Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2478 https://internationalpubls.com On Fractional Epidemic Model Order Shigella 1Mampi Saha, 2Kaveri Kanchan Kumari, 3Om Prakash, 4Asha Lata Keshri 1Assistant Professor , RTC Institute of Technology, Ranchi, India sajalmampi@gmail.com 2Research Scholar, Department of Mathematics, Ranchi University, Ranchi, India Kanchan_kaveri4@yahoo.in 3Research Scholar, Department of Mathematics, Ranchi University, Ranchi, India prakashcool.om12@gmail.com 4 Associate Professor, Department of Mathematics, Ranchi University, Ranchi, India ashlata.math@ranchiuniversity.ac.in Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: Introduction: We have adapted the continuous mathematical framework developed to examine the dynamics of a Shigella epidemic at a constant recruitment rate. They divided the population into seven groups in their model: susceptible (S), vaccination(V), exposΓ©(E), infected (I), isolated (G), hospitalized(H) and recovered (R), each with its own set of parameters. We examined a mathematical model of a Shigella outbreak in a community with a constant population using the SVEIGHR compartmental nonlinear deterministic approach. The model was subjected to analytical investigations utilizing the linearized stability method. The greatest eigenvalue of the next-generation matrix yields the fundamental reproductive number R0, which controls the spread of the disease. The generalized Routh-Hurwitz and Jacobian criterion are used to determine the threshold value R0. The model is unstable if, R0 > 1, and stable if, R0 < 1. Additionally, we determine the endemic and disease-free equilibrium points, which are helpful for a faster recovery. We have created graphs in Matlab to provide a more accurate model representation. Objectives: To describe the disease Shigella with the help of SVEIGHER compartmental nonlinear deterministic approach through factorial differential method and compare it to classical differential method and see the results. Results: From the above studies we conclude that fractional differential method is more effective than classical differential method. Conclusion : In this paper, we discuss SVEIGHR epidemic model for disease Shigella . This SVEIGHR model controlss the spreading of the mailto:sajalmampi@gmail.com mailto:Kanchan_kaveri4@yahoo.in mailto:prakashcool.om12@gmail.com mailto:ashlata.math@ranchiuniversity.ac.in Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2479 https://internationalpubls.com disease in Human population. By using Routh-Hurwitz Criteria we find all the eigen values for endemic point are negative which shows that the above model SVEIGHR (Susceptible Vacation