Dynamical Model for Determining Human Susceptibility to Dengue Fever

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1 American Journal of Applied ciences 8 (11): , 11 IN cience Publications Dynamical Model for Determining uman usceptibility to Dengue Feer 1 urapol Naowarat, 1 Thanon Korkiatsakul and I. Ming Tang 1 Department of Mathematics, Faculty of cience and Technology, uratthani Rajabhat Uniersity, urat Thani, 841, Thailand Department of Physics, Faculty of cience, Mahidol Uniersity, Rama 6 Road, Bangkok, 14, Thailand Abstract: Problem statement: Mathematical models are a useful tool for understanding and describing the transmission of diseases such as dengue feer, one of the most prealently emerging diseases common to tropical and subtropical areas throughout outh East Asia. By taking into account human susceptibility to disease, the dynamics of a dengue disease model is proposed. Approach: Using standard methods for analyzing a system, the stability of the model is determined by using Routh-urwitz criteria. Results and Conclusion: We can show that the basic reproductie number (R ), the threshold parameter, when R <1, the disease-free state is locally asymptotically stable. If R >1, the endemic equilibrium state is locally asymptotically stable. Numerical results illustrate the dynamics of the disease within the context of arying parameter alues. Key words: Dengue hock yndrome (D), Dengue aemorrhagic Feer (DF), Dengue Feer (DF), mathematical model, female aedes mosquitoes, sequential infections INTRODUCTION Dengue feer is an infectious disease caused by the dengue irus (genus flaiirus, family Flaiiridae) (WO, 9). There are four closely related serotypes of this irus known as DEN-1, DEN-, DEN-3 and DEN-4, all of which are transmitted to humans by the bite of infected female Aedes mosquitoes with Aedes aegypti being the principle ector and Aedes albopictus being a less common ector. Dengue feer occurs in tropical and subtropical regions around the world, predominantly in urban and semi-urban areas where mosquitoes can breed in water. Dengue feer has become a major international public health concern since it has been reported in oer 1 countries and is estimated to affect more than one hundred million people each year (WO, 9) with infants and, unlike many diseases, well-nourished children being most at risk (Ranjit and Kissoon, 1). For a yet-to-be explained reason, females are more susceptible to the disease than males (Guzman et al., 1). It is estimated that.5 billion people lie in dengue epidemic areas. The spectrum of illness of dengue ranges from mild infection Dengue Feer (DF), to seere deadly disease Dengue aemorrhagic Feer (DF) and Dengue hock yndrome (D) (WO, 9). All of the four dengue iruses co-circulate in many areas of Africa, the Americas and Asia, the dominant serotype has changed irregularly (Chikaki and Ishikawa, 9). Infection with one serotype confers permanent immunity against that serotype but only temporary and partial protection against the other three serotypes and secondary or sequential infections are possible after a short time (Rodenhuis-Zybert et al., 1). A person infected with one DEN irus produces lifelong immunity against with that serotype but no long-term cross - protection against the other three serotypes but they may be re-infected by the other three serotypes in about three months and will concurrently become more susceptible to deelop the more irulent DF form of the disease (Gubler and Kuno, 1997). Among many countries affected by the disease, all three forms of Dengue feer are endemic to Thailand. From , a total of,885 cases of DF, 65,81 cases of DF and 17,68 cases of D were reported (Kongnuy et al., 11). In many cases, the illness is asymptomatic and infection can only be determined through serologic tests. riprom et al. (3) sought to classify the primary and the secondary infections of DF in Thailand from erological tests established 1,8 confirmed cases diided into 14 due to primary infections, 91 due to Corresponding Author: urapol Naowarat, Department of Mathematics, Faculty of cience and Technology, uratthani Rajabhat Uniersity, urat Thani, 841, Thailand 111

