Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2392 https://internationalpubls.com Numerical Solution of Stochastic SEIQR epidemic model with Differential Transformation Method M. Priyadharshini1, J. Senbagamalar2*, and G. Sathishkumar 1 1 Department of Mathematics, Faculty of Science and Humanities, SRM Institute of Science and Technology, Chennai Ramapuram, Chennai-600 089, India. 2*Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai- 600 062, India E-mail : priya88.mathra@gmail.com * E-mail : senbagamalar2005@yahoo.com E-mail : gskmathspu@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: In this we solve the Policy Decision-Making for quarantine Policy. The quarantine variable in the SEIQR model enables the policy- makers to examine the relative impact of isolation measures on controlling disease transmission. Real-World Disease Modeling can be used to model different infectious diseases like COVID-19, SARS, MERS and influenza, where quarantine is an important factor. The Reproduction Number and Disease Dynamics, derivation of equilibrium points under which a disease dies out or continues to multiple. The research compares (DTM) with the (RK4) method to determine its accuracy and efficiency. The aim of this research is to examine the application of the differential transformation to determine the approximate solutions to the SEIQR epidemiology model. In solving differential and integral equations, numerical and semi- analytical techniques are employed respectively through this method. Through the application of the DTM (Differential Transformation Method), a SEIQR model was solved and two cases were discussed, one of which is endemic and another is disease-free case. In addition, we solve the solutions using Runge-Kutta method of order four. At last, we compare both the solutions obtained by using Differential Transformation Method (DTM) and the (RK4) method. The solution, so obtained is more precise to use than DTM. Keywords: Differential Transformation Method, SEIQR epidemiology model, RK4 Method, Transformed function, Numerical Solutions. 1. Introduction The study of Mathematics is the Science of order, relation and structure which is used across a variety of disciplines. It is possible to solve many problems mathematically. Solving and analyzing differential equations involves analytical and numerical methods. We use analytical Taylor series method; the differential transform method (DTM) solves integral equations as well as differential equations. As a result of all these activities, we should be able to gain information about how the disease is spreading throughout the population, how we can control the spread, and how to eradicate the disease completely [1]. It is most often infectious diseases that are modeled, i.e. disease that can Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2393 https://internationalpubls.com be transmitted from one person to another. In many diseases affect children, including measles, rubella, chicken pox and mumps, along with gonorrhea, syphilis and HIV/AIDS, as well as sexually transmitted diseases. Runge-Kutta methods are a class of methods that extrapolate the solution to the next time step by carefully using the information on the "slope" at many points. The fourth order