A mathematical model for virus infection in a system of interacting computers

A mathematical model for virus infection in a system of interacting computers

J. López Gondar; R. Cipolatti

Departamento de Matemática Aplicada, Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Rio de Janeiro, Brasil

E-mail: [e-mail] / [e-mail]

ABSTRACT

We introduce a simplified theoretical model to describe a virtual virus propagation process in a set of interacting computers. The propagation mechanisms considered here are those related to the reception of messages through internet as well as the ones concerning the simple exchange of files using recording devices as compact disks or the commonly used floppy disks. In spite of its inherent simplicity, this model provides a good idea of the infection process and trends. From the mathematical point of view, the nonlinear delay integral equation that we obtain here presents certain interesting features which are explored and enlightened in this paper.

Mathematical subject classification: 45G10, 49K25, 92D99.

Key words: virtual viruses, nonlinear delay integral equation.

1 Introduction

The infection of computers by virtual viruses is a present day problem. A lot of efforts have been (and will be) devoted to the development of virtual vaccines each time a new virus appears. Nevertheless, slow progress (if any) in understanding the propagation process, quantitatively, has been obtained up to now. In a certain sense, the propagation of virtual viruses in a system of interacting computers could be compared with a disease transmitted by vectors when dealing with public health. Concerning diseases transmitted by vectors, one has to take into account that the parasite spends part of its lifetime inhabiting the vector, so that the infection switches back and forth between host and vector. Almost the same occurs here and also a certain time interval elapses between successive contacts which, in this case, involves an electronic message (or a recording device) and two different computers. However, we have to point out that, in our simplified model, which ignores finer details, we suppose in the case of recording devices that they are simple carriers not permanently infected, which could transmit the virus only once (if it is so); i.e., after the first transmission of a certain program, the contents in the recording device are removed before it is used again. Under such hypothesis (among others considered below) we obtain a compact model restricted to one single integral equation.

The paper is organized as follows: in Section 2, we construct the theoretical model. Section 3 is devoted to performing a mathematical analysis of the integral equation obtained: existence, uniqueness and asymptotic behavior of the solutions, as well as stability of stationary solutions. Finally, Section 4 contains numerical experiments and our conclusions.

2 Modeling the dynamics of transmission

Before introducing the new model to describe the way by which some virtual virus infection transmitted by electronic messages (or by recording devices) could spread through a population of computers as a function of time, we shall start deducing a certain contact-propagation equation for a general infectious process having a natural (or induced) recovering time. We consider such an equation as a convenient point of reference for subsequent analysis. In the present approach we shall assume to deal with a new kind of virus infection for which there exists no vaccine available. Being so, the unique way to restore the functions of an infected computer is by deleting the hard disk (with which it becomes susceptible again). It should be pointed out that neither spatial dependence nor latent periods or immunity factors will be considered here.

Therefore, the total population is divided into two classes:

(I) the infective class. This includes all infected individuals because, in the absence of a latent period, any infected individual immediately becomes an infective one, displaying a potential capacity of transmitting the disease to other susceptible individuals;

(S) the susceptible class; i.e., those individuals capable of contracting the disease, becoming themselves infected.

The closed host population of total size N will be used as a normalization factor, so we set I + S = 1.

Let us denote by g(t) the rate of the infection process at time t, i.e., the number of new infected computers (population density of them) per unit time, at instant t. We also introduce a function of two variables P(t,t), P : L ® [0,1], where

which represents the probability of a computer infected at t to remain infectious at t. Such a function, of course, is related to the time elapsed between the detection of the infection and its eradication by erasing the hard disk. Considering a maximum recovering time T, the function P(t,t) is supposed to have the following general properties:

If we want to describe the density of infective individuals as a function of time, starting at t = 0, the only infected ones to be considered should be those in the interval [-T,t] (just because of the maximum recovering time T).

Dividing this interval into n equal parts, so that Dt = (t + T)/n and defining t₁ = -T, t₂ = -T + Dt, ..., t_n+1 = -T + nDt, i.e., t_j = -T + (j - 1)Dt, j = 1,..., n + 1, we should state, for Dt small enough, that the number of infected individuals in the interval [t_j, t_j+1] = [t_j, t_j + Dt] is given, approximately, as

Then, the number of infected individuals in this interval which remain infective at t is approximately,

and therefore, the total number of infective individuals at t may be written, approximately, as

which, in the limit Dt ® 0, gives

Taking into account that P(t,t) = 0 if t < t - T, we should write Eq. (2.6) as

The former equation contains the basic features for the evolution of an infection process in the framework of a SIS model. Depending on the specific transmission process, one should conceive, in each case, a suitable infection rate-function g(t). As it can be verified in references [4, 6], Eq. 2.7 and variations of it have already been used by other authors in describing the propagation of diseases.

Concerning the propagation of virtual viruses transmitted by electronic messages (a) or by recording devices (b), we shall establish the following hypothesis to construct g(t):

a) some finite time elapses between the act of sending and the reception of messages (t₀ on average);

b1) each recording device is used only twice in intervals of time t₀, on average;

b2) each device transmits the infection when, used by the first time in an infected computer, it is used by the second time in a susceptible one. Once this process occurs, previous contents are deleted.

Possible events concerning consecutive records and its corresponding probabilities are described below.

Among them we are only interested in the third one.

By a direct calculation, one can easily verify that the sum of the above probabilities gives just 1.

Of course, there exists some characteristic average frequency for the submission of electronic messages in the system of interacting computers considered here, as well as recording devices are subjected to a certain recording frequency. We shall denote such a characteristic frequency by N_r(t). Then, for a time interval Dt small enough, we may write the number of new infected computers which are increased to those already existing at t + t₀, approximately, by

Dividing by N Dt and taking the limit as Dt ® 0, we have

If we make the substitution t

t + t₀, Eq. (2.9) may be written as

or

where

Therefore, in the framework of our model and following Eq. (2.7), the density of infective computers at time t due to the virtual virus propagation process obeys the following law

where a(t) is given by (2.12).

Eq. (2.13) is a delay integral equation and the existence of solutions for t > 0 depends on data defined in the past -T - t₀< t < 0.

Although we are only considering t₀ > 0, it is interesting to point out that, if t₀ = 0 and P(t,t) = f(t - t), where f(x) is a real function, Eq. (2.13) reduces to the special case considered by Cooke and Yorke in [6].

3 Existence, uniqueness and asymptotic behavior

In this section we consider the problem of existence, uniqueness and asymptotic behavior of solutions for equation (2.13). We will say that a function I(t) is a solution generated by I₀ if I(t) = I₀(t) a.e. in [-T-t₀,0] and if I(t) also satisfies (2.13) for t > 0.

• Uniqueness:

We consider the set L defined by (2.1), a function P : L ® [0,1] such that (2.2) holds and I₀Î L^¥(-T - t₀,+¥).

In order to avoid technicalities (see Remark 3.2), we assume that

Lemma 3.1. Assume that a satisfies (3.1). If I₁ and I₂ are solutions of (2.13) generated by I₀(t), then I₁(t) = I₂(t) for all t > 0.

Proof. First of all it should be noticed that, if I_i is a solution of (2.13), then I_i is continuous in ]0,+¥[.

For 0 < t < t₀ we have

Let j be the function defined by j(t) = |I₁(t) - I₂(t)|. Since 0 < P(t,t) < 1, it follows that

where ||I₀|| = ess sup{|I₀(t)| ; -T - t₀< t < 0}.

Defining

it follows that y is differentiable in ]0,+¥[ and we have y'(t) = |a(t)|j(t).

After multiplying both sides of (3.2) by |a(t)|, we get

from which we obtain

Since y(0) = 0, we conclude that y(t) = 0 as well as j(t) = 0 for all 0 < t < t₀.

The same arguments can be repeated for (k - 1)t₀ < t < kt₀, for all k Î , and the proof is achieved.

Remark 3.2. We can obtain the same result of Lemma 3.1 with essentially the same proof under the weaker condition

• Existence and asymptotic behavior:

In order to prove the existence of solutions for Eq. (2.13), we assume that

In view of the model we have in mind, we restrict our analysis to the solutions that satisfy 0 < I(t) < 1, which allows us to consider the space E = L^¥(]-T - t₀,+¥[). This is a Banach space for the norm

In the sequel, we distinguish two cases, referred here as "Slow Infective Rate Case" and "General Infective Rate Case", respectively.

Let r(t) be the function defined by

Case 1: "Slow Infective Rate": r(t) < 1, "t > 0.

For a given I₀Î E such that 0 < I₀(t) < 1, we consider the set

which is a closed subset of E. We also consider the operator F: E(I₀) ® E(I₀) defined as follows: F[I](t) = I₀(t) almost everywhere in [-T - t₀,0] and, for t > 0,

The next theorem establishes the existence of a solution of Eq. (2.13) in E (which is unique from Lemma 3.1), as well as its asymptotic behavior in the case r(t) < 1 for t large enough.

Theorem 3.3. Assume a satisfies (3.3) and r(t) < 1, "t > 0. If I₀Î E is such that 0 < I₀(t) < 1, then the operator F defined by (3.6) has a unique fixed point Î E(I₀). In addition, if

then there exist C, M, L > 0 such that

Proof. We divide the proof into two steps. Let a,b be two real numbers such that -T - t₀ < a < b and consider the set

It is evident that E_a,b(I₀) is a nonempty closed subset of E and, for a > 0, E_a,b(I₀) Ì E(I₀).

We introduce the operator F: E_a,b(I₀) ® E_a,b(I₀) defined by F[I](t) = I₀(t) for t < a, F[I] = 0 for t > b and

Step 1: Let I, Î E_0,b(I₀) and consider the operator F defined in (3.8), with a = 0 and

Then, it is easily seen that F[I](t) - F[](t) = 0 a.e. in [-T - t₀,0] and

Since 0 < P(t,t) < 1 for all (t,t) Î L, it follows that

for every t Î ]0,b]. A recurrent argument using (3.10) and (3.11) gives, for t Î ]0,b],

from which we obtain easily

Choosing k large enough, it follows from (3.13) that F^k is a contraction in E_0,b(I₀) and the Banach fixed point theorem assures the existence of a unique fixed point I₁ for F^k in E_0,b(I₀). More precisely, there exists a unique I₁Î E_0,b(I₀) such that F^k[I₁] = I₁. Since F^k+1[I₁] = F^k[F[I₁]] = F[I₁], it follows from the uniqueness that F[I₁] = I₁ in E_0,b(I₀). Moreover, if Î E_0,b(I₀) is such that (t) = 0 for all t > 0, then

which implies that, for k Î and t Î ]0,b],

Taking the limit as k ® +¥, we obtain

Step 2: We consider now F: E_b,2b(I₁) ® E_b,2b(I₁), where b is defined in (3.9). Then, the same arguments of Step 1 hold and we obtain a fixed point I₂Î E_b,2b(I₁) such that

Arguing by induction, we obtain a sequence of functions in E(I₀) which satisfies the following properties: for each k Î ,

If we define

then it is easily seen that I_n ® uniformly on the compacts sets of . Hence, since

we have, after passing to the limit in both sides of (3.15), that is the unique solution of (2.13).

In addition, if lim sup_t®+¥r(t) < 1, then there exist 0 < r < 1 and M > 0 such that r(t) < r, "t > M. In particular, it follows from (3.14) that I_k0(t) < r||I_k0-1|| < r||I₀||, for k₀ > 1 + M/b and t Î ](k₀ - 1)b,k₀b].

Hence, we have for m Î ,

and the conclusion follows with

Remark 3.4. As an immediate consequence of the absolute continuity of the Lebesgue integral, it follows that is continuous in the interval ]0,+¥[. In order to assure its continuity in ]-T - t₀,+¥[, it suffices to assume that I₀ is continuous on [-T - t₀,0] and satisfies the following compatibility condition:

Case 2: "The General Infective Rate"

In this section we consider the existence and asymptotic behavior of solutions for Eq. (2.13) without the assumption r(t) < 1. Except for the arguments used to prove the asymptotic behavior, the existence of solutions in this case can be obtained with essentially the same proof as in the previous theorem. Indeed, with the notation introduced before, we have:

Theorem 3.5: Assume a satisfies (3.3). If I₀Î E, then Eq. (2.13) has a unique solution Î (-T - t₀,+¥) generated by I₀.

Proof. We argue as in the proof of Theorem 3.3.

Let a,b be two real numbers such that -T - t₀ < a < b and consider the set

It is obvious that F_a,b(I₀) is a nonempty closed subset of E.

We introduce the operator F: F_a,b(I₀) ® F_a,b(I₀) defined by F[I](t) = I₀(t) for t < a, F[I] = 0 for t > b and

Repeating the same arguments of steps 1 and 2 in the proof of Theorem 3.3 we show that F has a unique fixed point I_n Î F_nb,(n+1)b, where b is given by (3.9), satisfying the following properties: for each k Î ,

If we define

then it is easily seen that I_n ® uniformly on the compacts sets of . Since

we have, after passing to the limit in both sides of (3.17), that is the unique solution of (2.13) generated by I₀.

