Analysis of a dengue disease transmission model with clinical diagnosis in Thailand

Transcription

1 TEATOAL JOUAL OF MATEMATCAL MODELS AD METODS APPLED SCECES Analysis of a dengue disease transmission model with clinical diagnosis in Thailand. Kongnuy. E. aowanich, and P. Pongsumpun Corresponding author Abstract An S-- epidemiological model with clinical diagnosis of dengue transmission, Dengue Feer (), Dengue aemorrhagic Feer (), Dengue Shock Syndrome () dynamics in a population in Thailand is discussed. Our model consists of seen non-linear differential equations. The standard dynamical analysis is used for analyzing the behaior for the transmission of dengue disease. Local existence results are gien for the resulting ordinary differential system. The numerical results are discussed in terms of threshold parameters and the numerical simulations are shown to confirm our results. Keywords Clinical diagnosis, numerical simulation, ordinary differential system, S--. M. TODUCTO ATEMATCAL model hae been widely used in arious areas of infectious disease epidemiology. Mathematical modeling of dengue disease transmission in human and ector populations has been done since the beginning of last century. Some of the recent models could be seen in [-6]. Seeral studies on infection model within human hae been done for arious cases [7-8]. Mathematical models are used in comparing, planning, implementing, ealuating, and optimizing arious detection, preention, therapy, and control programs. Epidemiology modeling can contribute the design and analysis of epidemiological sureys, suggest crucial data that should be collected, identify trends, make general forecasts, and estimate the uncertainty in forecasts [9-]. The epidemiological systems often exists a peculiar equilibrium the disease free equilibrium, which corresponds to a steady state of the population without disease. The another one equilibrium is the endemic equilibrium state. t is the steady state solutions where the disease persists in the population. n the context of within host dengue iral infection, the basic reproductie number is defined as the aerage number of secondary infected monocytes generated by a single infected monocyte placed in an uninfected monocyte population []. n this paper, we are deeloping mathematical models to better understand the transmission and spread of dengue disease. Dengue is an acute feer caused by a Flaiirus. The disease can occur in three forms: Dengue Feer (), Dengue aemorrhagic Feer () and Dengue Shock Syndrome (). is an acute iral disease manifesting with myalgias, headache, retro-orbital pain, omiting, maculopapular rash, leucopenia and thrombocytopenia. is characterized by four major clinical features: high feer, hemorrhagic phenomena, hepatomegaly and signs of impending circulatory failure. The major pathophysiological abnormality differentiating from is the plasma leakage syndrome. The seerity of disease in depends on the quantum of plasma leakage. The patients are presented with shock due to excessie plasma loss are labeled as dengue shock syndrome (). / are potentially fatal conditions. As,, are endemic in Thailand, cases are reported eery years. A total of,885 cases of, 65,8 cases of and 7,67 cases of hae been reported during twele year reiew period. Between 997 and 8, the percentage of mortality, and cases reported.%,.8% and 58.%, respectiely. Fig., shows the percentage of cases by clinical diagnosis in Thailand between 997 and 8. Fig., shows the percentage of deaths by clinical diagnosis in Thailand during 997 to 8. Fig. The percentage of cases by clinical diagnosis in Thailand between 997 and 8 []. ssue, Volume 5, 59

