Australian Journal of Basic and Applied Sciences. Effect of Personal Hygiene Campaign on the Transmission Model of Hepatitis A

Transcription

1 Australian Journal of Basic and Applied Sciences, 9(13) Special 15, Pages: ISSN: Australian Journal of Basic and Applied Sciences Journal home page: wwwajbaswebcom Effect of Personal Hygiene Campaign on the ransmission Model of Hepatitis A 1 Jiraporn Boonsue Kanyarat Noochum and 3 Surapol Naowarat 1,,3 Department of Mathematics, Faculty of Science and echnology, Suratthani Rajabhat University, Surat hani, 841 hailand A R I C L E I N F O Article history: Received February 15 Accepted March 15 Available online 3 April 15 Keywords: Hepatitis A, Personal Hygiene Campaign, Basic Reproductive Number, Stability Analysis, Equilibrium Point A B S R A C In this paper, an SEIRV (Susceptible-Exposed-Infected-Recovered-Vaccinated) model for Hepatitis A is proposed and analyzed We proposed the mathematical model of Hepatitis A by nonlinear ordinary differential equations, describing the effect of personal hygiene campaign with taken into account he stability theory of differential equations is used for model analysis he analytic results showed that there were two equilibrium points; disease free equilibrium and endemic equilibrium point he qualitative results depend on the basic reproductive number R We obtained the basic reproductive number by using the next generation method Stabilities of the model are determined by Routh-Hurwitz criteria If R 1, then the disease free equilibrium point is local asymptotically stable, but If R 1, then the endemic equilibrium point is local asymptotically stable he graphical representations are provided to qualitatively support the analytical results It concluded that if the effective of personal hygiene campaign increase then the number of Hepatitis A infection will be decrease 15 AENSI Publisher All rights reserved o Cite his Article: Jiraporn Boonsue Kanyarat Noochum and Surapol Naowarat, Effect of Personal Hygiene Campaign on the ransmission Model of Hepatitis A Aust J Basic & Appl Sci, 9(13): 67-73, 15 INRODUCION Hepatitis is defined as inflammation of the liver It is a common medical condition caused by a wide range of viruses: hepatitis A, B, C, D and E Hepatitis A is considered as one of the eldest illnesses in man records, and it is a major cause of morbidity as a result of economic losses in many regions of the world (WHO, 14) Hepatitis A virus (HAV) is an RNA-containing virus of the Picornaviridae family he key feature is that it is a self-limiting disease Management of HAV should therefore be supportive he average HAV incubation time is 8 days, but it can vary from 15 to 45 days (WGO, 14) HAV infection is primarily transmitted by the fecal-oral route, by either person-to-person contact or consumption of contaminated food or water Although viremia occurs early in infection and can persist for several weeks after onset of symptoms, bloodborne transmission of HAV is uncommon HAV occasionally might be detected in saliva in experimentally infected animals, but transmission by saliva has not been demonstrated (CDC, 14) Most transmission is through contact with a household member, tourist travel to an endemic area, contact with a sex partner who has HAV, from an infection in an individual preparing food, or contact with babies in nurseries (Franco, 3) HAV has a global distribution and similar to many other enteric infectious diseases, it is typically an infection of younger patients and is related to conditions of cleanliness and hygiene (Hamad Ayed Al-Fahaad, 9) Nowadays, even in the absence of vaccination, hepatitis A is in a decaying plase, mainly determined by improved hygienic conditions derived from economic development and higher standards of living (Jacobsen and Koopman, 5) Mathematical models have become an important tool for understanding the spread and control of disease Mariana Alves and Claudia (5) proposed the simple model of hepatitis A to simulate hepatitis A population dynamics under different levels of endemicity David and Nikolaos (6) studied age-structured partial differential equation compartmental model to predict minimal vaccination strategies to eliminate hepatitis A in Bulgaria In this paper, we study with effect of personal hygiene campaign on the transmission of Hepatitis A Section II, we formulate a mathematical model of hepatitis A In section III, the model is analyzed Corresponding Author: Kanyarat Noochum, Department of Mathematics, Faculty of Science and echnology, Suratthani Rajabhat University, Surat hani, 841 hailand nokmath@hotmailcom

