Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2128 https://internationalpubls.com Stability Analysis by Routh Hurwitz on Dengue Epidemic Model Kaveri Kanchan Kumari 1 , Pawan Kumar2, Sujata Bala3, Rahidas Kumar4 1Research Scholar, University Department of Mathematics, Ranchi University, Ranchi 2Assistant Professor, Dept. of ECE, Cambridge Institute of Technology, Tatisilwai 3Assistant Professor, K.C.B. College Bero 4Assistant Professor, Dept. of Science and Humanities, R.V.S. College of Engineering and Technology, Jamshedpur 1Kanchan_kaveri4@yahoo.in , 2pk3ap.10@gmail.com , 3mailmeonsujatbala@gmail.com , 4kumarahidas@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: Dengue is a serious disease, according to the WHO. Humans contract this illness when female mosquitoes, particularly the Aedes aegypti/yellow fever mosquito, bite them. Humans can contract this disease when bitten by an infected mosquito carrying any of the four virus strains designated by DENV-1 to DENV-4. As an extension of the SIR model, we suggested the SIHRV (Susceptible Infectious Hospital Recovery Vaccine) model for this purpose. The threshold number is determined by applying the Routh-Hurwity and Jacobian criteria. The model is unstable if and stable if . In order to improve recovery, we also determine the endemic and disease- free equilibrium points. Keywords: Mathematical model, vaccination, Endemic equilibrium, Routh-Hurwitz criteria. AMS Classification 2010 : 93A30, 92B05, 92D30 Introduction Dengue fever is a major problem in many industrialized and developing countries' tropical and subtropical regions. Every year, millions of cases from all around the world are reported to the World Health Organization (WHO). The female Yellow Fever (Aedes aegypti) The main vector of this disease is the mosquito. Any one of the four virus strains—designated DENV-1 through DENV4—carried by mosquito bites can infect humans. Humans only get a temporary cross- immunity to the other strains of dengue after being bitten by one, but they are permanently immune to that strain. Usually appearing 4–10 days after the virus enters the bitten person's bloodstream, dengue symptoms can last for up to a week or two. Dengue fever can now be prevented by receiving the New Serotype Dengue Fever Vaccine. Preventive measures are the alternative, essentially costly or impractical, way to stay unaffected. This work uses generalized fractional order derivatives to generate the suggested Dengue Epidemic Model. It is demonstrated that fractional order equations are more suitable than integer order equations for simulating biological, economic, and social systems when memory effects are important. Humans are therefore frequently chosen by mosquitoes based on their prior encounters with them, especially on their location and degree of defense. Thus, a perfect model mailto:Kanchan_kaveri4@yahoo.in mailto:pk3ap.10@gmail.com mailto:3mailmeonsujatbala@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2129 https://internationalpubls.com to enable a nation to be disease free can be built based on the examination of the transmission history and fractional order calculus. Numerous authors [1,2] have written about the various kinds of models that are employed in computation. Sulami [3] , Hamdam [4], Defterli [5] Discuss the SIR model by fractional order within the Caputo fractional operator. SIHRV Model for transmission of viruses Condition : Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2130 https://internationalpubls.com Description of parameters used in above ODE: A is influx rate. is total population of human. is total population of mosquito. is total number of susceptible human population. is total number of infective human population. is total number of hospitalized human population. is total number of recovered human population. is total number of vaccinated human population. death rate in human population. is total number of susceptible mosquito population. is total number of infective mosquito population death rate in mosquito population. Transfer coefficient from human to mosquito. Transfer coefficient from mosquito to human. is recovery rate by infective human population. is the hospitalized rate by infective human population. is recovery rate by hospitalized human population. is vaccinated rate by recovered human population. Some definitions of fractional derivatives [6,7] are implimented. The Riemann –Liouville derivatives of th order is defined as An alternative definition by Caputo is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2131 https://internationalpubls.com In this paper we will use Caputo fractional derivative for the system (1) All parameters are assumed to be non-negative from system (1) Let 3. Equilibrium Points : Then For positive equilibrium point The Jacobian matrix for system (1) for disease free equilibrium point is as follows Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2132 https://internationalpubls.com The characteristic equation is The four Characteristic roots are which are negative, and the cubic equation is Compare this cubic equation with Here . According to Routh array is stable. Endemic equilibrium : The Jacobian matrix for system (1) for endemic equilibrium point is as follows The characteristic equation is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2133 https://internationalpubls.com The four Characteristic roots are which are negative, and the cubic equation is Compare this cubic equation with Here . According to Routh array is stable. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2134 https://internationalpubls.com Proposition : (i) Routh Hurwitz criterion are satisfied for point , is locally asymptotically stable. (ii) Routh Hurwitz criterion are satisfied for point , is locally asymptotically stable. Conclusion : By using Routh-Hurwitz Criteria we find that the above model is stable in endemic equilibrium point. By simulation we observe stability of the model exist in endemic equilibrium point. In extension of this work we can also use harmonic mean type incidence rate for control the disease. Reference : [1] Kumari K. K., Singh A., Mahto. S., “ Mathematical Models for Stability in Computer Network”, Journal of Computer and Mathematical Sciences, Vol.10(1), pp.92-98 January 2019. [2] Kumari K. K., Singh A., Mahto. S., “ Time Delay Seirs E-Epidemic Model For Computer Network” , International Journal of Mathematical Archive-9(2), 2018, PP.265-273. [3] Al-Sulami H., El-Shahed M., Nieto J. J. and Shammakh W., “On Fractional order Dengue Epidemic Model”, Mathematical Problem in Engineering, 2014,pp-1-6. [4] Hamdan N. I. and Kilicman A., “A fractional order SIR epidemic model for dengue transmission”, Chaos, Solitons and Fractals 114(2018), pp-55-62. [5]Defterli O., “Modeling the impact of temperature on fractional order dengue model with vertical transmission”, An International Journal of Optimization and control: Theories & Applications, Vol. 10(1), 2020 pp-85-93. [6] I.Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999 [7] A.A.Kilbas, H.M. Srivastava and J.J.Trujillo, Theory and Application Fractional Differential Equation, Elseviesr, Amsterclam The Netherlands, 2006.