2 TEATOAL JOUAL OF MATEMATCAL MODELS AD METODS APPLED SCECES Fig. The percentage of deaths by clinical diagnosis in Thailand []. t can be seen in the Fig. that the percentage of deaths of cases is higher than and death cases. A basic S-- (Susceptible-nfected-ecoered) model is used for representing,, transmission system in this study. This paper is organized as follows. n the first section, we present the introduction that guided the dengue disease and model s structure. n section, we present the model s equations and definition of the ariables and parameters. n section, we deduce the basic reproductie number, which can be used for predicting all the possible behaiors of the system. Finally, the numerical solutions of this model are presented.. MATEMATCAL MODEL A. Model Formulation The model describes the dynamic of dengue in the two components of transmission, namely human hosts and ector. The total human population is denoted by, it is partitioned into fie classes, the susceptible, infectious with clinical diagnosis, infectious with clinical diagnosis, infectious with clinical diagnosis and recoered are denoted by S,,, and, respectiely. The total ector population is and the ector population is diided into two classes, the susceptible and infectious ector are denoted by S and,respectiely. The total human population size can be determined by = S The total ector population size can be determined by = S +. This model is shown in Fig.. Fig. Compartmental transmission model of,, system following a Susceptible-nfectious-ecoered structure in human population and Susceptible-nfectious structure in ector population. The parameters in our model are defined in Table. Table. Parameter inoled in the transmission of dengue, incorporated into the model shown in Fig.. A λ Symbol Meaning Total human population size Total ector population size The constant recruitment rate of mosquitoes Birth rate of human population The natural death rate of human population Death rate of human population (with ) Death rate of human population (with ) Death rate of human population (with ) Transmission probability from infectious ector to V human and human becomes to infectious with Transmission probability from infectious ector to V human and human becomes to infectious with Transmission probability from infectious ector to V human and human becomes to infectious with Transmission probability from infecious human to V ector and ector becomes to infectious The contact rate from infectious ector to susceptible human and human becomes to infectious human with b r The contact rate from infectious ector to susceptible human and human becomes to infectious human with The contact rate from infectious ector to susceptible human and human becomes to infectious human with The contact rate from infectious human to susceptible ector and ector becomes to infectious ector Death rate of ector Aerage rate of biting per ector per day uman recoery rate From the aeraging of the real data in Thailand between 997 to 8, we hae V > V > V and ssue, Volume 5, 595

3 TEATOAL JOUAL OF MATEMATCAL MODELS AD METODS APPLED SCECES > > B. Model Equations When n our model, the dynamic of epidemic model is established: ds d = λ ( + ( + + ) )S, =S ( D F + r), d $ d $ d $ = S$ $ ε $, () = S$ $ ε $, = ( $ + $ + $ )( $ ) $ where ε = + r, ε = + r and ε = + r. d ( + r), =S For the biological interest, the region of system () is restricted to d d ds d =S ( + r), () + ), = r( + = A ( ( + + ))S, + = ( + + )S. The first fie equations represent the susceptible, infectious with, infectious with, infectious with and recoered human population densities, respectiely. The sixth and seenth equations represent the susceptible and infectious ector population densities. Before we analyze the system (), we reduce the number of parameters by introducing S $ = ( S ), $ = ( ), = ( ) $, $ = ( ), = ( ), S = ( S ) $, Ω= (S $, $, $, $, $ ): S $, $, $, $, $, { } and all of the parameters used in system () are positie.. MATEMATCAL AALYSS A. Analysis of Models The local stability of an equilibrium state is determined from the Jacobian matrix of the right hand side of () ealuated at the equilibrium point. We obtain the Jacobian matrix φ φ S εη ε εη S ε εη εη S ε εη εηs ( ) ( ) ( ) ( + + ) The equilibrium states (S,,,, ) are found by setting the right hand side of () to zero, then we obtain: $ =. ( ) S =, +φ η =, + φ η =, +φ ds$ We get the following model d $ ( ( ) $ )S $, = = S$ $ ε$, η =, and + φ ( ) = for the disease free state or ( η+η+η) = for the endemic disease η+η+η +φ state where ( + + ) φ =, η =, η =, η =. ε ε ε ssue, Volume 5, 596

4 TEATOAL JOUAL OF MATEMATCAL MODELS AD METODS APPLED SCECES For this case there are two equilibrium states, those are, the disease free equilibrium E = (,,,,) when = which always exists and the endemic equilibrium E = (S,,, ( η + η + η), V ) when =. ( η + η + η ) + φ B. Analysis of Stability The local stability of an equilibrium state is determined from the aboe Jacobian matrix ealuated at equilibrium state ( η + η + η) around = and =. ( η + η + η ) + φ The local stability property of those equilibrium state is gien in the following proposition. i) aaa > a + aa. () t can be seen is always positie. a when a ( ) η+η+η For the second and fourth conditions, we show the map by putting the regions in a phase space and (a a a a a a ) V conditions in Fig.. V. phase space to confirm two Proposition f <, the equilibrium is locally asymptotically stable. f >, the equilibrium E is unstable and is locally asymptotically stable when E E ( η+η+η ) =. Proof The local stability of E is goerned by the linearization of system () at E. The eigenalue of the disease free state is λ= and the other eigenalues are the roots of Fig. The second and fourth conditions which satisfies the outh- urwitz criteria. The alue of parameters are defined in Table. Moreoer, for >, we hae the characteristic equation λ + b λ + b λ + b λ + b λ+ b =. () 5 5 where λ + a λ + a λ + a λ+ a = () By the urwitz Criterion [], all the roots of the polynomial order fie hae negatie real part when i) bi > [i =,,,,5], () a =ε +ε +ε +, () ii) b bb > b + bb, () a = ( ηε +ηε +ηε ) +ε( ε +ε ) + ( ε +ε +ε), (5) a = ( εε +η ( ε +ε)) ε ( ηε ( ε +ε ) + ηε ( ε +ε))( εε +ε( ε +ε)), (6) a =εε ε( ( η +η +η) + ). (7) By the urwitz Criterion [], all the roots of the polynomial order four hae negatie real part when i) a >, (8) iii) (b b b )(b b b b b b ) > b (b b b ) + b b () The roots of polynomial () hae negatie real parts when they are corresponding aboe three conditions. We now map the regions in bi V phase space, (b b b b b b ) V phase space and ((bb b)(bbb 5 b bb) b(bb 5 b) bb) 5 V phase space in which the three aboe conditions are met and >. These are shown in the following figures. ii) a >, (9) iii) a, () ssue, Volume 5, 597

5 TEATOAL JOUAL OF MATEMATCAL MODELS AD METODS APPLED SCECES Table. Parameter inoled in the transmission of dengue, they are used in Fig., 5, 6 and 7. Symbol For Fig. and 5 For Fig. 6 For Fig. 7 6, 6,9 6,6,9 6, 6,9,,, λ /(65x7) /(65x7) /(65x7) b / / / /( 65x7) /( 65x7) /( 65x7) / / / V... V V V.. V... 5 ( V ( V ( VD F ( V ( V ( V ( V ( V ( V ( V ( V ( V x b )/ r / / / Fig. 5 The three aboe conditions which satisfies the outh-urwitz criteria. The alue of parameters are defined in Table. From the aboe figure, outh-urwitz Criterion are satisfied for >. Thus, the endemic equilibrium state is locally stable when > with ( η+η+η) =. (5) ssue, Volume 5, 598

6 TEATOAL JOUAL OF MATEMATCAL MODELS AD METODS APPLED SCECES C. umerical esults n this study, we consider the transmission of dengue disease when consider follow by the clinical diagnosis by using data in Thailand between 997 and 8. The alue of the parameters used in this study are = /(65 7) per day, this corresponds to a life expectancy of 7 years in Thai people. The mean life of ector is days so = / per day. is defined in (5). is the basic reproductie number. The trajectories of the numerical solutions of system () are shown in the following figures. Fig. 7 umerical solutions demonstrate time series when >. The alues of parameters are defined in Table. Fig. 6 umerical solutions demonstrate time series when <. The alue of parameters are defined in Table. 8a) 8b) Fig. 8 umerical solutions demonstrate time series when are different. The alue of parameters are defined in Table. ssue, Volume 5, 599

7 TEATOAL JOUAL OF MATEMATCAL MODELS AD METODS APPLED SCECES Table. The alue of parameters which we used for Fig. 8 and 9. Symbol For Fig.8a) For Fig. 8b) For Fig. 9 6,6,9 6,6,9 6,6,9,,, λ /(65x7) /(65x7) /(65x7) b / / / /(65x7) /(65x7) /(65x7) / / / V... V.5.5 V for Fig. 9a), V =.5 for Fig. 9b) V.. V =. for Fig. 9a), V for Fig. 9b). V ( V ( V ( V x b )/ ( V ( V ( V x b )/ ( V ( V ( V x b )/ ( V ( V ( V x b )/ r / / / <. From Fig. 7, the fraction of populations spiral to the endemic disease state (.78856,.76,.,.,.7995). Moreoer we compare the numerical solutions when is difference. We see the trajectories spiraling toward the different endemic disease states (.78856,.76,.,.,.7995) in Fig. 8a) and ( ,.6,.7,.,.9698) in Fig. 8b). Fig. 9, shows all proportions of population when the transmission probability from infectious ector to human and human becomes to infectious with ( V ) and transmission probability from infectious ector to human and human becomes to infectious with ( V ) are difference. V. DSCUSSO AD COCLUSO n this model, we hae used, it is assumed that the human population is constants. The epidemiological data from the Diision of Epidemiology, Ministry of Public ealth, Thailand between 997 and 8 are used. From our analysis, we obtain the following threshold number ( η+η+η) = b VV b VV b VV = + + (r + ) (r + ) (r + ). (6) The square root of (6) is the basic reproductie number. We can see from Fig. 6, S $, $, $, $ and $ approach to the disease free equilibrium state (,,,,) respectiely for ssue, Volume 5, 6

8 TEATOAL JOUAL OF MATEMATCAL MODELS AD METODS APPLED SCECES [] P. C. Park and V. ahn, Stability Theory, Prentice all, London, 99. M. Young, The techincal Writers andbook. Mill Valley, CA: Uniersity Science, a) 9b) Fig. 9 umerical solutions demonstrate time series when V and V are different. The alue of parameters are defined in Table. When V is higher, we can see the trajectories spiraling towards to the different endemic disease state, S $ decreases. While $, $, $ and $ increases. When V is higher, S $, $, $ and $ decrease while $ increase. n this study, it is assumed that the human population is constant. The effect of the non-constant human population is not taken into mathematical model. So on the further research, the non-constant human population should also be considered in this model.. Kongnuy is with the Department of Mathematics, Faculty of Science and Technology, ajamangala Uniersity of Technology Suarnabhumi, onthaburi oad, Muang, onthaburi,, Thailand (Corresponding author to proide phone: ; fax: ; rujirakung@ gmail.com). Educational background: Ph.D. (Applied Mathematics) 9 King Monkut s nstitute of Technology Ladkrabang, Bangkok, Thailand, M.Sc. (Mathematics) 5 amkhamhaeng Uniersity, Bangkok, Thailand, B.Sc. (Mathematics) 998 Price of Songkla Uniersity, Songkla, Thailand. E. aowanich is with the Department of Computer Science, Faculty of Science and Technology, ajamangala Uniersity of Technology Suarnabhumi, onthaburi oad, Muang, onthaburi,, Thailand (e- ekachai_nao@ yahoo.com). Educational background: M.Sc. (Computer mail: Sciences) 7 King Mongkut s Uniersity of Technology orth Bangkok, Thailand, B.Ed. (Computer Engineering) 999 ajamangala nstitute of Technology, orth Phra akhon. P. Pongsumpun., is with Department of Mathematics, Faculty of Science, King Mongkut s nstitute of Technology Ladkrabang, Chalongkrung road, Ladkrabang, Bangkok 5, Thailand, ( kppuntan@kmitl.ac.th). Educational background: Ph.D. (Mathematics) Mahidol Uniersity, Bangkok, Thailand, B.Sc. (Mathematics) Mahidol Uniersity, Bangkok, Thailand. ACKOWLEDGMET The authors would like to thank Prof. Dr.-Ming Tang at Mahidol Uniersity, Thailand. EFEECES [] L. Estea and C. Vargas, Analysis of a dengue disease transmission model, Math. Biosci, ol. 5, pp. -5, 998. [] L. Estea and C. Vargas, A model of dengue disease with ariable human population, J. Math. Biosci, ol. 8, pp. -, 999. [] L. Estea and C. Vargas, Coexistence of different serotypes of dengue irus, J. Math. Biosci, ol. 6, pp. -7,. [] Z. Feng and J. X. Velasco-ernandez, Competitie exclusion in a ector-host model for the dengue feer, J. Math. Biol, ol. 5, pp. 5-5, 997. [5]. uraini, E. Soewono and K. A. Sidarto, A mathematical model of dengue disease transmission with seere compartment, Bull. Malay. Math. Sci. Soc, ol., pp. 5-9, 7. [6] E. Soewono and A. K. Supriatna, A two-dimensional model for transmission of dengue feer disease, Bull. Malay. Math. Sci. Soc, ol., pp. 9-57,. [7]. W. ethcote, Three basic epidemiological models in Applied Mathematical Ecology. Berlin, Springer-Verlag, 989. [8]. W. ethcote and J. W. Van Ark, Modeling V Transmission and ADS in the United State, Lecture otes in Biomath. Berlin, Springer Verlag, 99. [9]. Bellomo,. K. Li and P. Maini, On the foundation of cancer modelling: Selected topics, speculations and perspecties, Math. Models. Methods Appl. Sci, ol. 8, pp , 8. [] M. A. owak, and. M. May, Virus dynamics, Mathematical Principles of mmunology and Virology, Oxford Uniersity Press,. [] O. Diekmann and J. A. P. eesterbeek, Mathematical Epidemiology of nfectious Disease, Model Buillding, Analysis and nterpretation, John Wiley Son:Chichester,. [] Epidemiologic Department Ministry of Public ealth, Annual Epidemiological Sureillance eport. Ministry of Public ealth oyal Thai Goernment, ssue, Volume 5, 6