2 68 Jiraporn Boonsue et al, 15 Australian Journal of Basic and Applied Sciences, 9(13) Special 15, Pages: by the standard modeling method and we present the numerical solutions in section IV II Model Formular: In this study, we formulate the SEIRV (Susceptible-Exposed-Infected-Recovered- Vaccinated) model for the dynamical transmission of Hepatitis A We assume that the total human population at time t Nt () is divided into four classes namely, susceptible S() t, exposed Et (), infectious It (), recovered Rt () and vaccinated Vt (), so that N( t) S( t) E( t) I( t) R( t) V( t) he interaction of the model is presented in flow chart in Fig1 V V S N S 1 IS E I E I R S E I R Fig 1: Flow chart for the dynamical transmission model of Hepatitis A he dynamical transmission model consists of system of non-linear differential equations is given as the following ds N S 1 IS S (1) de 1 IS E E () di E I I (3) dr I R (4) dv S V (5) Where : is the birth (death) rate of human population, is the vaccination rate of human population, is the effective of personal hygiene campaign of human population, is the effective contact rate of human population, is the progression rate from exposed to infectious class, is the recovery rate of human population III Model Analysis: Equilibrium Points he model will be analyzed to investigate the equilibrium points by using the standard method for analyzing our model he system has two possible equilibrium points, disease free equilibrium point and endemic equilibrium point he equilibrium points are found by setting the right hand side of equaltions (1)-(3) and (5) to zero 1) he disease free equilibrium point (DFE): In the case of the absence of the disease, that is I We obtained (,,, ) N,,, N E S E I V

3 69 Jiraporn Boonsue et al, 15 Australian Journal of Basic and Applied Sciences, 9(13) Special 15, Pages: ) he disease endemic equilibrium point (DEE): In the case where the disease is presented, that is I We obtained N N 1 I N E1 ( S, E, I, V ),, I, 1 I ( 1 I ) 1 I Where I 1 r r r r Basic Reproductive Number: he basic reproductive number R is defined as the expected number of secondary cases produced by a single infection in a completely susceptible population (Jones, 7) By using the next generation method and used spectral radius (Van den Driessche and Watmough, ) We rewritten the dx system in matrix form F( x) V ( x) Where Fx ( ) gives the rate of appeance of new infections in a compartment and V( x ) gives the transfer of individuals We obtained, S (1 ) IS S N (1 ) IS F x E E and V x I I E V S N Find the Jacobain matrix of Fxand ( ) V( x ) evaluated at N E,,, We obtained, (1 ) N FE ( ), V E 1 Find FV, we obtained, (1 ) N 1 FV (1 ) N 1 Find spectral radius of FV 1 denoted by ( FV ) FV N 1 1 We obtaind the basic reproductive number as shown, R 1 N (1 ) N ( ) Stability Analysis In this section, he stability of equilibrium can be analyzed using the Jacobian matrix of the model at the disease free equilibrium Referring to the results of Van den Driessche and Watmough (), the stability of this system as shown in the follow theorem

4 7 Jiraporn Boonsue et al, 15 Australian Journal of Basic and Applied Sciences, 9(13) Special 15, Pages: heorem 1: he disease free equilibrium of the system about the equilibrium E, is local asymptotically stable if R 1 and unstable if R 1 Proof: he Jacobain matrix of the model (Eq 1-3 and N 5) evaluated at N E,,,, is obtained as he eigenvalue of the Jacobian matrix J are det J I From this, we obtained by solving obtain the characteristic equation as follows: A B where A, 1 N B From the chareacterictic equation, we see that two eigenvalues are 1, ( ) he others two eigenvalues are the solution of quadratic equation A B he roots of this equation will have negative real parts if two conditions satisfied with the Routh- Hurwitz criteria (Alen, 6) hus, is local J 1 N 1 N asymptotically stable for R 1 if satisfies the following conditions : 1) A >, ) B < A B heorem : he endemic equilibrium of the system about the equilibrium E 1, is local asymptotically stable if R 1 and unstable if R 1 Proof: he Jacobain matrix of the model (Eq 1-3 and 5) evaluated at E1 S, E, I, V, is obtained as I I 1 1 S J 1 1 S 1 he characteristic equation of the Jacobian matrix J1 at 1 From this, we obtain the characteristic equation, 3 C D E Where 1 I C, E are obtained by solving J I 1 det 1 1 N D 1 I, I 1 N 1 NI E 1 I 1 1 I I From the chareacterictic equation, we see that asymptotically stable for R 1 if it satisfies the 1 and the other three eigenvalues of following conditions : 3 C D E will have negative real parts 1) C <, if they conditions satisfied with the Routh-Hurwitz ) E >, 3) CD + E > criteria (Alen, 6) hus, E 1 is local