Runge Kutta (RK4) method is explicit and one of the most used approaches for solving IVPs. There is a need for a method that precisely use the nonlinear terms easily without any restrictions and with less number of computations. Indeed, the so called Differential Transform Method (DTM) which gives a series solutions can overcome some of the above difficulties. The DTM is very effective numerical and analytical method for solving different types of differential equations as well as integral equations. This method converts the differential equations into recurrence relations and then by Taylor series expansion will give a different approach to obtain the convergent series solutions. The differential transformation method is very much familiar technique to solve all the kinds of differential equations. The method was originally developed by Zhou [2] for solving both linear and non-linear initial value problems in electrical circuits. Later, several researches have been conducted in applying differential transform method to different types of equations. These researches confirm the fact that this method is reliable, efficient as well as having a wider applicability. According to Hasan [3] the differential equation system was completed by using DTM and compared with RK method in 2008. In his opinion, DTM is one of the most accurate and easiest to explore the technologies. Despite its advantages, the only disadvantage of the DTM is that it produces an approximate solution in the truncated series form and also the convergence interval is pretty limited. In a tiny area, DTM produces a series of solution which is not reflect the real time behavior of the given problem, but does provide a good estimate to the correct solution. A number of differential algebraic equations have been solved by using this method, as well as SchrΓΆdinger equations [4], fractional differential equations [5,10] an equation of the type Lane- Emden and equations describing the unsteady movements of rotating spheres in inclined tubes. There are many advantages in this method, including the fact that it can be used directly with all ODEs without the requirement of perturbation, discretization or linearization. In [9] an artificial infectious disease optimization algorithm based on the SEIQR epidemic model is constructed. This method have an advantage in the way it is able to reduce the amount of computations needed to compute the series of solution while even maintaining accuracy and providing the series solution’s fast convergence as well. The Runge-Kutta method has been used by many authors to solve nonlinear ODEs [6, 7, 8]. In this paper, we apply the application of the Differential Transform Method to the proposed model and verify that Maple 18 and Matlab’s classical (RK4) method is valid for solving the model. Throughout this paper, the following topics are discussed. The approach of differential transformation method is covered in Section 2. To construct the SEIQR model in a dynamical system is discussed in Section 3. The equilibrium point of the model is analyzed in Section 4. In Section 5, an application of the differential transformation method and graphical representation for SEIQR https://www.sciencedirect.com/topics/computer-science/optimization-algorithm Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2394 https://internationalpubls.com model is seen. In Section 6, the comparison of the values of Runge-Kutta method of fourth order and DTM are shown in the table. 2. Methodology The purpose of this Section is to provide insight into the methodology we used to solve our problem. 2.1. Method of Differential Transformation If a function h(x) has kth derivatives, then it can be written in Taylor series expans ion about a point at x = x0 as β„Ž(π‘₯) = βˆ‘ π‘₯π‘˜ π‘˜! [ π‘‘π‘˜β„Ž(π‘₯) 𝑑π‘₯π‘˜ ] π‘₯=π‘₯0 ∞ π‘˜=0 . Table 1: A description of the fundamental operations of Differential Transformation Methods (DTMs) S.No Original Function Transformed Function 1 h(x) = m(x) Β± n(x) H(k) = M (k) Β± N (k) 2 h(x) = cm(x) H(k) = cM (k), where c is constant. 3 β„Ž(π‘₯) = π‘‘π‘š(π‘₯) 𝑑π‘₯ H(k) = (k + 1)M (k + 1) 4 β„Ž(π‘₯) = 𝑑2π‘š(π‘₯) 𝑑π‘₯2 H(k) = (k + 1)(k + 2)M (k + 2) 5 β„Ž(π‘₯) = π‘‘π‘Ÿπ‘š(π‘₯) 𝑑π‘₯π‘Ÿ H(k) = (k + 1)(k + 2) + ... + (k + l)M (k + l) 6 h(x) = 1 H(k) = Ξ΄(k) 7 h(x) = x H(k) = Ξ΄(k βˆ’ 1), where Ξ΄ is Kronecker delta. 8 h(x) = eΞ»x 𝐻(π‘˜) = πœ†π‘˜ π‘˜! 9 H(x) = m(x)n(x) 𝐻(π‘˜) = βˆ‘ 𝑀(𝑙)𝑁(π‘˜ βˆ’ 𝑙) π‘˜ 𝑙=0 10 h(x) = (1 + x)r 𝐻(π‘˜) = π‘Ÿ(π‘Ÿ βˆ’ 1)(π‘Ÿ βˆ’ 2). . . (π‘Ÿ βˆ’ π‘˜ + 1) π‘˜! If h(x) is differentially transformed, it has the following definition 𝐻(π‘₯) = 1 π‘˜! [ π‘‘π‘˜β„Ž(π‘₯) 𝑑π‘₯π‘˜ ] π‘₯=π‘₯0 . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2395 https://internationalpubls.com The inverse of differential transformation method of H(x) is β„Ž(π‘₯) = βˆ‘ π‘₯π‘˜π»(π‘˜) . ∞ π‘˜=0 Table-1 will provide a comprehensive overview of the st rong fundamental mathematical operations that are accomplished with differential transforms. 3. Stochastic SEIQR model To understand the dynamics of epidemics in mathematical terms, several models incorporating quarantine have been developed. According to this model, the whole population can be classified into five groups: susceptible, exposed, infectious, quarantined, and recovered. Table 2: Model Variables and Description [11] Variable Description S(t) Susceptible population E(t) Exposed population I(t) Infected population Q(t) Quarantined population R(t) Recovered population In this study, we examine the SEIQR model as described below, 𝑑𝑠 𝑑𝑑 = 𝑏 βˆ’ πœ‡ 𝑆 βˆ’ 𝛽 𝑆𝐼 𝑁 𝑑𝐸 𝑑𝑑 = 𝛽 𝑆𝐼 𝑁 βˆ’ (𝛾 + πœ‡)𝐸 𝑑𝐼 𝑑𝑑 = 𝛾𝐸 βˆ’ (πœ‰ + πœ‚ + 𝛼1 + πœ‡)𝐼 (3.1) 𝑑𝑄 𝑑𝑑 = πœ‚πΌ βˆ’ (𝛿 + 𝛼2 + πœ‡)𝑄 𝑑𝑅 𝑑𝑑 = πœ‰πΌ + 𝛿𝑄 βˆ’ πœ‡π‘…. In total, the size of the population N(t) = S(t) + E(t) + I(t) + R(t). Obviously the region D = {(S, E, I, Q, R)/S β‰₯ 0, E β‰₯ 0, I β‰₯ 0, Q β‰₯ 0, R β‰₯ 0, S + E + I + Q + R ≀ b/Β΅} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2396 https://internationalpubls.com is a collection of model (3.1) that is positively invariant. The reproduction number of the model is π‘πŸŽ = 𝛃𝐛𝛄 𝛍(𝛄+𝛍)(𝛏+𝛍+π›‚πŸ+π›ˆ) . Table 3: Model Parameter and Units Parameter Description Units b Birth rate Individual/day Ξ² Transmission co-efficient (IndividualΓ— day)βˆ’1 Β΅ Natural death rate 1/day Ξ³ Infective individuals of exposed people 1/day Ξ· Infective individuals of quarantined recovery rate 1/day Ξ΅ Recovered rate of infective people 1/day Ξ±1 Disease induced death rate of infected people 1/day Ξ±2 Disease induced death rate of quarantined people 1/day Ξ΄ Recovered people quarantined people rate 1/day 4. Equilibrium Analysis The following section discusses the endemic equilibrium as well as the disease-free equilibrium that will prevail in the future. There are two ways to analyze the qualitative system: 1. Disease-free equilibrium 2. Endemic equilibrium. 