Chaos, Solitons and Fractals

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1 Chaos, Solitons and Fractals 42 (2009) Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: Solutions of the SIR models of epidemics using HAM Fadi Awawdeh *, A. Adawi, Z. Mustafa Department of Mathematics, Hashemite University, Jordan article info abstract Article history: Accepted 3 April 2009 In this paper, we investigate the accuracy of the Homopy Analysis Method (HAM) for solving the problem of the spread of a non-fatal disease in a population. The advantage of this method is that it provides a direct scheme for solving the problem, i.e., without the need for linearization, perturbation, massive computation and any transformation. Mathematical modeling of the problem leads to a system of nonlinear ODEs. MATLAB 7 is used to carry out the computations. Graphical results are presented and discussed quantitatively to illustrate the solution. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction In the modeling transmission dynamics of a communicable disease, it is common to divide the population into disjoint classes (compartments) whose sizes change with time. The infection status of any individual in a population can be Susceptible, when the person is healthy and susceptible to the disease (dened by S), Exposed, when the person is in a latent period but n yet infectious (dened by E), Infected, when the individual carries the disease and is infectious (dened by I), or Removed, when the person has recovered and is at least temporarily immune or has died because of disease (dened by R). In some diseases such as HIV, there is no recovery. In her diseases, if an infected person recovered he/she may be susceptible again. A sequence of letters, such as SEIR, describes the movement of individuals between the classes: susceptible become latent, then infectious and finally recover with immunity. To model diseases which confer permanent immunity and which are endemic because of births of new susceptible, SIR or SEIR models with vital dynamics are suitable. Vital dynamics is needed to avoid explosion of the population size. Models of SEIRs or SIRs types are used to model diseases with temporary immunity and in cases where there is no immunity, models are named SIS or SEIS. The last S points the individual becoming susceptible again, after recovery. Such models may be appropriate for gonorrhea, for instance. Epidemic models has been widely used in different forms for studying epidemiological processes such as the spread of influenza [9] and SARS [2,8,11] and even for the spread of rumors [13,14]. Epidemic models are also applied to modeling of STI epidemics, but n all epidemic models are suitable for STIs since the sexual network plays an important role in spread of disease. Pair-formation models are a type of ordinary differential equation models that have sometimes been used to study STI transmission in populations. They incorporate the repeated contacts within partnerships which happen frequently in real sexual networks. They were first developed in 1988 by Dietz and Hadeler [3] to study STIs in monogamous partnerships. In this model if two susceptible individuals form a pair then they can be considered temporarily immune as long as they do n separate and have no contacts with her partners. This aspect influences transmission dynamics considerably, especially when the disease is first introduced, since the vast majority of existing pairs are susceptible. * Corresponding author. addresses: awawdeh@hu.edu.jo (F. Awawdeh), adawi@hu.edu.jo (A. Adawi), zmagablh@hu.edu.jo (Z. Mustafa) /$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi: /j.chaos

2 3048 F. Awawdeh et al. / Chaos, Solitons and Fractals 42 (2009) This is a simple model, due to Kermack and MacKendrick [5], of an epidemic of an infectious disease in a population. We assume the population consists of three types of individuals, whose numbers are dened by the letters S; I and R (which is why this is called an SIR model). All these of course are functions of time. SðtÞ is the number of susceptible, who do n have the disease but could get it. IðtÞ is the number of infectives, who have the disease and can transmit it to hers. RðtÞ is the number of removed, who cann get the disease or transmit it: either they have a natural immunity, or they have recovered from the disease and are immune from getting it again, or they have been placed in isolation, or they have died. The mathematical model does n distinguish between these possibilities. The following simple SIR model [1,4,10,12] is tested to show the efficiency of the HAM to solve such models. Assume there is a steady constant rate between susceptible and infectives and that a constant proportion of these constant result in transmission. Then in time dt; ds of the susceptible become infective, where ds ¼ bsidt; and b is a positive constant. If c > 0 is the rate at which current infectives become isolated, then di ¼ bsidt cidt; and the number of new isolates dr is given by dr ¼ cidt: If we let dt! 0, then the following nonlinear system of ODEs determines the progress of the disease: ds dt ¼ bsi; di ¼ bsi ci; dt ð2þ dr dt ¼ ci with initial conditions Sð0Þ ¼N S ; Ið0Þ ¼N I ; Rð0Þ ¼N R : In this paper, the HAM [6,7] is used to solve the epidemic models. An example is tested, and the obtained results suggest that newly improvement technique introduces a promising tool and powerful improvement for solving nonlinear equations. Unlike all previous techniques, the proposed technique provides us with a family of solution expressions in auxiliary parameter h. As a result, the convergence region and rate of solution depend upon the auxiliary parameter h and thus can be greatly enlarged by means of choosing a proper value of h. This provides a convenient way to adjust and control convergence region and rate of solution given by the technique. Liao [6] suggested the so-called h-curves to determine the value of h. 2. Basic idea of the HAM Consider N½yðtÞŠ ¼ 0; where N is any operator, yðtþ is unknown function and t the independent variable. Let y 0 ðtþ dene an initial guess of the exact solution yðtþ, h 0 an auxiliary parameter, HðtÞ 0 an auxiliary function, and L an auxiliary linear operator with the property L½yðtÞŠ ¼ 0 when yðtþ ¼0. Then using q 2½0; 1Š as an embedding parameter, we construct such a homopy ð1 qþl½/ðt; qþ y 0 ðtþš qhhðtþn½/ðt; qþš ¼ b H½/ðt; qþ; y 0 ðtþ; HðtÞ; h; qš: It should be emphasized that we have great freedom to choose the initial guess y 0 ðtþ, the auxiliary linear operator L, the non-zero auxiliary parameter h, and the auxiliary function HðtÞ. Enforcing the homopy (4) to be zero, i.e., bh½/ðt; qþ; y 0 ðtþ; HðtÞ; h; qš ¼0 ð1þ ð3þ ð4þ we have the so-called zero-order deformation equation ð1 qþl½/ðt; qþ y 0 ðtþš ¼ qhhðtþn½/ðt; qþš: When q ¼ 0, the zero-order deformation equation (5) becomes /ðt; 0Þ ¼y 0 ðtþ; and when q ¼ 1, since h 0 and HðtÞ 0, the zero-order deformation equation (5) is equivalent to /ðt; 1Þ ¼yðtÞ: ð5þ ð6þ ð7þ

3 F. Awawdeh et al. / Chaos, Solitons and Fractals 42 (2009) Thus, according to (6) and (7), as the embedding parameter q increases from 0 to 1, /ðt; qþ varies continuously from the initial approximation y 0 ðtþ to the exact solution yðtþ. Such a kind of continuous variation is called deformation in homopy. By Taylor s theorem, /ðt; qþ can be expanded in a power series of q as follows: /ðt; qþ ¼y 0 ðtþþ X1 y m ðtþq m where y m ðtþ ¼ 1 o m /ðt; qþ m! oq m : If the initial guess y 0 ðtþ, the auxiliary linear parameter L, the non-zero auxiliary parameter h, and the auxiliary function HðtÞ are properly chosen so that 1. The solution /ðt; qþ of the zero-order deformation equation (5) exists for all q 2½0; 1Š. 2. The deformation derivative om /ðt;qþ j oq m exists for m ¼ 1; 2; The power series (8) of /ðt; qþ converges at q ¼ 1. Then, we have under these assumptions the solution series /ðt; 1Þ ¼y 0 ðtþþ X1 y m ðtþ: For brevity, define the vector! y nðtþ ¼fy 0 ðtþ; y 1 ðtþ; y 2 ðtþ;...; y n ðtþg: According to the definition (9), the governing equation of y m ðtþ can be derived from the zero-order deformation equation (5). Differentiating the zero-order deformation equation (5) m times with respective to q and then dividing by m! and finally setting q ¼ 0, we have the so-called mth-order deformation equation where and L½y m ðtþ v m y m 1 ðtþš ¼ hhðtþr m ðy m 1 ðtþþ 1 R m ðy m 1 ðtþþ ¼ ðm 1Þ! v m ¼ 0; m 6 1 : 1; m > 1 o m 1 N½/ðt; qþš oq m 1 Ne that the high-order deformation equation (12) is governing by the linear operator L, and the term R m ðy m 1 ðtþþ can be expressed simply by (13) for any nonlinear operator N. According to the definition (13), the right-hand side of Eq. (12) is only dependent upon y m 1 ðtþ. Thus, we gain y 1 ðtþ; y 2 ðtþ;... by mean of solving the linear high-order deformation equation (12) one after the her in order. 3. Solution of the SIR model by HAM In order to explicitly construct approximate non-perturbative solutions of the system described by Eqs. (1) (3), HAM well addressed in [6,7] is employed. The advantage of this method is that it provides a direct scheme for solving the problem. To apply the HAM, we choose S 0 ðtþ ¼N S ; I 0 ðtþ ¼N I ; R 0 ðtþ ¼N R as initial approximations of SðtÞ; IðtÞ and RðtÞ. Let q 2½0; 1Š be the so-called embedding parameter. The HAM is based on a kind of continuos mappings SðtÞ!/ 1 ðt; qþ; IðtÞ!/ 2 ðt; qþ; RðtÞ!/ 3 ðt; qþ such that, as the embedding parameter q increases from 0 to 1, / i ðt; qþ varies from the initial approximation to the exact solution. To ensure this, choose such auxiliary linear operators as L i ½/ i ðt; qþš ¼ o/ iðt; qþ ; i ¼ 1; 2; 3 with the property L i ½C i Š¼0 ð8þ ð9þ ð10þ ð11þ ð12þ ð13þ

4 3050 F. Awawdeh et al. / Chaos, Solitons and Fractals 42 (2009) where C i are integral constants. We define the nonlinear operators N 1 ½/ i ðt; qþš ¼ o/ iðt; qþ þ b/ i ðt; qþ/ 2 ðt; qþ; N 2 ½/ i ðt; qþš ¼ o/ iðt; qþ b/ 1 ðt; qþ/ i ðt; qþ c/ i ðt; qþ; N 3 ½/ i ðt; qþš ¼ o/ iðt; qþ c/ 2 ðt; qþ: Let h i 0 and H i ðtþ 0 dene the so-called auxiliary parameter and auxiliary function, respectively. Using the embedding parameter q, we construct a family of equations ð1 qþl½/ 1 ðt; qþ S 0 ðtþš ¼ qh 1 H 1 ðtþn 1 ½/ 1 ðt; qþš; ð1 qþl½/ 2 ðt; qþ I 0 ðtþš ¼ qh 2 H 2 ðtþn 2 ½/ 2 ðt; qþš; ð1 qþl½/ 3 ðt; qþ R 0 ðtþš ¼ qh 3 H 3 ðtþn 3 ½/ 3 ðt; qþš; subject to the initial conditions / 1 ð0; qþ ¼S 0 ; / 2 ð0; qþ ¼I 0 ; / 3 ð0; qþ ¼R 0 : By Taylor s theorem, we expand / i ðt; qþ by a power series of the embedding parameter q as follows: / 1 ðt; qþ ¼S 0 ðtþþ X1 / 2 ðt; qþ ¼I 0 ðtþþ X1 / 3 ðt; qþ ¼R 0 ðtþþ X1 S m ðtþq m ; I m ðtþq m ; R m ðtþq m where S m ðtþ ¼ 1 o m / 1 ðt; qþ m! oq m ; I m ðtþ ¼ 1 o m / 2 ðt; qþ m! oq m ; R m ðtþ ¼ 1 o m / 3 ðt; qþ m! oq m : From the so-called mth-order deformation equations (12) and (13), we have L½S m ðtþ v m S m 1 ðtþš ¼ h 1 H 1 ðtþr m ðs m 1 ðtþþ; L½I m ðtþ v m I m 1 ðtþš ¼ h 2 H 2 ðtþr m ði m 1 ðtþþ; L½R m ðtþ v m R m 1 ðtþš ¼ h 3 H 3 ðtþr m ðr m 1 ðtþþ; S m ð0þ ¼0; I m ð0þ ¼0; R m ð0þ ¼0: ð14þ By the h-curves [6], it is reasonable to use h i ¼ 1. Using H i ðtþ ¼1, the mth-order deformation equation (14) for m P 1 becomes Z " # t Xm 1 S m ðtþ ¼v m S m 1 ðtþ ðsþþb S k ðsþi m 1 k ðsþ ds; I m ðtþ ¼v m I m 1 ðtþ R m ðtþ ¼v m R m 1 ðtþ S 0 m 1 0 Z t I 0 m 1 0 Z t 0 k¼0 " # Xm 1 ðsþ b S k ðsþi m 1 k ðsþþci m 1 ðsþ ds; k¼0 R 0 m 1 ðsþ ci m 1ðsÞ ds: 4. Numerical results and discussion For numerical results the following values, for parameters, are considered: N S N I N R b c

5 F. Awawdeh et al. / Chaos, Solitons and Fractals 42 (2009) Fig. 1. Pls of 20 terms approximations for SðtÞ; IðtÞ and RðtÞ versus time. Five and nine terms approximations for SðtÞ, IðtÞ and RðtÞ are calculated and presented below. Five terms approximations: S 5 ðtþ ¼ X5 I 5 ðtþ ¼ X5 R 5 ðtþ ¼ X5 Nine terms approximations: S 9 ðtþ ¼ X9 S m ðtþ ¼20 3t 0:045t 2 þ 0:028t 3 þ 0: t 4 0: t 5 ; I m ðtþ ¼15 þ 2:7t þ 0:018t 2 0:02817t 3 0: t 4 þ 0: t 5 ; R m ðtþ ¼10 þ 0:3t 0:054t 2 0:00012t 3 þ 0: t 4 0: t 5 : S m ðtþ ¼20 3t 0:045t 2 þ 0:028t 3 þ 0: t 4 0: t 5 0: t 6 þ 0: t 7 þ 0: t 8 0: t 9 ; I 9 ðtþ ¼ X9 I m ðtþ ¼15 þ 2:7t þ 0:018t 2 0:02817t 3 0: t 4 þ 0: t 5 þ 0: t 6 0: t 7 0: t 8 þ 0: t 9 ; R 9 ðtþ ¼ X9 R m ðtþ ¼10 þ 0:3t 0:054t 2 0:00012t 3 þ 0: t 4 0: t 5 þ 0: t 6 þ 0: t 7 0: t 8 0: t 9 : The HAM yields rapidly convergent series solution by using a few iterations. For the convergence of the HAM the reader is referred to [6]. Fig. 1 shows pls of 20 terms approximations of SðtÞ; IðtÞ and RðtÞ. Fig. 1 illustrates the case when we introduce a small number of infectives I 0 ¼ 1 into a susceptible population. An epidemic will occur and the number of infectives increases; the maximum infective population (climax of epidemic) I max ¼ 240:2564 will occur when S has decreased to the value 100. As time goes on (t!1) you travel along the curve to the right, eventually approaching I ¼ 0 and the disease died out. The epidemic will end as I! 0 with S approaching some positive value S 1 ¼ 3:4699. S 1 is the eventual population who were never infective. lim IðtÞ ¼lim I 20ðtÞ ¼0 and lim SðtÞ ¼lim S 20 ðtþ ¼S 1 ¼ 3:4699: t!1 t!1 t!1 t!1 5. Discussion and conclusion The analytical approximations to the solutions of the epidemic models are reliable and confirm the power and ability of the HAM as an easy device for computing the solution of nonlinear problems. The method avoids the difficulties and massive computational work that usually arise from parallel techniques and finite-difference method.

6 3052 F. Awawdeh et al. / Chaos, Solitons and Fractals 42 (2009) References [1] Biazar J. Solution of the epidemic model by Adomian decomposition method. Appl Math Comput 2006;173: [2] Bai Y, Jin Z. Prediction of SARS epidemic by BP neural networks with online prediction strategy. Chaos, Solitons & Fractals 2005;26: [3] Dietz K, Hadeler KP. Epidemiological models for sexually transmitted diseases. J Math Biol 1988;26:1 25. [4] Gao S, Teng Z, Xie D. Analysis of a delayed SIR epidemic model with pulse vaccination. Chaos, Solitons & Fractals 2009;40: [5] Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc Lond A 1927;115: [6] Liao SJ. Beyond perturbation: introduction to the homopy analysis method. Boca Raton: Chapman & Hall/CRC Press; [7] Liao SJ. On the homopy analysis method for nonlinear problems. Appl Math Comput 2004;174: [8] Lipsitch M, Cohen T, Cooper B, Robins JM, Ma S, James L. Transmission dynamics and control of severe acute respiratory syndrome. Science 2003;300: [9] Liu Z, Lai YC, Ye N. Propagation and immunization of infection on general networks with bh homogeneous and heterogeneous components. Phys Rev 2003;E67: [10] Pang G, Chen L. A delayed SIRS epidemic model with pulse vaccination. Chaos, Solitons & Fractals 2007;34: [11] Riley S. Transmission dynamics of the etiological agent of SARS in Hong Kong: impact of public health interventions. Science 2003;300: [12] Satsuma J, Willox R, Ramani A, Grammaticos B, Carstea AS. Extending the SIR epidemic model. Physics A 2004: [13] Zanette DH. Critical behavior of propagation on small-world networks. Phys Rev 2001;E64:050901(R). [14] Zanette DH. Dynamics of rumor propagation on small-world networks. Phys Rev 2002;E65: