<?xml version="1.0" encoding="UTF-8"?><!DOCTYPE article PUBLIC "-//NLM//DTD Journal Publishing DTD v3.0 20080202//EN" "http://dtd.nlm.nih.gov/publishing/3.0/journalpublishing3.dtd">
<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" dtd-version="3.0" xml:lang="en" article-type="research article">
 <front>
  <journal-meta>
   <journal-id journal-id-type="publisher-id">
    jis
   </journal-id>
   <journal-title-group>
    <journal-title>
     Journal of Information Security
    </journal-title>
   </journal-title-group>
   <issn pub-type="epub">
    2153-1234
   </issn>
   <issn publication-format="print">
    2153-1242
   </issn>
   <publisher>
    <publisher-name>
     Scientific Research Publishing
    </publisher-name>
   </publisher>
  </journal-meta>
  <article-meta>
   <article-id pub-id-type="doi">
    10.4236/jis.2024.154024
   </article-id>
   <article-id pub-id-type="publisher-id">
    jis-135045
   </article-id>
   <article-categories>
    <subj-group subj-group-type="heading">
     <subject>
      Articles
     </subject>
    </subj-group>
    <subj-group subj-group-type="Discipline-v2">
     <subject>
      Computer Science 
     </subject>
     <subject>
       Communications
     </subject>
    </subj-group>
   </article-categories>
   <title-group>
    An Agent Based Model for Ransomware Detection and Mitigation in a Cloud System
   </title-group>
   <contrib-group>
    <contrib contrib-type="author" xlink:type="simple">
     <name name-style="western">
      <surname>
       N’golo
      </surname>
      <given-names>
       Konate
      </given-names>
     </name>
    </contrib>
    <contrib contrib-type="author" xlink:type="simple">
     <name name-style="western">
      <surname>
       Tenan
      </surname>
      <given-names>
       Yeo
      </given-names>
     </name>
    </contrib>
   </contrib-group> 
   <aff id="affnull">
    <addr-line>
     aUFR Mathematiques et Informatiques, Université Felix-Houphouet Boigny, Abidjan, Cote d’Ivoire
    </addr-line> 
   </aff> 
   <pub-date pub-type="epub">
    <day>
     01
    </day> 
    <month>
     08
    </month>
    <year>
     2024
    </year>
   </pub-date> 
   <volume>
    15
   </volume> 
   <issue>
    04
   </issue>
   <fpage>
    419
   </fpage>
   <lpage>
    432
   </lpage>
   <history>
    <date date-type="received">
     <day>
      29,
     </day>
     <month>
      May
     </month>
     <year>
      2024
     </year>
    </date>
    <date date-type="published">
     <day>
      29,
     </day>
     <month>
      May
     </month>
     <year>
      2024
     </year> 
    </date> 
    <date date-type="accepted">
     <day>
      29,
     </day>
     <month>
      July
     </month>
     <year>
      2024
     </year> 
    </date>
   </history>
   <permissions>
    <copyright-statement>
     © Copyright 2014 by authors and Scientific Research Publishing Inc. 
    </copyright-statement>
    <copyright-year>
     2014
    </copyright-year>
    <license>
     <license-p>
      This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
     </license-p>
    </license>
   </permissions>
   <abstract>
    The increasing trend toward dematerialization and digitalization has prompted a surge in the adoption of IT service providers, offering cost-effective alternatives to traditional local services. Consequently, cloud services have become prevalent across various industries. While these services offer undeniable benefits, they face significant threats, particularly concerning the sensitivity of the data they handle. Many existing mathematical models struggle to accurately depict the complex scenarios of cloud systems. In response to this challenge, this paper proposes a behavioral model for ransomware propagation within such environments. In this model, each component of the environment is defined as an agent responsible for monitoring the propagation of malware. Given the distinct characteristics and criticality of these agents, the impact of malware can vary significantly. Scenario attacks are constructed based on real-world vulnerabilities documented in the Common Vulnerabilities and Exposures (CVEs) through the National Vulnerability Database. Defender actions are guided by an Intrusion Detection System (IDS) guideline. This research aims to provide a comprehensive framework for understanding and addressing ransomware threats in cloud systems. By leveraging an agent- based approach and real-world vulnerability data, our model offers valuable insights into detection and mitigation strategies for safeguarding sensitive cloud-based assets.