4.1 Disease Free Equilibrium In the event of disease death naturally, an equilibrium resulting from the above system takes the form of a disease-free population or a disease-free equilibrium, S = b/Β΅, E = 0, I = 0, Q = 0, R = 0, Hence (𝑆, 𝐸, 𝐼, 𝑄, 𝑅) = ( B πœ‡ , 0, 0, 0, 0). The disease-free equilibrium is locally stable when R0 < 1, but it is unstable when R0 > 1. 4.2 Endemic Equilibrium In the case of R0 > 1, the endemic equilibrium is always stable. It will take the form of an endemic equilibrium if there is a disease-free equilibrium by the population and diseases remain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2397 https://internationalpubls.com unstable. We set the RHS of equation (3.1) equal to zero. The disease endemic equilibrium (Sβˆ—, Eβˆ—, Iβˆ—, Qβˆ—, Rβˆ—) with the positive components, where π‘†βˆ— = 1 𝑅0 , πΈβˆ— = 𝑏 πœ‡ βˆ’ (πœ–+πœ‚+πœ‡+𝛼1)πœ‡ 𝛽𝛾 , πΌβˆ— = 𝑏𝛾 (𝛾+πœ‡)(πœ–+πœ‚++𝛼1) βˆ’ πœ‡ 𝛽 , π‘„βˆ— = πœ‚πΌβˆ— (𝛿 + 𝛼2 + πœ‡) & π‘…βˆ— = 𝐻 + π›Ώπœ‚ πœ‡(𝛿 + 𝛼2 + πœ‡) βˆ’ πœ– 𝛽 where H = 𝑏𝛾 (𝛾+πœ‡)(πœ–+πœ‚++𝛼1) (Ο΅ βˆ’ π›Ώπœ‚ πœ‡(𝛿+𝛼2+πœ‡) ). 5. An application of the DTM Based on the transformed function of the given function in table 1, equation (1) has the following recurrence relation: 𝑆(1 + π‘˜) = 1 π‘˜+1 [𝑏 βˆ’ 𝛽 𝑁 βˆ‘ 𝐼(π‘˜ βˆ’ π‘Ÿ) βˆ’ πœ‡ 𝑆(π‘˜)π‘˜ π‘Ÿ=0 ] 𝐸(1 + π‘˜) = 1 π‘˜+1 [ 𝛽 𝑁 βˆ‘ 𝐼(π‘˜ βˆ’ π‘Ÿ) βˆ’ 𝛾𝐸(π‘˜) βˆ’ πœ‡ 𝑆(π‘˜)π‘˜ π‘Ÿ=0 ] 𝐼(1 + π‘˜) = 1 π‘˜+1 [𝛾𝐸(π‘˜) βˆ’ πœ–πΌ(π‘˜) βˆ’ πœ‚πΌ(π‘˜) βˆ’ 𝛼1𝐼(π‘˜) βˆ’ πœ‡πΌ(π‘˜)] (5.1) 𝑄(1 + π‘˜) = 1 π‘˜+1 [πœ‚πΌ(π‘˜) βˆ’ 𝛿𝑄(π‘˜) βˆ’ 𝛼2𝑄(π‘˜) βˆ’ πœ‡π‘„(π‘˜)] 𝑅(1 + π‘˜) = 1 π‘˜+1 [πœ‰πΌ(π‘˜) βˆ’ 𝛿𝑄(π‘˜) βˆ’ πœ‡π‘…(π‘˜)] with the initial points S(0) = 500, E(0) = 100, I(0) = 100, Q(0) = 200, R(0) = 100 and the parameters N = 1000, b = 5, Ξ³ = 0.75, Ξ±1 = 0.01, Ξ±2 = 0.01, Ξ· = 0.5, Β΅ = 0.5, Ξ² = 1.5, Ο΅ = 0.5, Ξ΄ = 0.75. Based on the equation, we can derive the DTM series solution of the SEIQR model as follows: 𝑆(𝑑) = βˆ‘ 𝑆(π‘˜)π‘‘π‘˜ π‘˜ 𝑙=0 𝐸(𝑑) = βˆ‘ 𝐸(π‘˜)π‘‘π‘˜ π‘˜ 𝑙=0 𝐼(𝑑) = βˆ‘ 𝐼(π‘˜)π‘‘π‘˜ π‘˜ 𝑙=0 𝑄(𝑑) = βˆ‘ 𝑆(π‘˜)π‘‘π‘˜ π‘˜ 𝑙=0 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2398 https://internationalpubls.com 𝑅(𝑑) = βˆ‘ 𝑅(π‘˜)π‘‘π‘˜ π‘˜ 𝑙=0 . By applying the condition in (5.1), we get S(1) = βˆ’245, S(2) = 129, S(3) = βˆ’60.52083, S(4) = 17.6802, E(1) = βˆ’50 E(2) = βˆ’34, E(3) = 27.66875, E(4) = βˆ’32.6418, I(1) = βˆ’125 I(2) = 75.625, I(3) = βˆ’46.5646, I(4) = 22.7660, Q(1) =βˆ’250 Q(2) = 126.25, Q(3) = βˆ’40.4208, Q(4) = 2.3646, R(1) = 150 R(2) = βˆ’162.5, R(3) = 71.25, R(4) = βˆ’22.3057. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2399 https://internationalpubls.com Figure 1: A relationship between S(t), E(t), I(t), Q(t) and R(t) population and time in years. Hence the closed form of a solution, with k = 4, is as follows: 𝑆(𝑑) = βˆ‘ 𝑆(π‘˜)π‘‘π‘˜ π‘˜ 𝑙=0 = 500 βˆ’ 245𝑑 + 129𝑑2 βˆ’ 60.52083𝑑3 + 17.6802𝑑4 + . .. 𝐸(𝑑) = βˆ‘ 𝐸(π‘˜)π‘‘π‘˜ = 100 βˆ’ 50𝑑 – 34𝑑2 + 27.66875𝑑3 – 32.6418𝑑4 + β€¦π‘˜ 𝑙=0 𝐼(𝑑) = βˆ‘ 𝐼(π‘˜)π‘‘π‘˜ π‘˜ 𝑙=0 = 100 βˆ’ 125𝑑 + 75.625𝑑2 βˆ’ 46.5646𝑑3 + 22.76660𝑑4 + . .. 