2 Am. J. Applied ci., 8 (11): , 11 secondary infections and 577 undetermined. The predominant irus was DEN-1 (16), followed by DEN- (11), DEN-3 (7) and DEN-4 (17). Multiple iruses were found in 3 patients. alstead maintained that the mutation of the irus could hae produced iruses with greater irulence and therefore greater epidemic potential. Gien the disease s widespread prealence in Thailand, the need to better understand the epidemiology of Dengue feer is, therefore, most urgently needed. imulation approaches using mathematical models hae already proen themseles as important tools for understanding the spread and control of diseases as in the study of Adetunde (9) and Koriko and Yusuf (8) proposed mathematical models of Tuberculosis disease, determine the disease free and endemic equilibrium point and analyzed the stability. Naowarat et al. (11) proposed and analyzed the mathematical model to control the transmission of Chikungunya Feer. Dengue models proposed by Estaa and agus (1998) and by Yaacob (7) may hae een greater efficacy than those mentioned aboe as they permit both more than one serotype and the potential for re-infection to be incorporated within their models parameters. MATERIAL AND METOD Model formulation: In our model, we assume that human population and mosquito population are constant denoted by N and N, respectiely. The dynamics of the disease is depicted in the compartment diagram, as shown in Fig. 1. The human population is partitioned into two compartments, the susceptible human ( ) and the infected human ( I ) compartment. ince there is not enough information known about the degree of immunity conferred after recoery beyond what was mentioned earlier (i.e., only one serotype confers permanent immunity to itself but only temporary immunity to the other three), we assume that when a susceptible human is infected with one type of DEN irus they can be infected by the other types. Likewise, for the purposes of this study we hae omitted the recoered or immune human compartment. The mosquito population, in turn, is partitioned into two compartments. The susceptible mosquito mosquito ( ) and the infected ( I ) compartment. The recoered mosquito does not, of course, exist, as once infected as a carrier it will remain infected oer the course of its two weeks life span. The transmission dynamics of Dengue feer are described in the following differential equations, premised on the assumptions and exclusions outlined below Eq. 1: d bβ λ N I, di bβ I ( r ) I, d bβ A I, di bβ I I With two conditions: N I N I (1) The third equation can be canceled since the mosquito population is constant. i.e., N I. The number of dependent ariables is limited to three, designated as, I, I. To analyze the model we can normalize the Eq. 1 and define new ariables:,i, N N I I I and I N A /( ) N A /( ) Fig. 1: Flow chart for the transmission of dengue disease ince the total human and mosquito populations are constant, the time rate of change of the human 11

3 d di population is equal to zero, i.e.,. This dt dt means that the birth rate and the death rate of the human population is equal; that is λ. The total A number of mosquito at equilibrium is equal to. Where:,(I ),(I ) The number of susceptible (infected) human population The number of susceptible (infected) mosquito population λ,( ) The birth (death) rate of human population A The recruitment rate of mosquito population m The number of other animals that the mosquitoes can feed on N, (N ) The number of human(mosquito) b ( ) γ r γ population The biting (death) rate of mosquito population The transmission rate of DEN irus from infected mosquito to human population i.e., Abβ γ (N m) The recoery rate of human population The transmission rate of DEN irus from infected human population to susceptible b N mosquito population, i.e., γ N m The reduced models are depicted as following Eq. : d Abβ (1 ) I, dt (N m) di Abβ I ( r )I, dt (N m) β di b N (1 I )I I dt N m Am. J. Applied ci., 8 (11): , 11 () Analysis of the model: Equilibrium points: The model will be analyzed to inestigate the equilibrium points and its stability. The system has two possible equilibrium points: the disease free equilibrium point and an endemic equilibrium point. Two equilibrium points are found by setting the right hand side of Eq. to zero we obtained. Endemic equilibrium (E 1 ): In the other