   </abstract>
   <kwd-group> 
    <kwd>
     Cloud Computing
    </kwd> 
    <kwd>
      Information Security
    </kwd> 
    <kwd>
      Multi-Agent System
    </kwd> 
    <kwd>
      IaaS
    </kwd> 
    <kwd>
      Malware Propagation
    </kwd>
   </kwd-group>
  </article-meta>
 </front>
 <body>
  <sec id="s1">
   <title>1. Introduction</title>
   <p>From the basic software to the full infrastructure, cloud systems provide services at different levels to support organizational processes. Cloud computing enables organizations to scale their IT resources up or down quickly and easily, without the need for costly hardware upgrades. This can help organizations to respond more quickly to changing business needs and market conditions <xref ref-type="bibr" rid="scirp.135045-1">
     [1]
    </xref>. Unlike local infrastructures which require a major investment, organizations only pay for the resources needed in a cloud system. However depending on the type of cloud, providers in either organization should invest a lot in security. According to <xref ref-type="bibr" rid="scirp.135045-2">
     [2]
    </xref>, there are four types of cloud computing, as seen in <xref ref-type="fig" rid="fig1">
     Figure 1
    </xref>, which are used in different fields of life with specific rules and respective specifications. Those four types rely on different types of layers and have specific roles:</p>
   <p>According to Markets and Markets, global public cloud services have a compound annual growth rate of 17.5%. Therefore the cyber security landscape is characterized by the regular emergence of new types of cyber threats and trends which constantly sophisticated and diverse for both individuals and organizations.</p>
   <fig id="fig1" position="float">
    <label>Figure 1</label>
    <caption>
     <title>Figure 1. Cloud system environment.</title>
    </caption>
    <graphic mimetype="image" position="float" xlink:type="simple" xlink:href="https://html.scirp.org/file/7801030-rId13.jpeg?20240801083346" />
   </fig>
   <p>While traditional attacks relied on signature-based detection, which made them easier to identify and counter, newer attacks use artificial intelligence (AI) characteristics like machine learning (ML) and deep learning to make malware more persuasive and easy to spread. With the number of services connected, the attacker can weaponize cyber AI for cyber attacks. <xref ref-type="bibr" rid="scirp.135045-3">
     [3]
    </xref> define weaponized AI as malicious AI algorithms that can degrade the performance and disrupt the normal functions of benign AI algorithms, while providing technological edge attack scenarios in both cyberspace and physical spaces. Fighting cyber crimes require now a more comprehensive and safer approach <xref ref-type="bibr" rid="scirp.135045-1">
     [1]
    </xref>. New mathematical models and cyber defense tools are now oriented towards mathematical models <xref ref-type="bibr" rid="scirp.135045-4">
     [4]
    </xref>, and deep learning <xref ref-type="bibr" rid="scirp.135045-5">
     [5]
    </xref> <xref ref-type="bibr" rid="scirp.135045-6">
     [6]
    </xref>. This paper is organized as follows. Section II discusses some essential concepts related to the paper as the new trends in cloud computing and the related threats. Section III will be dedicated to the multi-agents model formulation and the rule definitions. Discussions and limitations of the current model are presented in section VI. Finally we conclude the paper in section V.</p>
  </sec><sec id="s2">
   <title>2. Literature Review</title>
   <sec id="s2_1">
    <title>2.1. Cloud Based System</title>
    <p>Due to the information systems heterogeneity, cloud systems involve now every component such as end-users, networks, access management, and infrastructures. Therefore before diving into security issues, we need to understand cloud-based systems new trends. The cloud computing services as represented in <xref ref-type="fig" rid="fig2">
      Figure 2
     </xref> have been offered into three common service models including Software-as-a-Service (SaaS), Platform-as-a-Service (PaaS), and Infrastructure-as-a-Service (IaaS) <xref ref-type="bibr" rid="scirp.135045-7">
      [7]
     </xref>:</p>
    <fig id="fig2" position="float">
     <label>Figure 2</label>
     <caption>
      <title>Figure 2. Cloud based architecture.</title>
     </caption>
     <graphic mimetype="image" position="float" xlink:type="simple" xlink:href="https://html.scirp.org/file/7801030-rId14.jpeg?20240801083347" />
    </fig>
    <p>Since the malware is more likely to disrupt IaaS, let’s dive into its new trends.</p>
    <p>The user in an IaaS is in charge of managing the operating system and software applications, while the underlying network in the cloud infrastructure service is controlled by the cloud service provider <xref ref-type="bibr" rid="scirp.135045-9">
      [9]
     </xref>. As seen in <xref ref-type="fig" rid="fig3">
      Figure 3
     </xref>, IaaS network users can install multiple operating systems on the virtual machine images.</p>
    <fig id="fig3" position="float">
     <label>Figure 3</label>
     <caption>
      <title>Figure 3. IasS network.</title>
     </caption>
     <graphic mimetype="image" position="float" xlink:type="simple" xlink:href="https://html.scirp.org/file/7801030-rId15.jpeg?20240801083347" />
    </fig>
    <p>This horizontal view of the cloud system can be segmented into five essential characteristics <xref ref-type="bibr" rid="scirp.135045-10">
      [10]
     </xref>:</p>
    <p>The connection between the user and IaaS is done through either virtual private servers, a storage or a network. Then the request is sent to the real server through the virtual services resources.</p>
    <p>IaaS security model: The security model in IaaS should take into account the three layers in a cloud architecture. Therefore, many components are used to monitor the environment <xref ref-type="bibr" rid="scirp.135045-11">
      [11]
     </xref>.</p>
   </sec>
   <sec id="s2_2">
    <title>2.2. Cyber Threats in Cloud System</title>
    <p>Since IaaS offers computing capabilities and essential storage as standardized services across the network <xref ref-type="bibr" rid="scirp.135045-12">
      [12]
     </xref>, the infrastructures face threats related to the underlying protocols. Therefore, appropriate safety measures should be taken care of.</p>
    <p>A threat is a process whereby an intruder gathers, identifies and determines the risk associated with each area. Each threat identified during this process is analyzed in the exploit database. The threat faced by the cloud environment can emerge either on the user side or the provider side.</p>
    <p>Since IaaS inherits data security’s features in the SaaS layer <xref ref-type="bibr" rid="scirp.135045-13">
      [13]
     </xref> and Security concerns associated with SaaS layer are almost data centric. Some concerns about data security are:</p>
    <p>The last question is a fundamental one, because good safety monitoring and control considerably enhance the third parties’ confidence. While discussing security issues, it’s important to note the impact of AI on cyber threats. Machine learning and artificial intelligence (AI) can be used to automate many cybersecurity tasks, such as intrusion detection, malware analysis and vulnerability assessment <xref ref-type="bibr" rid="scirp.135045-14">
      [14]
     </xref>. Since existing cyber defense infrastructures are becoming inadequate to address the increasing speed, and complex decision logic of AI-driven attacks <xref ref-type="bibr" rid="scirp.135045-15">
      [15]
     </xref>. We will introduce the AI driven propagation metrics to see their impact on the global infrastructure.</p>
   </sec>
   <sec id="s2_3">
    <title>2.3. Mathematical Model in Information Security</title>
    <p>Mathematical model to perform information system issues is a long problem discussed in the literature under different headings.</p>
   </sec>
  </sec><sec id="s3">
   <title>3. Methods</title>
   <sec id="s3_1">
    <title>3.1. Cloud Architecture</title>
    <p>As represented in <xref ref-type="fig" rid="fig3">
      Figure 3
     </xref>, Cloud physical architecture can be subdivided into provider and tenant parts.</p>
    <p>At the provider side: The hypervisor allows each machine to work independently regarding the CPU, memory and NIC. An intruder who targets the hypervisor may be able to corrupt any resource. The runtime space is listed below <xref ref-type="bibr" rid="scirp.135045-11">
      [11]
     </xref>:</p>
    <p>On the user side: Hypervisor provides a resource isolation to the tenant. Therefore a multi-tenancy occurs. Though it increases the architecture performance, it increases the probability that a legal and malicious user can be located in the same physical machine.</p>
   </sec>
   <sec id="s3_2">
    <title>3.2. Security Model</title>
    <p>IaaS particularly among clouds offers services that make it difficult to a global model for all the architecture. Therefore, the Model for IaaS Security and Privacy (MISP) is the one retained for this paper <xref ref-type="bibr" rid="scirp.135045-16">
      [16]
     </xref>. As represented in <xref ref-type="fig" rid="fig4">
      Figure 4
     </xref>, the security model is organized in cubical form with three planes defined as shown in <xref ref-type="fig" rid="fig4">
      Figure 4
     </xref>.</p>
    <fig id="fig4" position="float">
     <label>Figure 4</label>
     <caption>
      <title>Figure 4. IaaS’s security model.</title>
     </caption>
     <graphic mimetype="image" position="float" xlink:type="simple" xlink:href="https://html.scirp.org/file/7801030-rId16.jpeg?20240801083349" />
    </fig>
    <p>The first plan exhibits aspects of Infrastructure as a Service (IaaS), involving the cloud computing user and the Cloud Service Provider (CSP) as typical stakeholders. They typically collaborate to uphold the security and confidentiality of the infrastructure model.</p>
   </sec>
   <sec id="s3_3">
    <title>3.3. Multi Agent System Proposed</title>
    <p>The agents involved in a system are:</p>
    <p>The critical component of an IaaS cloud architecture is the cloud OS, which manages the physical and virtual structures and controls the supply of virtual resources in line with the needs of the user goods and services <xref ref-type="bibr" rid="scirp.135045-17">
      [17]
     </xref>. However, the OS cannot be taken in the context of a cloud system without the VM.</p>
    <p>The Virtual Machine Agent (VMA) provides virtualized computing resources on-demand. It’s in charge of running applications and services, allocating and managing scalability and flexibility in resources allocation, and ensuring isolation and security of a virtualized environment.</p>
    <p>The Computer Agents (CA) executes computational tasks and processing data. The computer agent interacts with other agents in the cloud system. Since it hosts applications and launches tasks, it can be the attack source.</p>
    <p>The Mobiles Phones Agent (MPA) accesses cloud-based applications and data remotely, ensuring the security and privacy of data transmitted to and from the cloud.</p>
    <p>Although IaaS security is an ongoing process, its implementation must correspond to the architecture and security policy. However, the security model retains as much as generic to be implemented in different environments.</p>
    <p>The IDS Agent (IDSA) analysis workflow follows those specifications:</p>
   </sec>
   <sec id="s3_4">
    <title>3.4. Dynamic Equation of VM</title>
    <p>Each agent 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mi>
          A 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
      </mrow> 
     </math> is represented by a state vector 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mi>
          x 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
      </mrow> 
     </math> where 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <mi>
         i 
       </mi> 
       <mo>
         ∈ 
       </mo> 
       <mn>
         1 
       </mn> 
       <mo>
         , 
       </mo> 
       <mo>
         ⋯ 
       </mo> 
       <mn>
         , 
       </mn> 
       <mi>
         N 
       </mi> 
      </mrow> 
     </math> and t represent the time. The dynamics of resources allocations of 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <mi>
         V 
       </mi> 
       <msub> 
        <mi>
          M 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
      </mrow> 
     </math> are based on its own resource demand and the resource demands of other agents (VM, CA and PA). The VMi is therefore formulated as:</p>
    <p>
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mover accent="true"> 
         <mi>
           x 
         </mi> 
         <mo>
           ˙ 
         </mo> 
        </mover> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
       <mo>
         = 
       </mo> 
       <mfrac> 
        <mn>
          1 
        </mn> 
        <mrow> 
         <msub> 
          <mi>
            m 
          </mi> 
          <mi>
            i 
          </mi> 
         </msub> 
        </mrow> 
       </mfrac> 
       <mstyle displaystyle="true"> 
        <msub> 
         <mo>
           ∑ 
         </mo> 
         <mrow> 
          <mi>
            j 
          </mi> 
          <mo>
            ≠ 
          </mo> 
          <mi>
            i 
          </mi> 
         </mrow> 
        </msub> 
        <mrow> 
         <msub> 
          <mi>
            α 
          </mi> 
          <mrow> 
           <mi>
             i 
           </mi> 
           <mi>
             j 
           </mi> 
          </mrow> 
         </msub> 
         <mrow> 
          <mo>
            ( 
          </mo> 
          <mrow> 
           <msub> 
            <mi>
              x 
            </mi> 
            <mi>
              i 
            </mi> 
           </msub> 
           <mrow> 
            <mo>
              ( 
            </mo> 
            <mi>
              t 
            </mi> 
            <mo>
              ) 
            </mo> 
           </mrow> 
           <mo>
             − 
           </mo> 
           <msub> 
            <mi>
              x 
            </mi> 
            <mi>
              j 
            </mi> 
           </msub> 
           <mrow> 
            <mo>
              ( 
            </mo> 
            <mi>
              t 
            </mi> 
            <mo>
              ) 
            </mo> 
           </mrow> 
          </mrow> 
          <mo>
            ) 
          </mo> 
         </mrow> 
        </mrow> 
       </mstyle> 
      </mrow> 
     </math> (1)</p>
    <p>where</p>
    <p>Let’s denote 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mi>
          μ 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
      </mrow> 
     </math>, the impact of malware on 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <mi>
         V 
       </mi> 
       <msub> 
        <mi>
          M 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
      </mrow> 
     </math> at time t. The dynamic equation of 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <mi>
         V 
       </mi> 
       <msub> 
        <mi>
          M 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
      </mrow> 
     </math> can therefore be expressed as</p>
    <p>
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mover accent="true"> 
         <mi>
           x 
         </mi> 
         <mo>
           ˙ 
         </mo> 
        </mover> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
       <mo>
         = 
       </mo> 
       <mfrac> 
        <mn>
          1 
        </mn> 
        <mrow> 
         <msub> 
          <mi>
            m 
          </mi> 
          <mi>
            i 
          </mi> 
         </msub> 
        </mrow> 
       </mfrac> 
       <mstyle displaystyle="true"> 
        <msub> 
         <mo>
           ∑ 
         </mo> 
         <mrow> 
          <mi>
            j 
          </mi> 
          <mo>
            ≠ 
          </mo> 
          <mi>
            i 
          </mi> 
         </mrow> 
        </msub> 
        <mrow> 
         <msub> 
          <mi>
            α 
          </mi> 
          <mrow> 
           <mi>
             i 
           </mi> 
           <mi>
             j 
           </mi> 
          </mrow> 
         </msub> 
         <mrow> 
          <mo>
            ( 
          </mo> 
          <mrow> 
           <msub> 
            <mi>
              x 
            </mi> 
            <mi>
              i 
            </mi> 
           </msub> 
           <mrow> 
            <mo>
              ( 
            </mo> 
            <mi>
              t 
            </mi> 
            <mo>
              ) 
            </mo> 
           </mrow> 
           <mo>
             − 
           </mo> 
           <msub> 
            <mi>
              x 
            </mi> 
            <mi>
              j 
            </mi> 
           </msub> 
           <mrow> 
            <mo>
              ( 
            </mo> 
            <mi>
              t 
            </mi> 
            <mo>
              ) 
            </mo> 
           </mrow> 
          </mrow> 
          <mo>
            ) 
          </mo> 
         </mrow> 
        </mrow> 
       </mstyle> 
       <mo>
         + 
       </mo> 
       <msub> 
        <mi>
          μ 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
      </mrow> 
     </math> (2)</p>
    <p>Since the malware considered will be used in many scenarios, it’s formulated as:</p>
    <p>
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mi>
          μ 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
       <mo>
         = 
       </mo> 
       <msub> 
        <mi>
          β 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mo>
         ⋅ 
       </mo> 
       <msub> 
        <mi>
          δ 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
      </mrow> 
     </math> (3)</p>
    <p>where:</p>
    <p>
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mi>
          δ 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
       <mo>
         = 
       </mo> 
       <mrow> 
        <mo>
          { 
        </mo> 
        <mrow> 
         <mtable columnalign="left"> 
          <mtr columnalign="left"> 
           <mtd columnalign="left"> 
            <mrow> 
             <mo>
               + 
             </mo> 
             <mi>
               ∞ 
             </mi> 
            </mrow> 
           </mtd> 
           <mtd columnalign="left"> 
            <mrow> 
             <mi>
               t 
             </mi> 
             <mo>
               = 
             </mo> 
             <mn>
               0 
             </mn> 
             <mtext>
                 
             </mtext> 
             <mrow> 
              <mo>
                ( 
              </mo> 
              <mrow> 
               <mtext>
                 infected 
               </mtext> 
              </mrow> 
              <mo>
                ) 
              </mo> 
             </mrow> 
             <mn>
               , 
             </mn> 
            </mrow> 
           </mtd> 
          </mtr> 
          <mtr columnalign="left"> 
           <mtd columnalign="left"> 
            <mn>
              0 
            </mn> 
           </mtd> 
           <mtd columnalign="left"> 
            <mrow> 
             <mi>
               t 
             </mi> 
             <mo>
               ≠ 
             </mo> 
             <msub> 
              <mi>
                t 
              </mi> 
              <mn>
                0 
              </mn> 
             </msub> 
             <mrow> 
              <mo>
                ( 
              </mo> 
              <mrow> 
               <mtext>
                 not 
               </mtext> 
               <mtext>
                   
               </mtext> 
               <mtext>
                 infected 
               </mtext> 
              </mrow> 
              <mo>
                ) 
              </mo> 
             </mrow> 
            </mrow> 
           </mtd> 
          </mtr> 
         </mtable> 
        </mrow> 
       </mrow> 
      </mrow> 
     </math> (4)</p>
    <p>Therefore 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mi>
          δ 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
       <mo>
         = 
       </mo> 
       <msub> 
        <mi>
          δ 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mrow> 
         <mi>
           t 
         </mi> 
         <mo>
           − 
         </mo> 
         <msub> 
          <mi>
            t 
          </mi> 
          <mn>
            0 
          </mn> 
         </msub> 
        </mrow> 
        <mo>
          ) 
        </mo> 
       </mrow> 
      </mrow> 
     </math>. Incorporating this into the dynamic equations for 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <mi>
         V 
       </mi> 
       <msub> 
        <mi>
          M 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
      </mrow> 
     </math>, we get</p>
    <p>
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mover accent="true"> 
         <mi>
           x 
         </mi> 
         <mo>
           ˙ 
         </mo> 
        </mover> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
       <mo>
         = 
       </mo> 
       <mfrac> 
        <mn>
          1 
        </mn> 
        <mrow> 
         <msub> 
          <mi>
            m 
          </mi> 
          <mi>
            i 
          </mi> 
         </msub> 
        </mrow> 
       </mfrac> 
       <mstyle displaystyle="true"> 
        <msub> 
         <mo>
           ∑ 
         </mo> 
         <mrow> 
          <mi>
            j 
          </mi> 
          <mo>
            ≠ 
          </mo> 
          <mi>
            i 
          </mi> 
         </mrow> 
        </msub> 
        <mrow> 
         <msub> 
          <mi>
            α 
          </mi> 
          <mrow> 
           <mi>
             i 
           </mi> 
           <mi>
             j 
           </mi> 
          </mrow> 
         </msub> 
         <mrow> 
          <mo>
            ( 
          </mo> 
          <mrow> 
           <msub> 
            <mi>
              x 
            </mi> 
            <mi>
              i 
            </mi> 
           </msub> 
           <mrow> 
            <mo>
              ( 
            </mo> 
            <mi>
              t 
            </mi> 
            <mo>
              ) 
            </mo> 
           </mrow> 
           <mo>
             − 
           </mo> 
           <msub> 
            <mi>
              x 
            </mi> 
            <mi>
              j 
            </mi> 
           </msub> 
