(( rf   s f   wf )N R (R0  1)  h R0 )   N (R 1) * wf h R 0

i fw  , (  h )(( rf   s f   wf )N R (R0 1)  h R0 ) * R0 1 iR  , R0 The disease endemic steady state has biological meaning when R0  1. To determine the local stability of each steady state, we find the eigenvalues and then check the sign of the real parts. If they give negative sign of the real part, then that steady state is local stability. The eigenvalues () are solutions of the characteristic equation: det(DF(Ei )  I)  0 ; i 1,2. (23) where DF(Ei ) is the Jacobian matrix of each steady state and I is the identity matrix. For the disease free steady state E1 : the Jacobian matrix is given by

  0 0 0 0 0 0 0 (     )N   h rm sm wm R   0   h 0 0 0 0 0 0 smNR   0 0    0 0 0 0 0  N   h rm R   0 0 0   h 0 0 0 0 wmNR    J1  0 0 0 0  h 0 0 0 (r f  s f  w f )NR   0 0 0 0 0   h 0 0 s NR   f   0 0 0 0 0 0   h 0 r f NR    0 0 0 0 0 0 0   h w f NR    0 0 0 0 0 0 0 0 dR  RNR  (24) The eigenvalues are solutions of the characteristic equation 2 6 (h  ) (  h  ) ( R N R  h  )  0. (25)

Thus, 1,2  h , 3,4,5,6,7,8    h , 9   R N R  h . (26)  R N R We can see that all eigenvalues have negative real parts for R0  1 where R0  . d R For the disease endemic steady state E2 : Using the same method as above, the eigenvalues are solutions of the characteristic equation

(     )N (R  1) 6 s f rf w f R 0 2 (  h  ) (h   ) (d R (1  R0 )  )  0. (27) R0 Thus, (b+ b + b )NR ( - 1) sf r f w f R 0 l1,2,3,4,5,6= - g - mh, l 7 = - m h - , l 8 = -d R ( R 0 - 1). (28) R0

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 R N R We can see that all eigenvalues have negative real parts for R0 1 where R0  . d R 4. NUMERICAL SOLUTIONS

In this study, we show the numerical solutions of our model. The parameters are defined corresponding to the real situations.

Fig.5 Time series solutions of susceptible male proportion, infectious male proportion in summer season, infectious male proportion in rainy season, infectious male proportion in winter season, susceptible female proportion, infectious female proportion in summer season, infectious female proportion in rainy season, infectious female proportion in winter season and infectious rat proportion, respectively, for R0 > 1. The parameters are h 1/(365* 65) corresponding to the 65 life cycle years of human,  1/15 satisfy to 15 days of recovering for each patient, d R 1/(365*1.5) satisfy to 1.5 life cycle years of rat,   0.02,   0.07,   0.02, s m r m w m N 1,500,   0.01,   0.035,   0.01,   0.0000012are arbitrarily chosen. R s f r f w f R

R0  0.99 . The solutions converge to the disease free steady state: E1 = (1,0,0,0,1,0,0,0,0)

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Fig.6 Time series solutions of susceptible male proportion, infectious male proportion in summer season, infectious male proportion in rainy season, infectious male proportion in winter season, susceptible female proportion, infectious female proportion in summer season, infectious female proportion in rainy season, infectious female proportion in winter season and infectious rat proportion, respectively, for R0 > 1. The parameters are h  1/(365* 65) ,   0.02, s m   0.07,   0.02, N 1,500,  1/15,   0.01,   0.035,   0.01, r m w m R s f r f w f

 R  0.00012, d R 1/(365*1.5), R0 100 . The solutions converge to the disease endemic steady state:

E2  (0.00000028,0.0002,0.0004,0.00013,0.0000004,0.00006,0.0003,0.00009,0.989853) Moreover, we consider the behavior solutions of our model for the different transmission rate of Leptospirosis between rat and total number of rat.

Case I When there is the different transmission rate of Leptospirosis between rat.

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Fig.7 Time series solutions of susceptible male proportion, infectious male proportion in summer season, infectious male proportion in rainy season, infectious male proportion in winter season, susceptible female proportion, infectious female proportion in summer season, infectious female proportion in rainy season, infectious female proportion in winter season and infectious rat proportion, respectively, for the different transmission rate of Leptospirosis between rat. The other parameters are same as fig.6.

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Case II When there is the different total number of rat.

Fig.8 Time series solutions of susceptible male proportion, infectious male proportion in summer season, infectious male proportion in rainy season, infectious male proportion in winter season, susceptible female proportion, infectious female proportion in summer season, infectious female proportion in rainy season, infectious female proportion in winter season and infectious rat proportion, respectively, for the different total number of rat. The other parameters are same as fig.6.

5. DISCUSSION AND CONCLUSION

From fig.5 and fig.6, we can see that the solutions converge to the disease free steady state and endemic steady state, respectively. The numerical solutions are showed to confirm our analytical results. Next, we consider behavior

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solutions when there is the different transmission rate between rat and different total number of rat. We can see that the peak of each population group is smaller and the length of outburst is longer when there is the smaller transmission rate between rat. The peak of each population group is higher when there is the greater number of rat. From our analytical results, we obtain the threshold number, namely R0. R0 is defined as follows  R N R R0  . (29) d R

Fig. 9 Bifurcation diagram of the solutions of eqs. (14) to (22) for the different values of R0 .o-o-o denote the stable solutions and x-x-x denote the unstable solutions. Furthermore, we consider the bifurcation diagram of each population group as shown in fig.9.If the threshold number is greater than 1, the fraction of susceptible male and female population are decrease. The fractions of infectious male population in summer season, infectious male population in rainy season, infectious male population in winter season, infectious female population in summer season, infectious female population in rainy season, infectious female population in winter season and infectious rat population are

255 Pongsumpun,2014 increase. These behaviors occur because there are enough susceptible male and female population be infected from infectious rat. The threshold number is used for reducing the transmission of many diseases [8]-[11]. From the threshold number as in eqs.(29), we can see that the transmission rate of Leptospirosis and total number of rat are influence to the transmission of this disease. Therefore, if we can reduce the number of rat, then the transmission of this disease will disappear.

Acknowledgment The authors declare that they have no conflicts of interest in this research.

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