Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2023 https://internationalpubls.com Stability of Malicious Object in SIEQAR Model Kaveri Kanchan Kumari 1 , Ashish Kumar Jha2 1Research Scholar, University Department of Mathematics, Ranchi University, Ranchi 2Associate Professor, , University Department of Mathematics, Ranchi University, Ranchi 1Kanchan_kaveri4@yahoo.in , 2jhaak25@gmail.com Article History: Received: 12-01-2025 Revised: 15-02-2025 Accepted: 01-03-2025 Abstract: The proposed model is SIEQAR ( Susceptible-Infected-Exposed-Quarantine-Antidotal- Recovered) which is extension of SAIR model. In this model we discussed Basic Reproduction number for MFE ( Malware Free Equilibrium) point. We discussed about Local stability at that point, also Endemic equilibrium point ids discussed. Keywords: Quarantine, Antidotal, Malware Free Equilibrium Μ§Endemic Equilibrium, reproduction number.. 1. Introduction: Computer virus is nothing but it is a code. Malware is computer program which destroy the important files from computer. Computer virus is similar like biological virus[1]. More than 1000 papers are discussed for many type of model [2-3]. Common models are Susceptible- Infectious- Susceptible (SIS) model[4-5], Susceptible- Infected- Recovered (SIR) model[6-7], Kaveri et.al.[8,9] discussed the different type of model. 2000 clasification: 92D30 2. Formulation of model: mailto:Kanchan_kaveri4@yahoo.in mailto:jhaak25@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2024 https://internationalpubls.com 𝑑𝑆 𝑑𝑑 = 𝑇 βˆ’ 𝛽𝑆𝐼 + 𝛼𝐸 + 𝛾𝑅 βˆ’ πœ‡π‘† 𝑑𝐼 𝑑𝑑 = 𝛽𝑆𝐼 + 𝛿𝐸 βˆ’ (πœ‡ + πœ“ + 𝜎)𝐼 𝑑𝐸 𝑑𝑑 = 𝐡 βˆ’ (𝛼 + πœ‡ + 𝛿 + πœƒ)𝐸 𝑑𝑄 𝑑𝑑 = 𝜎𝐼 βˆ’ (πœ‰ + πœ‡ + πœ‚)𝑄 𝑑𝐴 𝑑𝑑 = πœ‰π‘„ + πœƒπΈ βˆ’ (πœ‡ + πœ™)𝐴 𝑑𝑅 𝑑𝑑 = πœ‚π‘„ + πœ“πΌ + πœ™π΄ βˆ’ (πœ‡ + 𝛾)𝑅 } … (i) All parameters are positive. i.e., 𝑆 β‰₯ 0, 𝐼 β‰₯ 0, 𝐸 β‰₯ 0, 𝑄 β‰₯ 0, 𝐴 β‰₯ 0, 𝑅 β‰₯ 0. ∴ total population 𝑁 = 𝑆 + 𝐼 + 𝐸 + 𝑄 + 𝐴 + 𝑅. Then, 𝑑𝑁 𝑑𝑑 = 𝑑𝑆 𝑑𝑑 + 𝑑𝐼 𝑑𝑑 + 𝑑𝐸 𝑑𝑑 + 𝑑𝑄 𝑑𝑑 + 𝑑𝐴 𝑑𝑑 + 𝑑𝑅 𝑑𝑑 . β‡’ 𝑑𝑁 𝑑𝑑 = 𝑇 + 𝐡 βˆ’ πœ‡π‘ Solving lim π‘‘β†’βˆž sup(𝑆 + 𝐼 + 𝐸 + 𝑄 + 𝐴 + 𝑅) ≀ 𝑇 + 𝐡 πœ‡ Hence the feasible region for system (1) is, Ξ© = {(𝑆, 𝐼, 𝐸, 𝑄, 𝐴, 𝑅): 𝑆, 𝐼, 𝐸, 𝑄, 𝐴, 𝑅 β‰₯ 0; 𝑆 + 𝐼 + 𝐸 + 𝑄 + 𝐴 + 𝑅 ≀ 𝑇 + 𝐡 πœ‡ } The Malware Free Equilibrium (MFE) of system (1) is denoted by πΈπ‘œ 𝑖. 𝑒. , πΈπ‘œ = (𝑆, 𝐼, 𝐸, 𝑄, 𝐴, 𝑅) = ( 𝑇 + 𝐡 πœ‡ , 0, 0, 0, 0, 0) Description of parameters used in above ODE system. 𝑆 is Number of susceptible nodes. 𝐼 is Number of infected nodes. 𝐸 is Number of exposed nodes. 𝑄 is Number of quarantined nodes 𝐴 is Number of antidotal nodes 𝑅 is Number of recovered nodes 𝛽 is Coefficient of transmission for susceptible individuals 𝜎 is rate of quarantine for infectious individuals Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2025 https://internationalpubls.com πœ“ is rate of recovery for infectious individuals πœ‰ is rate of antidotal for quarantine individuals πœ™ is rate of recovery for antidotal individuals πœ‚ is rate of recovery for quarantine individuals πœƒ is rate of antidotal for exposed individuals 𝛿 is rate of infective for exposed individuals 𝛾 is rate of recovered individuals to susceptible 𝛼 is rate of exposed individuals to susceptible πœ‡ is death rate due to other than the malicious objects Basic Reproduction Number: For convenient we take four classes for calculation of basic reproduction number, from ODE system (1). 𝑑𝐼 𝑑𝑑 = 𝛽𝑆𝐼 + 𝛿𝐸 βˆ’ (πœ‡ + πœ“ + 𝜎)𝐼 𝑑𝐸 𝑑𝑑 = 𝐡 βˆ’ (𝛼 + πœ‡ + 𝛿 + πœƒ)𝐸 𝑑𝑄 𝑑𝑑 = 𝜎𝐼 βˆ’ (πœ‰ + πœ‡ + πœ‚)𝑄 𝑑𝐴 𝑑𝑑 = πœ‰π‘„ + πœƒπΈ βˆ’ (πœ‡ + πœ™)𝐴 By using next generation matrix 𝑑π‘₯ 𝑑𝑑 = 𝑓 βˆ’ 𝑣. Where, 𝑓 = [ 𝛽𝑆𝐼 0 0 0 ] and 𝑣 = [ (πœ‡ + πœ“ + 𝜎)𝐼 βˆ’ 𝛿𝐸 (𝛼 + πœ‡ + 𝛿 + πœƒ)𝐸 βˆ’ 𝐡 (πœ‰ + πœ‡ + πœ‚)𝑄 βˆ’ 𝜎𝐼 (πœ‡ + πœ™)𝐴 βˆ’ πœ‰π‘„ βˆ’ πœƒπΈ ] 𝐹 is Jacobian of 𝑓 at 𝑀𝐹𝐸 is [ 𝛽 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] 𝑉 is Jacobian of 𝑣 at 𝑀𝐹𝐸 is [ πœ‡ + πœ“ + 𝜎 βˆ’π›Ώ 0 𝛼 + πœ‡ + 𝛿 + πœƒ 0 0 0 0 βˆ’πœŽ 0 0 βˆ’πœƒ πœ‰ + πœ‡ + πœ‚ 0 βˆ’πœ‰ πœ‡ + πœ™ ] Then, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2026 https://internationalpubls.com π‘‰βˆ’1 = [ 1 πœ‡ + πœ“ + 𝜎 𝛿 (πœ‡ + πœ“ + 𝜎)(𝛼 + πœ‡ + 𝛿 + πœƒ) 0 1 (𝛼 + πœ‡ + 𝛿 + πœƒ) 0 0 0 0 𝜎 (πœ‡ + πœ“ + 𝜎)(πœ‰ + πœ‡ + πœ‚) πœŽπ›Ώ (πœ‡ + πœ“ + 𝜎)(𝛼 + πœ‡ + 𝛿 + πœƒ)(πœ‰ + πœ‡ + πœ‚) πœŽπœ‰ (πœ‡ + πœ“ + 𝜎)(πœ‰ + πœ‡ + πœ‚)(πœ‡ + πœ™) (πœ‡ + πœ“ + 𝜎)πœƒ(πœ‰ + πœ‡ + πœ‚) + π›ΏπœŽπœ‰ (πœ‡ + πœ“ + 𝜎)(𝛼 + πœ‡ + 𝛿 + πœƒ)(πœ‰ + πœ‡ + πœ‚)(πœ‡ + πœ™) 1 (πœ‰ + πœ‡ + πœ‚) 0 πœ‰ (πœ‰ + πœ‡ + πœ‚)(πœ‡ + πœ™) 1 (πœ‡ + πœ™)] πΉπ‘‰βˆ’1 = [ 𝛽 πœ‡ + πœ“ + 𝜎 𝛽𝛿 (πœ‡ + πœ“ + 𝜎)(𝛼 + πœ‡ + 𝛿 + πœƒ) 0 0 0 0 0 0 0 0 0 0 0 0 0 0] The dominant eigen values 𝑅0 = 𝛽 (πœ‡ + πœ“ + 𝜎) 3. Local Stability at MFE (Malware Free Equilibrium ) Point: For equilibrium points in the steady state of ODE system (i). 𝑇 βˆ’ 𝛽𝑆𝐼 + 𝛼𝐸 + 𝛾𝑅 βˆ’ πœ‡π‘† = 0 𝛽𝑆𝐼 + 𝛿𝐸 βˆ’ (πœ‡ + πœ“ + 𝜎)𝐼 = 0 𝐡 βˆ’ (𝛼 + πœ‡ + 𝛿 + πœƒ)𝐸 = 0 𝜎𝐼 βˆ’ (πœ‰ + πœ‡ + πœ‚)𝑄 = 0 πœ‰π‘„ + πœƒπΈ βˆ’ (πœ‡ + πœ™)𝐴 = 0 πœ‚π‘„ + πœ“πΌ + πœ™π΄ βˆ’ (πœ‡ + 𝛾)𝑅 = 0 } … (ii) Theorem 1: The πΈβˆ—of system (i) is locally asymptotically stable (LAS) if 𝑅0 < 1. Proof: The Jacobian of system (i) 𝐽1 = [ βˆ’πœ‡ 0 0 0 0 0 0 π›½π‘†βˆ— βˆ’ (πœ‡ + πœ“ + 𝜎) 0 𝜎 0 πœ“ 0 𝛿 βˆ’(𝛼 + πœ‡ + 𝛿 + πœƒ) 0 πœƒ 0 0 0 0 βˆ’(πœ‰ + πœ‡ + πœ‚) πœ‰ πœ‚ 0 0 0 0 βˆ’(πœ‡ + πœ™) πœ™ 𝛾 0 0 0 0 βˆ’(πœ‡ + 𝛾)] The characteristic equation of 𝐽1 is |𝐽1 βˆ’ πœ†πΌ| = 0 β‡’ | | βˆ’(πœ‡ + πœ†) 0 0 0 0 0 0 π›½π‘†βˆ— βˆ’ (πœ‡ + πœ“ + 𝜎 + πœ†) 0 𝜎 0 πœ“ 0 𝛿 βˆ’(𝛼 + πœ‡ + 𝛿 + πœƒ + πœ†) 0 πœƒ 0 0 0 0 βˆ’(πœ‰ + πœ‡ + πœ‚ + πœ†) πœ‰ πœ‚ 0 0 0 0 βˆ’(πœ‡ + πœ™ + πœ†) πœ™ 𝛾 0 0 0 0 βˆ’(πœ‡ + 𝛾 + πœ†) | | = 0 The eigen vlues of|𝐽1 βˆ’ πœ†πΌ| = 0 are Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2027 https://internationalpubls.com πœ†1 = βˆ’πœ‡ < 0 πœ†2 = 𝛽𝑆 βˆ— βˆ’ (πœ‡ + πœ“ + 𝜎) πœ†3 = βˆ’(𝛼 + πœ‡ + 𝛿 + πœƒ) < 0 πœ†4 = βˆ’(πœ‰ + πœ‡ + πœ‚) < 0 πœ†5 = βˆ’(πœ‡ + πœ™) < 0 πœ†6 = βˆ’(πœ‡ + 𝛾) < 0 Clearly it has five negative real roots and the root πœ†2 = 𝛽𝑆 βˆ— βˆ’ (πœ‡ + πœ“ + 𝜎) < 0 If π›½π‘†βˆ— < (πœ‡ + πœ“ + 𝜎). Clearly all roots are negative real roots so system (i) is Malware free equilibrium and is localy asymptotically stable. 