Infectious Isolated Hospitalization Recovered) is stable. Extending our work, we can also use harmonic mean type incidence rate for better stability and control the disease. Key word : Mathematical model, stability analysis, endemic equilibrium; Routh-Hurwitz criteria, Reproduction number. AMS Classification 2010 : 93A30, 92B05, 92D30 1. Introduction Gram-negative, non-spore-forming Shigella bacteria are the main cause of Shigella infection, commonly referred to as Shigellosis, a diarrheal illness. It is via fecal-oral matter, which is typically brought on by contaminated food, water, or close contact with infected people. These bacteria infect the intestinal lining. Tenesmus, fever, abdominal pain, and diarrhea-often bloody-are among the symptoms that start to show up 1-2 days after exposure and can linger for up to a week. Serious side effects include hemolytic uremic syndrome, sepsis, and reactive arthritis could happen. The majority of cases go away without special care, but rest and hydration are crucial. Antibiotics such as azithromycin or ciprofloxacin may be necessary in severe cases, though resistance is becoming a bigger problem. Although there isn't a licensed vaccine at the moment, a number of possibilities are being developed, such as conjugate and live attenuated vaccines. Reducing transmission still requires preventive actions like better sanitation and hand hygiene. Fractional calculus investigates function derivatives and integrals. However, in this area of mathematics, we are examining integrals and derivatives of non-integer order rather than the typical integer order. These might be of real or complex order and are referred to as fractional-order derivatives and fractional integrals. A mathematical tool that unifies and generalizes the derivative and integral of integer order to any arbitrary order is generally referred to as fractional calculus. Leibniz initially addressed the theory of fractional derivative in 1695. Because of this, mathematicians from the 18th to the 19th centuries were interested in studying this field. A well known scientist, Abel, in 1823 was the first scientist to apply Fractional calculus for investigating tautochrone problems. Fractional calculus has been the subject of worldwide attention in the last decades due to its broad range in biology, physics, chemistry, engineering, modeling etc. Fractional differential equations system allows more degree of freedom, memory effect, more adequate, more realistic interpretation of natural phenomena than integer-order derivatives in the model. They have become an excellent tool in modeling epidemiology. Hence, fractional derivatives based on epidemic system have also been used to deal with some epidemic behaviors. The statically data gathered during an actual disease epidemic cannot be adequately reproduced by traditional first-order differential equations, and they also do not yield satisfactory results. The research has examined a more specific and complex set of differential equations in order to provide Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2480 https://internationalpubls.com better conclusions that are more in line with reality. We suggest a significantly altered system of equations that makes use of the fractional order differential equation. Hypotheses of the model i. The recruitment process is constant and solely based on birth. ii. Every person is susceptible from birth. iii. Contact with the excrement of diseased people and tainted food or water can infect a person. iv. Infected people can pass either naturally or as a result of the illness. v. Only susceptible adults and children between the ages of 0 and 6 months are eligible for immunization. vi. No long-term recovery is possible. vii. The population is homogeneously mixed. viii. People engage with one another in a panmictic manner 2. Model Derivation : 𝑑𝑆 𝑑𝑑 = Ξ› βˆ’ 𝛾1𝑆 βˆ’ 𝛾2𝑆 βˆ’ 𝑑𝑆 𝑑𝑉 𝑑𝑑 = 𝛾2𝑆 + 𝜏2𝐼 βˆ’ 𝑑𝑉 𝑑𝐸 𝑑𝑑 = 𝛾1𝑆 βˆ’ 𝛿1𝐸 βˆ’ 𝛿2𝐸 βˆ’ 𝑑𝐸 𝑑𝐼 𝑑𝑑 = 𝛿1𝐸 βˆ’ 𝜏1𝐼 βˆ’ 𝜏2𝐼 βˆ’ 𝜏3𝐼 βˆ’ 𝑑𝐼 𝑑𝐺 𝑑𝑑 = 𝛿2𝐸 + 𝜏3𝐼 βˆ’ 𝜁1𝐺 βˆ’ 𝜁2𝐺 βˆ’ 𝑑𝐺 𝑑𝐻 𝑑𝑑 = 𝜏1𝐼 + 𝜁1𝐺 βˆ’ πœ‰π» βˆ’ 𝑑𝐻 𝑑𝑅 𝑑𝑑 = πœ‰π» + 𝜁2𝐺 βˆ’ 𝑑𝑅 } …. (1) 𝑉 𝑑𝑉 𝑑𝐻 𝐸 𝛾1𝑆 𝐼 𝐻 𝑅 𝐺 Ξ› 𝑆 𝑑𝐸 𝑑𝐼 𝑑𝑆 𝑑𝐺 𝑑𝑅 𝛿1𝐸 𝛿2𝐸 𝜏1𝐼 𝜏3𝐼 𝜁1𝐺 𝜁2𝐺 πœ‰π» 𝜏2𝐼 𝛾2𝑆 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2481 https://internationalpubls.com Condition : 𝑆 + 𝑉 + 𝐸 + 𝐼 + 𝐺 + 𝐻 + 𝑅 = 𝑁 where, and Variables Description 𝑆 The total number of susceptible population at time (𝑑) 𝑉 The total number of vaccination population at time (𝑑) 𝐸 The total number of exposed population at time (𝑑) 𝐼 The total number of infective population at time (𝑑) 𝐺 The total number of isolated population at time (𝑑) 𝐻 The total number of hospitalized population at time (𝑑) 𝑅 The total number of recovered population at time (𝑑) Parameters Description Values Ξ› All the new immigrants are susceptible and join the group at a constant rate 𝑁 The human population 10000 𝛾1 The exposed rate by susceptible population 0.25 𝛾2 The vaccination rate by susceptible population 0.75 𝛿1 The infected rate by exposed population 0.5 𝛿2 The isolated rate by exposed population 0.2 𝜏1 The hospitable rate by infected population 0.15 𝜏2 The vaccination rate by infected population 0.25 𝜏3 The isolated rate by infected population 0.2 𝜁1 The hospitable rate by isolated population 0.25 𝜁2 The recovery rate by isolated population 0.20 πœ‰ The recovery rate by hospitable population 0.25 𝑑 The natural death rate 0.000457 𝛽 The order of fractional 0 < 𝛽 ≀ 1 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2482 https://internationalpubls.com Many definitions of fractional derivatives [3,4] are used. The Riemann –Liouville derivatives of 𝑖th order is defined as π’Ÿ0+ 𝛽 𝑅𝐿 𝑓(𝑑) = 1 Ξ“(𝑛 βˆ’ 𝑖) ( 𝑑 𝑑𝑑 ) 𝑛 ∫ 𝑓(𝑠) (𝑑 βˆ’ 𝑠)π‘–βˆ’π‘›+1 𝑑 0 𝑑𝑠 where, 𝑛 = [𝑖] + 1 … (2) An alternative definition by Caputo was as follows π’Ÿπ‘‘ 𝛽 𝑓(𝑑) = 1 Ξ“(𝑛 βˆ’ 𝑖) ∫ 𝑓(𝑛)(𝑠) (𝑑 βˆ’ 𝑠)π‘–βˆ’π‘›+1 𝑑 0 𝑑𝑠 where, 𝑛 = [𝑖] + 1 … (3) In this paper we will use Caputo fractional derivative for the system (1) π’Ÿπ‘‘ 𝛽 𝑆 = Ξ› βˆ’ (𝛾1 + 𝛾2 + 𝑑)𝑆 π’Ÿπ‘‘ 𝛽 𝑉 = 𝛾2𝑆 + 𝜏2𝐼 βˆ’ 𝑑𝑉 π’Ÿπ‘‘ 𝛽 𝐸 = 𝛾1𝑆 βˆ’ (𝛿1 + 𝛿2 + 𝑑)𝐸 π’Ÿπ‘‘ 𝛽 𝐼 = 𝛿1𝐸 βˆ’ (𝜏1 + 𝜏2 + 𝜏3 + 𝑑)𝐼 π’Ÿπ‘‘ 𝛽 𝐺 = 𝛿2𝐸 + 𝜏3𝐼 βˆ’ (𝜁1 + 𝜁2 + 𝑑)𝐺 π’Ÿπ‘‘ 𝛽 𝐻 = 𝜏1𝐼 + 𝜁1𝐺 βˆ’ (πœ‰ + 𝑑)𝐻 π’Ÿπ‘‘ 𝛽 𝑅 = πœ‰π» + 𝜁2𝐺 βˆ’ 𝑑𝑅 } …. (4) All parameters are assumed to be non-negative from system (4), where 𝑑 β‰₯ 0 π’Ÿπ‘‘ 𝛽 𝑁 = Ξ› βˆ’ 𝑑𝑁 … (5) We solve the equation (5) by using Mittage-Leffler function, which is defined as 𝐸𝛽,𝛾(𝑧) ~ βˆ’βˆ‘ 𝑧𝑖 Ξ“(𝛾 βˆ’ 𝛽𝑖) 𝑛 𝑖=1 + 𝑂(|𝑧|βˆ’1βˆ’π‘›), (|𝑧| β†’ ∞, π›½πœ‹ 2 < |arg (𝑧)| ≀ πœ‹) Let Ξ© = {(𝑆, 𝑉, 𝐸, 𝐼, 𝐺, 𝐻, 𝑅) ∈ 𝑅7 +: 𝑆, 𝑉, 𝐸, 𝐼, 𝐺, 𝐻, 𝑅 β‰₯ 0, 𝑆 + 𝑉 + 𝐸 + 𝐼 + 𝐺 + 𝐻 + 𝑅 = Ξ›/𝑁} where Ξ© is a closed set. After evaluating the equation (5), we get 𝑁(𝑑) = 𝑁(0)𝐸𝛽,1(βˆ’π‘π‘‘ 𝛽) + Λ𝑑𝛽𝐸𝛽,𝛽+1(βˆ’π‘π‘‘ 𝛽). 3. Equilibrium Points : Let as assume that all equilibrium points are as follows: π’Ÿπ‘‘ 𝛽 𝑆 = 0, π’Ÿπ‘‘ 𝛽 𝑉 = 0, π’Ÿπ‘‘ 𝛽 𝐸 = 0, π’Ÿπ‘‘ 𝛽 𝐼 = 0, π’Ÿπ‘‘ 𝛽 𝐺 = 0, π’Ÿπ‘‘ 𝛽 𝐻 = 0, π’Ÿπ‘‘ 𝛽 𝑅 = 0 … (6) The disease free equilibrium 𝐸0 = (Ξ›/𝑁, 0,0,0,0,0,0) The basic reproduction number of the disease is 𝑅0 which is defined as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2483 https://internationalpubls.com 𝑅0 = Ξ› (𝛿1 + 𝛿2 + 𝑑)(𝜏1 + 𝜏2 + 𝜏3 + 𝑑) If 𝑅0 > 1, then model contains a unique positive equilibrium point 𝐸1 = (𝑆′, 𝑉′, 𝐸′ , 𝐼′, 𝐺′, 𝐻′, 𝑅′) where, 𝑆′ = Ξ› 𝛾1 + 𝛾2 + 𝑑 𝑉′ = Ξ›{𝛾2(𝛿1 + 𝛿2 + 𝑑)(𝜏1 + 𝜏2 + 𝜏3 + 𝑑) + 𝜏2𝛾1𝛿1} (𝛾1 + 𝛾2 + 𝑑)(𝛿1 + 𝛿2 + 𝑑)(𝜏1 + 𝜏2 + 𝜏3 + 𝑑) 𝐸′ = 𝛾1Ξ› (𝛾1 + 𝛾2 + 𝑑)(𝛿1 + 𝛿2 + 𝑑) 𝐼′ = 𝛿1𝛾1Ξ› (𝛾1 + 𝛾2 + 𝑑)(𝛿1 + 𝛿2 + 𝑑)(𝜏1 + 𝜏2 + 𝜏3 + 𝑑) 𝐺′ = 𝛾1Ξ›{𝛿2(𝜏1 + 𝜏2 + 𝜏3 + 𝑑) + 𝜏3𝛿1} (𝛾1 + 𝛾2 + 𝑑)(𝛿1 + 𝛿2 + 𝑑)(𝜏1 + 𝜏2 + 𝜏3 + 𝑑) 𝐻′ = {𝛾1𝛿1(𝜁1 + 𝜁2 + 𝑑) + 𝜁1[𝛿2(𝜏1 + 𝜏2 + 𝜏3 + 𝑑) + 𝜏3𝛿1]} (𝛾1 + 𝛾2 + 𝑑)(𝛿1 + 𝛿2 + 𝑑)(𝜏1 + 𝜏2 + 𝜏3 + 𝑑) 𝑅′ = 𝜁1𝐻′ + 𝜁2𝐺′ 𝑑 If 𝑅0 < 1, then model is stable. The Jacobian matrix 𝐽(𝐸0) for system (4) for diseases free equilibrium point is as follows: 𝐽(𝐸0) = [ βˆ’(𝛾1 + 𝛾2 + 𝑑) 0 0 𝛾2 βˆ’π‘‘ 0 𝛾1 0 βˆ’(𝛿1 + 𝛿2 + 𝑑) 0 𝜏2 0 0 0 0 0 0 0 0 0 0 0 0 𝛿1 βˆ’(𝜏1 + 𝜏2 + 𝜏3 + 𝑑) 0 0 0 0 0 𝛿2 0 0 0 0 0 0 𝜏3 𝜏1 0 βˆ’(𝜁1 + 𝜁2 + 𝑑) 0 0 𝜁1 βˆ’(πœ‰ + 𝑑) 0 𝜁2 πœ‰ βˆ’π‘‘ ] The characteristic equation |𝐽(𝐸0) βˆ’ πœ†πΌ| = 0 |𝐽(𝐸0) βˆ’ πœ†πΌ| = (𝛾1 + 𝛾2 + 𝑑 + πœ†)(𝑑 + πœ†)(𝛿1 + 𝛿2 + 𝑑 + πœ†)(𝜏1 + 𝜏2 + 𝜏3 + 𝑑 + πœ†)(𝜁1 + 𝜁2 + 𝑑 + πœ†)(πœ‰ + 𝑑 + πœ†)(𝑑 + πœ†) = 0 The characteristic root for |𝐽(𝐸0) βˆ’ πœ†πΌ| = 0 are πœ†1 = βˆ’(𝛾1 + 𝛾2 + 𝑑), πœ†2 = βˆ’π‘‘, πœ†3 = βˆ’(𝛿1 + 𝛿2 + 𝑑), πœ†4 = βˆ’(𝜏1 + 𝜏2 + 𝜏3 + 𝑑), πœ†5 = βˆ’(𝜁1 + 𝜁2 + 𝑑), πœ†6 = βˆ’(πœ‰ + 𝑑), πœ†7 = βˆ’π‘‘ According to the generalized Routh-Hurwitz and Jacobian criterion when all the roots are negative then point 𝐸0 is disease free equilibrium point, which is stable. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2484 https://internationalpubls.com 4. Numerical methods : The generalized Adams Bashforth Moulton method can be adopted to the numerical solutions of system (4). Evaluate the following nonlinear fractional differential equation to obtain the algorithm's approximate solution: π’Ÿπ‘‘ 𝛽 𝑦(𝑑) = 𝑓(𝑑, 𝑦(𝑑)), 0 ≀ 𝑑 < 𝑏 < ∞ π‘¦β„Ž(0) = 𝑦0 β„Ž, β„Ž = 0,1,2, … , 𝑝 βˆ’ 1, where 𝑝 = [𝛽] } (7) This equation is equivalent to Volterra integral equation: 𝑦(𝑑) = βˆ‘π‘¦0 β„Ž π‘βˆ’1 β„Ž=0 π‘‘β„Ž β„Ž! + 1 𝛀(𝛽) ∫ (𝑑 βˆ’ 𝑠)π›½βˆ’1 𝑑 0 𝑓(𝑠, 𝑦(𝑠))𝑑𝑠 … (8) By applying the predictor-correctors scheme to the fractional order shigella epidemic model and set π‘˜ = 𝑇/𝑁, (8) can be discretized as follows: π‘†π‘š+1 = 𝑆0 + π‘˜π›½ 𝛀(𝛽 + 2) (Ξ› βˆ’ (𝛾1 + 𝛾2 + 𝑑)π‘†π‘š+1 π‘ž ) + π‘˜π›½ 𝛀(𝛽 + 2) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (Ξ› βˆ’ (𝛾1 + 𝛾2 + 𝑑)𝑆𝑖) π‘‰π‘š+1 = 𝑉0 + π‘˜π›½ 𝛀(𝛽 + 2) (𝛾3π‘†π‘š+1 π‘ž + 𝜏2πΌπ‘š+1 π‘ž βˆ’ π‘‘π‘‰π‘š+1 π‘ž ) + π‘˜π›½ 𝛀(𝛽 + 2) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (𝛾3𝑆𝑖 + 𝜏2𝐼𝑖 βˆ’ 𝑑𝑉𝑖) πΈπ‘š+1 = 𝐸0 + π‘˜π›½ 𝛀(𝛽 + 2) (𝛾1π‘†π‘š+1 π‘ž βˆ’ (𝛿1 + 𝛿2 + 𝑑)πΈπ‘š+1 π‘ž ) + π‘˜π›½ 𝛀(𝛽 + 2) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (𝛾1𝑆𝑖 βˆ’ (𝛿1 + 𝛿2 + 𝑑)𝐸𝑖) πΌπ‘š+1 = 𝐼0 + π‘˜π›½ 𝛀(𝛽 + 2) (𝛿1πΈπ‘š+1 π‘ž βˆ’ (𝜏1 + 𝜏2 + 𝜏3 + 𝑑)πΌπ‘š+1 π‘ž ) + π‘˜π›½ 𝛀(𝛽 + 2) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (𝛿1𝐸𝑖 βˆ’ (𝜏1 + 𝜏2 + 𝜏3 + 𝑑)𝐼𝑖) πΊπ‘š+1 = 𝐺0 + π‘˜π›½ 𝛀(𝛽 + 2) (𝛿2πΈπ‘š+1 π‘ž + 𝜏3πΌπ‘š+1 π‘ž βˆ’ (𝜁1 + 𝜁2 + 𝑑)πΊπ‘š+1 π‘ž ) + π‘˜π›½ 𝛀(𝛽 + 2) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (𝛿2𝐸𝑖 + 𝜏3𝐼𝑖 βˆ’ (𝜁1 + 𝜁2 + 𝑑)𝐺𝑖) π»π‘š+1 = 𝐻0 + π‘˜π›½ 𝛀(𝛽 + 2) (𝜏1πΌπ‘š+1 π‘ž + 𝜁1πΊπ‘š+1 π‘ž βˆ’ (πœ‰ + 𝑑)π»π‘š+1 π‘ž ) + π‘˜π›½ 𝛀(𝛽 + 2) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (𝜏1𝐼𝑖 + 𝜁1𝐺𝑖 βˆ’ (πœ‰ + 𝑑)𝐻𝑖) π‘…π‘š+1 = 𝑅0 + π‘˜π›½ 𝛀(𝛽 + 2) (πœ‰π»π‘š+1 π‘ž + 𝜁2πΊπ‘š+1 π‘ž βˆ’ π‘‘π‘…π‘š+1 π‘ž ) + π‘˜π›½ 𝛀(𝛽 + 2) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (πœ‰π»π‘– + 𝜁2𝐺𝑖 βˆ’ 𝑑𝑅𝑖) } (9) where, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2485 https://internationalpubls.com π‘†π‘š+1 π‘ž = 𝑆0 + 1 𝛀(𝛽) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (Ξ› βˆ’ (𝛾1 + 𝛾2 + 𝑑)𝑆𝑖) π‘‰π‘š+1 π‘ž = 𝑉0 + 1 𝛀(𝛽) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (𝛾3𝑆𝑖 + 𝜏2𝐼𝑖 βˆ’ 𝑑𝑉𝑖) πΈπ‘š+1 π‘ž = 𝐸0 + 1 𝛀(𝛽) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (𝛾1𝑆𝑖 βˆ’ (𝛿1 + 𝛿2 + 𝑑)𝐸𝑖) πΌπ‘š+1 π‘ž = 𝐼0 + 1 𝛀(𝛽) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (𝛿1𝐸𝑖 βˆ’ (𝜏1 + 𝜏2 + 𝜏3 + 𝑑)𝐼𝑖) πΊπ‘š+1 π‘ž = 𝐺0 + 1 𝛀(𝛽) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (𝛿2𝐸𝑖 + 𝜏3𝐼𝑖 βˆ’ (𝜁1 + 𝜁2 + 𝑑)𝐺𝑖) π»π‘š+1 π‘ž = 𝐻0 + 1 𝛀(𝛽) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (𝜏1𝐼𝑖 + 𝜁1𝐺𝑖 βˆ’ (πœ‰ + 𝑑)𝐻𝑖) π‘…π‘š+1 π‘ž = 𝑅0 + 1 𝛀(𝛽) βˆ‘π‘π‘–,π‘š+1 π‘š 𝑖=0 (πœ‰π»π‘– + 𝜁2𝐺𝑖 βˆ’ 𝑑𝑅𝑖) } (10) 𝑏𝑖,π‘š+1 = { 𝑛𝛽 βˆ’ (π‘š βˆ’ 𝛽)(π‘š + 1), 𝑖 = 0 (𝑛 βˆ’ 𝑖 + 2)𝛽+1 + (𝑛 βˆ’ 𝑖)𝛽+1, 1 ≀ 𝑖 ≀ π‘š βˆ’2(𝑛 βˆ’ 𝑖 + 2)𝛽+1, 𝑖 = π‘š + 1 𝑐𝑖,π‘š+1 = π‘˜π›½ 𝛽 ((𝑛 βˆ’ 𝑖 + 2)𝛽 + (𝑛 βˆ’ 𝑖)𝛽), 1 ≀ 𝑖 ≀ π‘š } (11) 5. Simulation : 𝐚𝐭 𝜷 = 𝟏 𝐚𝐭 𝜷 = 𝟎. πŸ— Figure 1 Figure 2 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2486 https://internationalpubls.com 𝐚𝐭 𝜷 = 𝟎. πŸ– Figure 3 In figure, blue line represents susceptible, orange dash line represents infectious and pink dash line represents hospitalization. In all the three figures, parameters used are same except 𝛽. Figure 1 shows sharp fall in the values of S in a relatively small period of time compared to figure 2 and figure 3. While, in figure 1 both values of I and V show striking rise during the period of 30 days, there is no notable changes in the value of I and V in figure 2 and 3. Nevertheless, the conversion of a classical model into a fractional model makes it highly perceptive to the order of differentiation 𝛽 i.e, a minute change can bring about great changes in the final result. It is evident from the numerical values from figure 2 and 3 that it depends continuously on the fractional derivative 𝛽. It is apparent that figure 2 and 3 are more realistic than figure 1 owing to the fact that it takes significant time for the rate of susceptibility or recovery to decrease or increase. 6. Reference : [1] Al-Sulami H., El-Shahed M., Nieto J. J. and Shammakh W., β€œOn Fractional order Dengue Epidemic Model”, Mathematical Problem in Engineering, 2014,pp-1-6. [2] Defterli O., β€œModeling 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. [3] Gorenflo R., Loutschko J. and Luchko Y., β€œComputation of the Mittage- Lefflerfunction 𝐸𝛼,𝛽(𝑧) and its derivatives”, [4] Hamdan N. I. and Kilicman A., β€œA fractional order SIR epidemic model for dengue transmission”, Chaos, Solitons and Fractals 114(2018), pp-55-62. [5] Kilbas A. A., Srivastava H. M. and Trujillo J. 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