5 71 Jiraporn Boonsue et al, 15 Australian Journal of Basic and Applied Sciences, 9(13) Special 15, Pages: IV Numerical Results: In this section, we present the numerical simulation of the our model he parameter values that we used in the numerical simulations are given in able 1 able 1: Parameter values used in the numerical simulations Parameter Description Value N otal number of population 1, people Birth (death) rate 3 day 1 Vaccination rate 1 Personal hygiene campaign rate 99 Effective contact rate 8 Progression rate from exposed to infectious class 5 Recovery rate 6 day 1 Stability of disease free state: Using the values of parameters as show in able 1 We obtained the eigenvalues and the basic reproductive number as follows; 3, , , R ,4 Since all eigenvalues are to be negative and the basic reproductive number is less than one, the disease free equilibrium E will be local asymptotically stable as shown in Fig (a) (b) (c) (d) (e) Fig : ime series of (a) unvaccinated susceptible S, (b) unvaccinated exposed E, (c) unvaccinated infectious I, (d) unvaccinated recovered R and (e) vaccinated susceptible V We have see that the fraction of populations approach to the disease free equilibrium point E 9,,,,1 Stability of endemic state: We change the value of the personal hygiene campaign to 8 and keep the other values of parameters to be those given in able 1 We obtained the eigenvalues and the basic reproductive number as follows; 1 3,, , 4 643, R Since all eigenvalues are to be negative and the basic reproductive number is greater than one, the endemic equilibrium E 1 will be local asymptotically stable as shown in Fig 3

6 7 Jiraporn Boonsue et al, 15 Australian Journal of Basic and Applied Sciences, 9(13) Special 15, Pages: (a) (b) (c) (d) (e) Fig 3: ime series of (a) unvaccinated susceptible S, (b) unvaccinated exposed E, (c) unvaccinated infectious I, (d) unvaccinated recovered R and (e) vaccinated susceptible V We have see that the fractions of populations approach to the endemic equilibrium point E1 7,5,6,98,7 Conclusion: We proposed the mathematical model of hepatitis A with the effect of personal hygiene campaign of human population We analyzed model by the standard modeling method he population are assumed to be constant sizes he basic reproductive number of this disease is R 1 N, it indicates the average number of secondary patients that one patient can produce if introduced into a susceptible After that, we found the local asymptotic stability of the DFE and DEE For the disease free equilibrium R 1 is asymptotically stable and R 1 the endemic equilibrium is asymptotically stable We found the value of R were , 1651when 99,8 respectively It seen that the infected human will decrease when the effect of personal hygiene campaign is increase ACKNOWLEDGEMEN he author is grateful to the Department of Mathematics, Faculty of Science and echnology, Suratthani Rajabhat University for providing the facilities to carry out the research REFERENCES Allen, LJS, 6 An Introduction t to Mathematics Biology Peason/Prince Hall, Upper Saddle River, New Centers for Disease Control and Prevention, 14 Hepatitis A Available: HAV/indexhtm [Accessed : December, 14] David, G and S Nikolaos, 6 A Statistical Method for Modelling Hepatitis A Vacination in Bulgaria Proceedings of the 7th WSEAS International Conference on Mathematics & Computers in Biology & Chemistry, Cavtat, Croatia, pp: 19-5 Franco, E, C Giambi, R Ialacci, RC Coppola, AR Zanetti, 3 Risk groups for hepatitis A virus infection Vaccine, 1: 4-33 Hamad Ayed Al-Fahaad, 9 Preventive Measure and Precautions for Hepatitis A outbreak control in Yadamah, Najran, Saudi Arabia Journal of Medical, 15: 34 Jacobsen, KH and JS Koopman, 5 he effects of socioeconomic development on worldwide hepatitis A virus seroprevalence patterns International Journal of Epidemiology, 34: 6-69 Jersey James Holland Jones, 7 Notes On R Department of Anthropological Sciences Stanford University

7 73 Jiraporn Boonsue et al, 15 Australian Journal of Basic and Applied Sciences, 9(13) Special 15, Pages: Mariana Alves, DG and C Claudia, 5 Experiments with mathematical models to simulate hepatitis A population dynamics under different levels of endemicity Artigo Article, 1(5): Van den Driessche P and J Watmough, Reproductive number and sub-threshold endemic equilibria for compartmental mosdels of disease transmission, math Biosci, 18: 9-48 World Gastroenterology Organisation, 14 Hepatitis A Available: hepatitis/whocdscsredc7/en/index1html [Accessed : December, 14] World Health Organization, 14 Hepatitis A Available: whocdscsredc7/en/index1html [Accessed : December, 14]