4. Endemic Equilibrium point : The endemic Equilibrium (EE) Point π‘ƒβˆ—(π‘†βˆ—, πΌβˆ—, πΈβˆ—, π‘„βˆ—, π΄βˆ—, π‘…βˆ—), which can be calculated by using system (ii) Thus, π‘†βˆ— = {𝑇(𝛼 + πœ‡ + 𝛿 + πœƒ) + 𝛼𝛽}(πœ‰ + πœ‡ + πœ‚)(πœ‡ + πœ™)(πœ‡ + 𝛾) +𝛾[πΌβˆ—(𝛼 + πœ‡ + 𝛿 + πœƒ)[{πœ‚πœŽ + πœ“(πœ‰ + πœ‡ + πœ‚)}(πœ‡ + πœ™) + πœ™πœ‰πœŽ] + πœƒπ΅(πœ‰ + πœ‡ + πœ‚)] (𝛼 + πœ‡ + 𝛿 + πœƒ)(πœ‰ + πœ‡ + πœ‚)(πœ‡ + πœ™)(πœ‡ + 𝛾)(π΅πΌβˆ— + πœ‡) , πΌβˆ— = 𝛿𝐸 (πœ‡ + πœ“ + 𝜎 βˆ’ π›½π‘†βˆ—) , πΈβˆ— = 𝐡 𝛼 + πœ‡ + 𝛿 + πœƒ , π‘„βˆ— = πœŽπΌβˆ— (πœ‰ + πœ‡ + πœ‚) , π΄βˆ— = πœ‰πœŽπΌβˆ— + πœƒπ΅(πœ‰ + πœ‡ + πœ‚) (𝛼 + πœ‡ + 𝛿 + πœƒ)(πœ‰ + πœ‡ + πœ‚)(πœ‡ + πœ™) , π‘…βˆ— = πΌβˆ—(𝛼 + πœ‡ + 𝛿 + πœƒ)[{πœ‚πœŽ + πœ“(πœ‰ + πœ‡ + πœ‚)}(πœ‡ + πœ™) + πœ™πœ‰πœŽ] + πœƒπ΅(πœ‰ + πœ‡ + πœ‚) (𝛼 + πœ‡ + 𝛿 + πœƒ)(πœ‰ + πœ‡ + πœ‚)(πœ‡ + πœ™)(πœ‡ + 𝛾) From above calculation, the EE state exist. 5. Conclusion:In above paper, the proposed model is SIEQAR ( Susceptible-Infected-Exposed- Quarantine-Antidotal-Recovered) which is extension of SAIR model.This model is useful for getting Malware Free Equilibrium (MFE) point. We discussed about Local stability of MFE point and Endemic Equilibrium (EE) point. locally asymptotically stable (LAS) if 𝑅0 < 1 MFE point. In 2-dimenssion and 3- dimension several graphs of parameters are discussed. The above paper is useful for controlling the virus in all computer network. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 9s (2025) 2028 https://internationalpubls.com Refrences: [1]Han X, Tan Q. Dynamical behavior of computer virus on internet. Appl Math comput 2010; 217 (6); 2520(6). [2] Serazzi G, Zanero s. Computer virus propagation models. In International workshop on modeling, analysis and simulation of computer and telecommunication systems. Berlin, Heidelberg: springer, 2003, p. 26-50 [3]Yaun H, Chen G. 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