case when the disease is presented, I, I, we obtained Eq. 3: Disease free equilibrium (E ): In the absence of disease, that is 1, 1. We obtained 1, then the ( r ) ( r ) 4 ( r )(1 R ) λ 3 disease free equilibrium is E (1,,) 113 I * β M β MR R 1 β MR * β(r 1) I R ( β M) * Where: b N r β β,m (N m) R b ββ NA /( ) (N m) (r ) Then the endemic equilibrium point is E 1(,1,1 ). (3) Local asymptotical stability: The local stability of an equilibrium point is determined from the Jacobian matrix of the system of ordinary differential Eq. ealuated at each equilibrium point. The Jacobian matrix at E is shown as Eq. 4: J ( r ) bβn N m Abβ (N m) Abβ (N m) (4) The eigenalues of the J are obtained by soling det (J λ I). We obtained the following characteristic Eq. 5, we get λ 1 λ λ (5) ( r ) ( r )(1 R ) We obtain: ( r ) ( r ) 4 ( r )(1 R ) λ

4 Am. J. Applied ci., 8 (11): , 11 Which the last two eigenalues is negatie if R <1. Disease endemic equilibrium: To determine the stability of the endemic equilibrium point, E 1, by finding the eigenalues of Jacobian matrix at E 1, as follow Eq. 6: β MR MR β M ( ) ( ) β M β β MR M(R 1) MR β M J1 M ( ) (6) β M β β MR β β MR β M ( ) R ( ) R β M β MR The characteristic Eq. 7 is calculated by setting det (J 1 -λ1). We obtain: λ 3 Aλ 3 BλC (7) Where: ( β MR ) R ( β M) A M β M β MR M( β MR ) M β(r 1) B R β M β MR C M(R 1) By using Routh-urwitz criteria, the endemic equilibrium point is locally stable if the following conditions are satisfied: 1. A>. C> 3. AB>C It can be easily seen that A >, C > with R > 1. Consider: ( β MR ) R ( β M) AB ( M ) β M β MR M( β MR ) M β(r 1) ( R ) β M β MR AB > MR > C Thus E 1 is locally asymptotically stable. Table 1: Parameter alues leading to disease-free state Parameters alue A 4. b.5 day 1.5 day 1 1/(6 365) day 1 m. r.148 day 1 β.75 β 1. N 1. tability of disease-free state: From the alues of the parameters listed in Table 1, the calculated eigenalues and basic reproductie number are: λ , λ , λ and R ince these alues leads to all of the eigenalues to be negatie and the basic reproductie number to be less than one, the equilibrium state will be the disease - free state, E as shown in Fig.. tability of endemic state: Next we change the alue of the recruitment rate of mosquito ( A ) from 4 to be equal to 5 and keep the other alues of the parameters the same. With these alues, we obtain: λ λ i, λ i and R 1.8>1. This along the fact that the real parts of the eigenalues are all negatie leads to the eqilibrium state to be the endemic state, E 1 and this state will be locally asymptotically stable. The fact that λ and λ 3 are complex conjugates means that the tempolary behaior of the population will exhibit oscillatory behaiors. The time series solutions for susceptible human, infected human and infected mosquito as shown in Fig. 3. Furthermore, in Fig. 4a and b, the trajectories of the solutions in the (,1 ) plane and (1, 1 ) plane, respectiely the trajactories spiraling into the equilibrium endemic state; Fig. 4c It shown that the infected human with * restpect to R, as I where R <1, the disease freeequilibrium is stable; If R >1 the endemic equilibrium * is stable the solution I >, which approach to the endemic state. DICUION We establish the threshold parameter for this model is: R b ββ NA / ( ) γ (N m) ( ) REULT The quantity R ɶ R is the basic reproductie number of the disease. It represents the aerage number of The system of Eq. was soled numerically using secondary cases that one case can produce if introduced the alues of the parameters are gien in Table 1. into a sesceptible population (Anderson and May,199). 114

5 Am. J. Applied ci., 8 (11): , 11 (a) (b) (c) Fig. : Numerical solution of system (), time series of (a), (b) I and (c) I with A4, b.5,.5, 1/(6 365), r.148, β.75,β 1., N 1, eigenalues are λ , λ , λ and R <1. The numerical solutions conerge to the disease - free state E (1,, ) (a) (b) (c) Fig. 3: Numerical solution of system (), time series of (a), (b) I and (c) I with A5, b.5,.5, 1/(6 365), r.148, β.75,β 1.,N 1, eigenalues are λ , λ i, λ i andr 1.8>1 (a) (b) (c) Fig. 4: Trajectories of human - mosquito population of system The numerical solutions conerge to the endemic state E 1 (.95856,.89441, ). 115 Our simulation results show that the basic reproductie number will be increase if the recruitment

6 rate of mosquito is increase. i.e., we find that R.1168 for the recruitment rate of mosquito A1 and R for A7. The normalized indiidual populations are shown when the alues of the basic reproductie number are different, as shown in the Fig. and 3. ere we can see that the normalized indiidaul populations conerge to the disease- free equilibrium point for R <1. In cases of R >1, the normalized indiidual populations oscillate to the endemic equilibrium point. The basic reproductie numbers are used for controlling the disease (Pongsumpun and Tang, 8) and (Pongsumpun, 1) by decreasing the carry capacity of the enironment for mosquitoes by frequent reduction of mosquito breeding sites, seems to be a more efficatie way of controlling the disease: (), in the (a)(,i ) and (b) (I, I ) plane, the solution approach to the endemic state; (c) Bifurcation diagram for equilibrium of system (), demonstrate the infected human with restpect to R, represents the stable solutions and represents the unstable solutions for R <1, E will be stable, R >1, E 1 will be stable, with the same set of parameters used in Fig. 3 CONCLUION In this study, we hae proposed and analyzed the dynamical transmission of Dengue feer by taking into account the role played without immunity in human population. We found that there are two equilibrium states, a disease-free state and endemic state. This will decrease the basic reproductie number to below one. Consequently, we can reduce the human sucecptibility to the disease and, in turn, this can reduce the outbreak of the disease. ACKNOWLEDGMENT urapol Naowarat would like to thank Assoc. Prof. Dr. Thongchai Kruahong, Asst. Prof. Dr. Duangrut Patamaraka for giing me suggestions, Asst. Prof. Dr. Narong Buddhichiwin and uratthani Rajabhat Uniersity for financial support according to Code: 55 A15913 REFERENCE Adetunde, I.A., 9. The mathematical models of the dynamical behaior of tuberculosis disease in the upper East region of the Northern Part of Ghana: A case study of bawku. Curr. Res. Tuberculosis., 1: 1-6. Anderson, R.M. and R.M. May., 199. Infectious Diseases of umans: Dynamics and Control. 1st Edn., Oxford Uniersity Press, Oxford, IBN: X, pp: 768. Am. J. Applied ci., 8 (11): , Chikaki, E. and. Ishikawa, 9. A dengue transmission model in Thailand considering sequential infections with all four serotypes. J. Infect. De. Ctries., 3: PMID: Estaa, L. and C. asgus, Analysis of a dengue disease transmission model. Math. Biosci., 15: DOI: 1.116/5-5564(98)13- Guzman, M.G.,.B. alstead,. Artsob, P. Buchy and J. Farrar et al., 1. Dengue: A continuing global threat. Nat. Re. Microbiol., 8: PMID: Kongnuy, R., E. Naowanich and P. Pongsumpun, 11. Analysis of a dengue disease transmission model with clinical diagnosis in Thailand. Int. J. Math. Models Methods Applied ci., 5: Koriko, O.K. and T.T. Yusuf, 8. Mathematical model to simulate tuberculosis disease population dynamics. Am. J. Applied. ci., 5: DOI: /ajassp Naowarat,., W. Tawarat and I.M. Tang, 11. Control of the transmission of Chikungunya feer epidemic through the use of adulticide. Am. J. Applied. ci., 8: DOI: /ajassp Pongsumpun, P. and I.M. Tang, 8. Effect of the seasonal ariation in the extrinsic incubation peroid on the long term behaior of the dengue hemorrhagic feer epidemic. World Acad. ci. Eng. Technol., 34: Pongsumpun, P., 1. Dynamical transmission model of chikungunya in Thailand. World Acad. ci. Eng. Technol., 68: Ranjit,. and N. Kissoon, 1. Dengue hemorrhagic feer and shock syndromes. Pediatr. Crit. Care. Med., 1: 9-1. PMID: Rodenhuis-Zybert, I.A., J. Wilschut and J.M. mith, 1. Dengue irus life cycle: iral and host factors modulating infectiity. Cell. Mol. Life. ci., 67: DOI: 1.17/s z riprom, M., P. Pongsumpan,. Yoksan, P. Barbazan and J.P. Gonzalez et al., 3. Dengue haemorrhagic feer in Thailand, : Primary or secondary infection. Dengue. Bull., 7: Gubler, D.J. and G. Kuno, Dengue Feer and Dengue emorrhagic Feer. 1st Edn., CAB International, Wallingford IBN-1: , pp: 478. WO, 9. Dengue: Guidelines for Diagnosis, Treatment, Preention and Control. 1st Edn., World ealth Organization, Genea, IBN: , pp: 147. Yaacob, Y., 7. Analysis of a dengue disease transmission model without immunity. Matematika, 3: