# A Perturbation and Symmetry-Based Analysis of Mobile Malware Dynamics in Smartphone Networks
## 1 Introduction  
The increase in malware-infected applications, particularly on the Google Play Store, poses a significant threat, often enabling botnet-driven Distributed Denial-of-Service (DDoS) attacks. Understanding the dynamics of malware propagation in Android devices is crucial for developing effective prevention and mitigation strategies. This article introduces a mathematical model based on an epidemic approach, similar to classical models for biological diseases or computer worms, to describe the malware spread in a homogeneous network of Android devices [1]. Smartphone malware and its evolution, infection vectors, potential damage, and propagation models have been previously investigated, with an emphasis on emergingtrends [2; 3]. Malware is broadly defined as software designed to disrupt, damage, or gain unauthorized access to a computer system [4; 5], encompassing various forms like viruses, trojans, and bots, all characterized by malicious intent without user consent [6]. The frequency and sophistication of malware infections are escalating [7]. Cyber-criminals continually develop new methods to exploit vulnerabilities, including targeted attacks for extortion using stolen data or asset control [8]. These tactics involve social engineering [9], phishing [10], and zero-day exploits [11]. Malware is also evolving with capabilities like timed concealment [12], mutation, encryption [13], and Advanced Persistent Threats (APTs) [14], rendering basic antivirus systems ineffective. Furthermore, attackers are increasingly targeting Internet of Things (IoT) environments due to their often weaker security [15]. In response, individuals and organizations are prioritizing advanced security measures, user education, and specialized cybersecurity services [7]. However, proactive collaborative strategies between industry, government, and researchers remain underdeveloped [16]. The current malware campaigns focus on profit generation through fraud, spam, authentication theft, data exfiltration, and botnet rentals [17]. Android OS malware is a prevalent threat due to the operating system's open-source nature, widespread use, and the high volume of new applications, making it an easy target [18; 19]. In 2023, nearly 33.8 million mobile attacks were thwarted, with adware being the most common (40.8%) [19], underscoring the need for robust evolving security. Malware can extract user data, control the OS, send unauthorized SMS messages, make calls, or install paid apps for illicit profit [20], motivating further study of Android OS malware. Mathematical models of malware propagation help to estimate the potential impact of attacks, inform countermeasures, and support the development of effective defense strategies. By categorizing devices into compartments such as susceptible, infected, and recovered, ordinary differential equations (ODEs) can be used to describe the temporal evolution of these populations. Unlike biological epidemics--where model construction is often hindered by limited or noisy data [21; 22; 23]--the behavior of smartphone malware is often more observable and analyzable. Moreover, the finite number of smartphones introduces a natural logistic constraint on the spread. Given the complexity of constructing precise ODE models, we adopt a stylized yet mathematically tractable formulation. Despite its simplicity, this approach allows parameters to reflect the key technological aspects of the malware's behavior. Our proposed work presents a mathematical model (\(M_{\mathit{sipr}}\)) for mobile malware spread with the following main contributions:  
* Quarantine modeling: Includes a quarantine compartment and dual recovery paths to reflect real containment behavior. * Stability and \(R_{0}\) analysis: Derive the basic reproduction number and determine when malware dies out or persists. * Perturbation and sensitivity: Identify infection and quarantine rates as key parameters affecting outbreak size and timing.