           <mrow> 
            <mo>
              ( 
            </mo> 
            <mi>
              t 
            </mi> 
            <mo>
              ) 
            </mo> 
           </mrow> 
          </mrow> 
          <mo>
            ) 
          </mo> 
         </mrow> 
        </mrow> 
       </mstyle> 
       <mo>
         + 
       </mo> 
       <msub> 
        <mi>
          β 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <msub> 
        <mi>
          δ 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mrow> 
         <mi>
           t 
         </mi> 
         <mo>
           − 
         </mo> 
         <msub> 
          <mi>
            t 
          </mi> 
          <mn>
            0 
          </mn> 
         </msub> 
        </mrow> 
        <mo>
          ) 
        </mo> 
       </mrow> 
      </mrow> 
     </math> (5)</p>
   </sec>
   <sec id="s3_5">
    <title>3.5. Dynamic Equation of Computer Agent</title>
    <p>
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mover accent="true"> 
         <mi>
           y 
         </mi> 
         <mo>
           ˙ 
         </mo> 
        </mover> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
       <mo>
         = 
       </mo> 
       <mfrac> 
        <mn>
          1 
        </mn> 
        <mrow> 
         <msub> 
          <mi>
            m 
          </mi> 
          <mi>
            i 
          </mi> 
         </msub> 
        </mrow> 
       </mfrac> 
       <mstyle displaystyle="true"> 
        <msub> 
         <mo>
           ∑ 
         </mo> 
         <mrow> 
          <mi>
            j 
          </mi> 
          <mo>
            ≠ 
          </mo> 
          <mi>
            i 
          </mi> 
         </mrow> 
        </msub> 
        <mrow> 
         <msub> 
          <mi>
            γ 
          </mi> 
          <mrow> 
           <mi>
             i 
           </mi> 
           <mi>
             j 
           </mi> 
          </mrow> 
         </msub> 
         <mrow> 
          <mo>
            ( 
          </mo> 
          <mrow> 
           <msub> 
            <mi>
              y 
            </mi> 
            <mi>
              i 
            </mi> 
           </msub> 
           <mrow> 
            <mo>
              ( 
            </mo> 
            <mi>
              t 
            </mi> 
            <mo>
              ) 
            </mo> 
           </mrow> 
           <mo>
             − 
           </mo> 
           <msub> 
            <mi>
              x 
            </mi> 
            <mi>
              j 
            </mi> 
           </msub> 
           <mrow> 
            <mo>
              ( 
            </mo> 
            <mi>
              t 
            </mi> 
            <mo>
              ) 
            </mo> 
           </mrow> 
          </mrow> 
          <mo>
            ) 
          </mo> 
         </mrow> 
        </mrow> 
       </mstyle> 
       <mo>
         + 
       </mo> 
       <msub> 
        <mi>
          β 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <msub> 
        <mi>
          δ 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mrow> 
         <mi>
           t 
         </mi> 
         <mo>
           − 
         </mo> 
         <msub> 
          <mi>
            t 
          </mi> 
          <mn>
            0 
          </mn> 
         </msub> 
        </mrow> 
        <mo>
          ) 
        </mo> 
       </mrow> 
      </mrow> 
     </math> (6)</p>
    <p>where</p>
   </sec>
   <sec id="s3_6">
    <title>3.6. Mathematical Formulation of Malware Propagation</title>
    <p>The mathematical of malware propagation is formulated as the one in <xref ref-type="bibr" rid="scirp.135045-18">
      [18]
     </xref> <xref ref-type="bibr" rid="scirp.135045-19">
      [19]
     </xref>:</p>
    <p>
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <mi>
         N 
       </mi> 
       <mo>
         = 
       </mo> 
       <msub> 
        <mi>
          S 
        </mi> 
        <mn>
          1 
        </mn> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
       <mo>
         + 
       </mo> 
       <msub> 
        <mi>
          S 
        </mi> 
        <mi>
          f 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
       <mo>
         + 
       </mo> 
       <msub> 
        <mi>
          S 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
       <mo>
         + 
       </mo> 
       <msub> 
        <mi>
          C 
        </mi> 
        <mi>
          a 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          ( 
        </mo> 
        <mi>
          t 
        </mi> 
        <mo>
          ) 
        </mo> 
       </mrow> 
      </mrow> 
     </math> (7)</p>
    <p>where 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mi>
          S 
        </mi> 
        <mi>
          f 
        </mi> 
       </msub> 
      </mrow> 
     </math> are susceptible devices undergoing concurrent attacks but not yet infected (victim of attack Type 2), and 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mi>
          S 
        </mi> 
        <mn>
          1 
        </mn> 
       </msub> 
      </mrow> 
     </math> are vulnerable devices attacked for the first time (victim of attack Type 1); 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mi>
          S 
        </mi> 
        <mi>
          i 
        </mi> 
       </msub> 
      </mrow> 
     </math> are vulnerable devices undergoing simultaneous attacks, one of which has already been successful (attack type 3 victim); and 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mi>
          C 
        </mi> 
        <mi>
          a 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          [ 
        </mo> 
        <mi>
          T 
        </mi> 
        <mo>
          ] 
        </mo> 
       </mrow> 
      </mrow> 
     </math> are devices that have already contracted an infection and are further attacking the network (attack type 4 victim). Additionally, 
     <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> 
       <msub> 
        <mi>
          C 
        </mi> 
        <mi>
          a 
        </mi> 
       </msub> 
       <mrow> 
        <mo>
          [ 
        </mo> 
        <mi>
          T 
        </mi> 
        <mo>
          ] 
        </mo> 
       </mrow> 
      </mrow> 
     </math> stands for any device whose state—such as permanently immune devices or malware-inaccessible devices—cannot alter following a malware assault.</p>
   </sec>
  </sec><sec id="s4">
   <title>4. Results and Discussions</title>
   <p>To investigate malware propagation in the clouds the different agents are subdivided as stated in Equation (7) (susceptibles, vulnerables, infected and computer agent). <xref ref-type="fig" rid="fig5">
     Figure 5
    </xref> shows that the malware spread faster in Computer Agent rather than inside the IaaS during a single threat from any part of the system. the computer is used by humans, who are the weakest link in the security chain. Furthermore, security patches are more likely to be applied to core network infrastructures rather than to computer</p>
   <p>Moreover, in <xref ref-type="fig" rid="fig6">
     Figure 6
    </xref>, the Cloud’s components performances decrease significantly. This is due to the fact that malware consumes cloud’s internal resources. And, some particular malware increases file sizes.</p>
   <fig id="fig5" position="float">
    <label>Figure 5</label>
    <caption>
     <title>Figure 5. Malware propagation in cloud system.</title>
    </caption>
    <graphic mimetype="image" position="float" xlink:type="simple" xlink:href="https://html.scirp.org/file/7801030-rId101.jpeg?20240801083351" />
   </fig>
   <fig id="fig6" position="float">
    <label>Figure 6</label>
    <caption>
     <title>Figure 6. Clouds SYSTEM global performances under multiple infections.</title>
    </caption>
    <graphic mimetype="image" position="float" xlink:type="simple" xlink:href="https://html.scirp.org/file/7801030-rId102.jpeg?20240801083351" />
   </fig>
   <p>
    <xref ref-type="fig" rid="fig7">
     Figure 7
    </xref> shows the combined effect of responsiveness coefficient, interaction strength, and initial infection points on the mean resource allocation over time. Moreover, higher interaction strengths indicate a stronger influence between VMs and Computer Agents, leading to faster spread of malware. The mesh grid visualizes the combined effect of responsiveness coefficient, interaction strength, and initial infection points on the mean resource allocation over time.</p>
   <p>
    <xref ref-type="fig" rid="fig8">
     Figure 8
    </xref> shows that higher interaction strengths indicate a stronger influence between VMs and Computer Agents, leading to faster spread of malware while Lower interaction strengths result in more isolated resource allocation patterns, limiting the impact of malware propagation.</p>
   <p>
    <xref ref-type="fig" rid="fig9">
     Figure 9
    </xref> shows that earlier initial infection points lead to quicker initiation of malware propagation. On the other hand later initial infection points delay the onset of malware propagation, giving more time for security measures to be deployed.</p>
   <fig id="fig7" position="float">
    <label>Figure 7</label>
    <caption>
     <title>Figure 7. Resource allocations by responsiveness coefficient.</title>
    </caption>
    <graphic mimetype="image" position="float" xlink:type="simple" xlink:href="https://html.scirp.org/file/7801030-rId103.jpeg?20240801083351" />
   </fig>
   <fig id="fig8" position="float">
    <label>Figure 8</label>
    <caption>
     <title>Figure 8. Resource allocations by interaction strength.</title>
    </caption>
    <graphic mimetype="image" position="float" xlink:type="simple" xlink:href="https://html.scirp.org/file/7801030-rId104.jpeg?20240801083351" />
   </fig>
   <fig id="fig9" position="float">
    <label>Figure 9</label>
    <caption>
     <title>Figure 9. Resource allocation by initial infection points.</title>
    </caption>
    <graphic mimetype="image" position="float" xlink:type="simple" xlink:href="https://html.scirp.org/file/7801030-rId105.jpeg?20240801083351" />
   </fig>
  </sec><sec id="s5">
   <title>5. Conclusion</title>
   <p>The formulated model shows malware propagation in a cloud system. Therefore, It can be used to adjust the interaction strength values between the different agents that could significantly impact the overall security posture of the system, with stronger interactions potentially increasing vulnerability to rapid malware dissemination. Moreover the timing of initial infection points can determine the window of opportunity for security defenses to detect and mitigate malware threats. This indicator can be used to select adequate protection tools. Finally Identifying clusters of high resource allocation can guide security practitioners in prioritizing response efforts and implementing targeted security measures to contain malware outbreaks.</p>
  </sec>
 </body><back>
  <ref-list>
   <title>References</title>
   <ref id="scirp.135045-ref1">
    <label>1</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Islam, R., Patamsetti, V., Gadhi, A., Gondu, R.M., Bandaru, C.M., Kesani, S.C. and Abiona, O. (2023) International Journal of Communications. Network and System Sciences Scientific Research Publishing. Scientific Research Publishing.
