A SCIRS Model for Malware Propagation in Wireless Networks

Abstract

The main goal of this work is to propose a novel mathematical model to simulate malware spreading in wireless networks considering carrier devices (those devices that malware has reached but it is not able to carry out its malicious purposes for some reasons: incompatibility of the host’s operative system with the operative system targeted by the malware, etc.) Specifically, it is a SCIRS model (Susceptible-Carrier-Infectious-Recovered-Susceptible) where reinfection and vaccination are considered. The dynamic of this model is studied determining the stability of the steady states and the basic reproductive number. The most important control strategies are determined taking into account the explicit expression of the basic reproductive number.

Appendix: Proofs of Theorems

1.1 Proof of Theorem 1

The system of differential equations governing the dynamic of the model (1)–(4) can be reduced to the following system taking into account that the total number of devices remains constant ($N=S\left( t\right) +C\left( t\right) +I\left( t\right) +R\left( t\right)$):

$$\begin{aligned} S'\left( t\right)= & {} -a\cdot S\left( t\right) \cdot I\left( t\right) -v \cdot S\left( t\right) +\epsilon \cdot R\left( t\right) , \end{aligned}$$

(19)

$$\begin{aligned} C'\left( t\right)= & {} a \cdot \left( 1-\delta \right) S\left( t\right) \cdot I\left( t\right) -b_C \cdot C\left( t\right) , \end{aligned}$$

(20)

$$\begin{aligned} I'\left( t\right)= & {} a \cdot \delta \cdot S\left( t\right) I\left( t\right) -b_I \cdot I\left( t\right) , \end{aligned}$$

(21)

The Jacobian matrix associated to this system of ordinary differential Eqs. (19)–(21) in the disease-free steady state is given by:

$$\begin{aligned} J\left( E^*_0\right) =\left( \begin{array}{ccc} -v &{} 0 &{} -\dfrac{a N \epsilon }{v + \epsilon } \\ 0 &{} -b_C &{} \dfrac{a N \left( 1-\delta \right) \epsilon }{v+ \epsilon }\\ 0 &{} 0 &{} - b_I+\dfrac{a N \delta \epsilon }{v+ \epsilon } \end{array} \right) , \end{aligned}$$

(22)

so that a simple calculus shows that its eigenvalues are:

$$\begin{aligned} \lambda _1=-b_C, \quad \lambda _2= -v, \quad \lambda _3= \frac{a N \delta \epsilon -b_I \left( v+\epsilon \right) }{v + \epsilon }=b_I\left( R_0-1\right) . \end{aligned}$$

(23)

Consequently $\text {Re}\left( \lambda _1\right) =-b_C<0$ and $\text {Re}\left( \lambda _2\right) =-v<0$. Moreover, $\text {Re}\left( \lambda _3\right) =b_I\left( R_0-1\right) <0$ iff $R_0 \le 1$, thus finishing. On the other hand, the global stability of the disease-free steady state is also satisfied when $R_0 \le 1$; this result is easily obtained.

1.2 Proof of Theorem 2

The Jacobian matrix of the system (19)–(21) in the endemic equilibrium steady state $E^*_1$ is:

$$\begin{aligned} J\left( E^*_1\right) =\left( \begin{array}{ccc} -v-\frac{b_C (-b_I v-b_I \epsilon +a N\delta \epsilon )}{b_I b_C+\delta \epsilon b_C+b_I \epsilon -b_I\delta \epsilon } &{} 0 &{} -\frac{b_I}{\delta } \\ \frac{b_C (1-\delta ) (-b_I v-b_I \epsilon +a N \delta \epsilon )}{b_I b_C+\delta \epsilon b_C+b_I \epsilon -b_I \delta \epsilon } &{} -b_C &{} \frac{b_I (1-\delta )}{\delta } \\ \frac{b_C \delta (-b_I v-b_I \epsilon +a N \delta \epsilon )}{b_I b_C+\delta \epsilon b_C+b_I \epsilon -b_I \delta \epsilon } &{} 0 &{} 0 \\ \end{array} \right) , \end{aligned}$$

(24)

such that the explicit expression of its characteristic polynomial is the following:

$$\begin{aligned} p\left( \lambda \right)= & {} p_0\lambda ^3+p_1 \lambda ^2+p_2 \lambda +p_3\\= & {} \lambda ^3+ \lambda ^2 \frac{b_C \delta \epsilon (a N+b_C+v)+b_I \left( b_C^2-\delta \epsilon (b_C+v)+v \epsilon \right) }{b_I (b_C-\delta \epsilon +\epsilon )+b_C \delta \epsilon } \nonumber \\&+\,\,\lambda \frac{b_C \left( b_I \epsilon (a \delta n-b_C-\delta v+v)+b_C \delta \epsilon (a N+v)-b_I^2 (v+\epsilon )\right) }{b_I (b_C-\delta \epsilon +\epsilon )+b_C \delta \epsilon } \nonumber \\&-\,\,\frac{b_I b_C^2 (b_I (v+\epsilon )-a \delta n \epsilon )}{b_I (b_C-\delta \epsilon +\epsilon )+b_C \delta \epsilon }. \nonumber \end{aligned}$$

(25)

Note that $p_0>0$, $p_1>0$ if $R_0>\delta$, $p_2>0$ if $R_0>1$, and $p_3>0$ iff $R_0>1$; as a consequence, the coefficients of the characteristic polynomial of $J\left( E_1^*\right)$ are positive if $R_0>1$. Now, by applying the Routh-Kurwitz stability criterion, the real part of the eigenvalues of $p\left( \lambda \right)$ will be negative when the following conditions hold:

$$\begin{aligned} \varDelta _1=p_1>0, \varDelta _2=\left| \begin{array}{cc} p_1 &{} p_3\\ 1 &{} p_2 \end{array} \right|>0, \varDelta _3=\left| \begin{array}{ccc} p_1 &{} p_3 &{}0\\ 1 &{} p_2 &{}0\\ 0 &{} p_1 &{} p_3 \end{array} \right| =p_3 \varDelta _2>0. \end{aligned}$$

(26)

Note that $p_1>0$ if $R_0>1$ as was mentioned previously. On the other hand as

$$\begin{aligned} \varDelta _2=\frac{b_C \varOmega \varTheta }{b_C \delta \epsilon + b_I\left( b_P+\epsilon \left( 1-\delta \right) \right) ^2}, \end{aligned}$$

(27)

where

$$\begin{aligned} \varOmega= & {} b_Ib_CvR_0+b_Cv\delta \epsilon +b_Ib_C\epsilon +b_I\epsilon v \left( 1-\delta \right) ,\end{aligned}$$

(28)

$$\begin{aligned} \varTheta= & {} b_ib_C^2+b_ib_CvR_0+b_C^2\delta \epsilon +b_Cv\delta \epsilon +b_ib_C\epsilon \left( R_0-\delta \right) \nonumber \\&+\,\,b_iv\epsilon \left( 1-\delta \right) +b_I^2\left( v+\epsilon \right) \left( R_0-1\right) , \end{aligned}$$

(29)

then $\varDelta _2>0$ if $R_0>1$ since $b_C>0$, $b_C \delta \epsilon + b_I\left( b_P+\epsilon \left( 1-\delta \right) \right) ^2>0$, $\varOmega >0$, and $\varTheta >0$ if $R_0>1$. Finally, taking into account the last results, it is easy to check that $\varDelta _3=p_3 \varDelta _2>0$ when $R_0>1$. Consequently, the endemic steady state is locally asymptotically stable. The global stability follows from simple but tedious computations.