𝑄(𝑑) = βˆ‘ 𝑆(π‘˜)π‘‘π‘˜ = 200 βˆ’ 250𝑑 + 126.25𝑑2 βˆ’ 40.4208𝑑3 + 2.3646𝑑4 + . . .π‘˜ 𝑙=0 𝑅(𝑑) = βˆ‘ 𝑅(π‘˜)π‘‘π‘˜ = 100 + 150𝑑 βˆ’ 162.5𝑑2 + 71.25𝑑3 βˆ’ 22.3057𝑑4 + . . .π‘˜ 𝑙=0 6. The Comparison between RK4 method and DTM Now take the solution of the SEIQR model and compute it by the DTM and RK4 method. The values are discussed in the Table 4. Table 4: Comparison between RK4 and DTM ti 0 0.25 0.5 0.75 1 Susceptible RK4 500 444.3097 395.7261 353.2100 315.8906 DTM 500 445.9359 403.2899 368.8744 341.1594 Exposed RK4 100 86.8303 75.0636 64.6330 55.4509 DTM 100 85.6798 67.9184 44.7196 11.0269 Infected RK4 100 87.9395 76.9315 66.9970 58.1147 DTM 100 72.8379 52.0085 36.3481 26.8270 Quarantined RK4 200 191.1697 183.4350 176.6936 170.8434 DTM 200 144.7683 101.6576 67.2112 38.1938 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 10s (2025) 2400 https://internationalpubls.com Recovered RK4 100 82.9566 66.7665 51.4759 37.1038 DTM 100 128.3698 141.8871 144.0946 136.4443 It is evident from Table 4 that Susceptible and Recovered populations are increasing in our cases. It is noticeable that Exposed, Infected and Quarantined populations are decreasing. 7. Conclusion The SEIQR model with starting conditions has successfully been solved approximately in this study using the differential transformation method (DTM). We have applied the presented methods directly, without linearizing, discretizing, or perturbing. In this method, results show excellent agreement, indicating reliability and effectiveness. This tool can be used in a wide range of fields of study to solve linear problems and non-linear too. We used the Differential Transformation Method (DTM) in this work to analyze the Stochastic SEIQR Model that explains infectious disease transmission with quarantine strategies. The method offered is an efficient approximate solution method that does not need linearization, discretization, or perturbation. Comparison between the (DTM) and (RK4) methods confirms the precision and efficacy of the DTM approach to nonlinear epidemiological models. This paper suggesting that quarantine playing a vital role in the control of the spread of infection, and the model can predict disease development under varying parameter conditions. Moreover, the equilibrium analysis shows that the disease-free equilibrium is stable if R0 < 1 and an endemic equilibrium exists and indicating the threshold value for disease control if R0> 1. SEIQR model and DTM framework can be generalized to other contagious diseases, such as COVID-19, SARS, and influenza, in support of public health decision-making and policy design. Research should be done on fractional-order models, stochastic phenomena, and vaccine strategies inorder to extend this approach towards applications in real epidemiology. Refrences [1] J. Biazar, β€œ Solution of the epidemic model by Adomian decomposition method”, [2] Applied Mathematics and Computation, 173(2) (2006), 1101-1106. http://dx.doi.org/10.1016/j.amc.2005.04.036 [3] J.K. Zhou, β€œDifferential Transformation and its Applications for Electrical Circuits”, Huazhong University Press, Wuhan, China, (1986) (in Chinese). [4] I.H. 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