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref2">
    <label>2</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Ullah, A., Nawi, N.M. and Ouhame, S. (2021) Recent Advancement in VM Task Allocation System for Cloud Computing: Review from 2015 to2021. Artificial Intelligence Review, 55, 2529-2573. &gt;https://doi.org/10.1007/s10462-021-10071-7
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref3">
    <label>3</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Yamin, M.M., Ullah, M., Ullah, H. and Katt, B. (2021) Weaponized AI for Cyber Attacks. Journal of Information Security and Applications, 57, Article ID: 102722. &gt;https://doi.org/10.1016/j.jisa.2020.102722
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref4">
    <label>4</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Balarezo, J.F., Wang, S., Chavez, K.G., Al-Hourani, A. and Kandeepan, S. (2022) A Survey on DOS/DDOS Attacks Mathematical Modelling for Traditional, SDN and Virtual Networks. Engineering Science and Technology, an International Journal, 31, Article ID: 101065. &gt;https://doi.org/10.1016/j.jestch.2021.09.011
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref5">
    <label>5</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Almalaq, A., Albadran, S. and Mohamed, M. (2022) Deep Machine Learning Model-Based Cyber-Attacks Detection in Smart Power Systems. Mathematics, 10, Article No. 2574. &gt;https://doi.org/10.3390/math10152574
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref6">
    <label>6</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Aldhyani, T.H.H. and Alkahtani, H. (2023) Cyber Security for Detecting Distributed Denial of Service Attacks in Agriculture 4.0: Deep Learning Model. Mathematics, 11, Article No. 233. &gt;https://doi.org/10.3390/math11010233
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref7">
    <label>7</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Gourisaria, M.K., Samanta, A., Saha, A., Patra, S.S. and Khilar, P.M. (2020) An Extensive Review on Cloud Computing. In: Raju, K.S., et al., Eds., Data Engineering and Communication Technology, Springer, 53-78. &gt;https://doi.org/10.1007/978-981-15-1097-7_6
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref8">
    <label>8</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Laato, S., Mäntymäki, M., Islam, A.K.M.N., Hyrynsalmi, S. and Birkstedt, T. (2022) Trends and Trajectories in the Software Industry: Implications for the Future of Work. Information Systems Frontiers, 25, 929-944. &gt;https://doi.org/10.1007/s10796-022-10267-4
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref9">
    <label>9</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Soh, J., Copeland, M., Puca, A. and Harris M. (2020) Microsoft Azure: Planning, Deploying, and Managing the Cloud. Springer.
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref10">
    <label>10</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Sunyaev, A. (2020) Cloud Computing. In: Internet Computing. Springer, 195-236. &gt;https://doi.org/10.1007/978-3-030-34957-8_7
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref11">
    <label>11</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Giannakou, A., Rillling, L., Pazat, J.-L., Majorczyk, F. and Morin, C. (2015) Towards Self Adaptable Security Monitoring in IaaS Clouds. 2015 15th IEEE/ACM International Symposium on Cluster, Cloud and Grid Computing, Shenzhen, China, 2015, 737-740. &gt;https://doi.org/10.1109/CCGrid.2015.133
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref12">
    <label>12</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Tabrizchi, H. and Kuchaki Rafsanjani, M. (2020) A Survey on Security Challenges in Cloud Computing: Issues, Threats, and Solutions. The Journal of Supercomputing, 76, 9493-9532. &gt;https://doi.org/10.1007/s11227-020-03213-1
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref13">
    <label>13</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Parast, F.K., Sindhav, C., Nikam, S., Yekta, H.I., Kent, K.B. and Hakak, S. (2022) Cloud Computing Security: A Survey of Service-Based Models. Computers &amp; Security, 114, Article 102580.
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref14">
    <label>14</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Admass, W.S., Munaye, Y.Y. and Diro, A.A. (2024) Cyber Security: State of the Art, Challenges and Future Directions. Cyber Security and Applications, 2, Article ID: 100031. &gt;https://doi.org/10.1016/j.csa.2023.100031
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref15">
    <label>15</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Guembe, B., Azeta, A., Misra, S., Osamor, V.C., Fernandez-Sanz, L. and Pospelova, V. (2022) The Emerging Threat of AI-Driven Cyber Attacks: A Review. Applied Artificial Intelligence, 36, Article ID: 2037254. &gt;https://doi.org/10.1080/08839514.2022.2037254
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref16">
    <label>16</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Sahu, I.K. and Nene, M.J. (2021) Model for IaaS Security Model: MISP Framework. 2021 International Conference on Intelligent Technologies (CONIT), Hubli, 25-27 June 2021, 1-6. &gt;https://doi.org/10.1109/conit51480.2021.9498375
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref17">
    <label>17</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Hu, V.C., Iorga, M., Bao, W., Li, A., Li, Q.H., Gouglidis, A., et al. (2020) General Access Control Guidance for Cloud Systems, NIST Special Publication, 800-210. &gt;https://doi.org/10.6028/NIST.SP.800-210
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref18">
    <label>18</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Carnier, R.M., Li, Y., Fujimoto, Y. and Shikata, J. (2024) Deriving Exact Mathematical Models of Malware Based on Random Propagation. Mathematics, 12, Article No. 835. &gt;https://doi.org/10.3390/math12060835
    </mixed-citation>
   </ref>
   <ref id="scirp.135045-ref19">
    <label>19</label>
    <mixed-citation publication-type="other" xlink:type="simple">
     Aslan, O., Ozkan-Okay, M. and Gupta, D. (2021) Intelligent Behavior-Based Malware Detection System on Cloud Computing Environment. IEEE Access, 9, 83252-83271. &gt;https://doi.org/10.1109/access.2021.3087316
    </mixed-citation>
   </ref>
  </ref-list>
 </back>
</article>