Remark 3.6. It is clear that the set E(I₀) is not an invariant set for the operator F if r(t) does not satisfy r(t) < 1. For instance, let I(t) = c for all t > -T - t₀, where 0 < c < 1, P(t,t) = f(t - t), being f the characteristic function of the interval [0,T] and a(t) = 1. Then

and we have F[I](t) > 1 if T > . In spite of this, we can show that the solution may belong to E(I₀). This is proved in the next theorem.

We assume that f: [0,+¥[ ® [0,1] is such that

Theorem 3.7. Assume that a Î E, a(t) > 0 and let P(t,t) = f(t - t), with f satisfying (3.18). If I₀Î E with 0 < I₀(t) < 1, then the solution

of (2.13) generated by I₀ satisfies 0 < (t) < 1, "t > -T - t₀.

Proof. We proceed in three steps.

Step 1: In addition to the hypothesis, we assume that

We prove firstly that, under (3.19), if I₀Î C([-T - t₀,0]) is such that 0 < I₀(t) < 1 in [-T -t₀,0] and Î is the solution of Eq. (2.13) generated by I₀, then 0 < (t) < 1 for all t > -T - t₀.

Indeed, consider

which is a closed subset of [0,+¥[. If G ¹ Æ, let t₁ = inf G. Since (t) = I₀(t) for t < 0, then t₁ > 0 and 0 < (t) < 1 for all t < t₁.

Suppose that (t₁) = 1. Since we are assuming (3.19), it follows that (t) is differentiable for t > 0, t ¹ t₀. In particular, if t₁¹ t₀, then

and we can pick up d > 0 such that (t₁ - d) > 1, which is a contradiction. Besides,

and we have the same contradiction if t₁ = t₀.

Suppose that (t₁) = 0. It follows from the intermediate value theorem for integrals that there exists t Î ]t₁-T,t₁[ such that

and we have also a contradiction.

Therefore, G = Æ and we have the conclusion.

Now let I₀Î E such that 0 < I₀(t) < 1 a.e. in ]-T - t₀,0[ and Î the solution generated by I₀.

Since C([-T - t₀,0]) is dense in L¹(-T - t₀,0), it follows that, for a given e > 0, there exists continuous such that 0 < (t) < 1 for t Î [-T - t₀,0] and

If we denote by e the solution of Eq. (2.13) generated by , then 0 < e(t) < 1 for t > 0. Moreover, if 0 < t < b, we have

From the Gronwall inequality we get

Then, e converges to as e ® 0, uniformly in [0,b], and we have, in particular, 0 < (t) < 1 "t Î [0,b].

Arguing by induction on the intervals [kb,(k + 1)b], k > 0, we conclude that 0 < (t) < 1 for t > 0.

Step 2: In addition to the hypothesis, we assume that

Let Î be the solution of Eq. (2.13) generated by I₀. For e > 0 given, we can construct (for instance, by smoothing step functions) a function f_e satisfying (3.19)-(ii) and such that

Let eÎ be the solution of Eq. (2.13) (with P_e(t,t) = f_e(t-t)) generated by I₀. It follows from step 1 that 0 < e(t) < 1, for all t > 0.

Besides, if 0 < t < b, we have

Hence, the Gronwall inequality implies

_e ® as e ® 0, uniformly on ] 0,b], and we have the same conclusion as in step 1.

Step 3: Let be the solution of Eq. (2.13) generated by I₀. Since we are assuming a Î E, for each e > 0, we can construct a function a_e satisfying (3.21) and such that

Then, the same arguments of step 2 allows to show that 0 < < 1 for all t > 0 and this completes the proof.

Remark 3.8. Concerning the asymptotic behavior of the solutions of Eq. (2.13), it should be noticed that, if r(t) is defined by (3.4) and satisfies

we expect to have the following global asymptotics for the solution generated by 0 < I₀< 1:

Actually, r_T < 1 is a particular case of Slow Infective Rate and the asymptotic behavior is obtained in Theorem 3.3.

Although it seems to be difficult to prove the global behavior (3.23) in general, some simple evidences can be obtained in the special case where P(t,t) = f(t-t), with f satisfying (3.18) and a(t) º a is constant. Indeed, it is easily seen that, in this case we have r_T = af(t)dt and

are the only stationary solutions of Eq. (2.13).

Assuming that I_¥ is one of the stationary solutions of Eq. (2.13) and considering (t) = I_¥ + X(t), the solution generated by I₀(t) = I_¥ + X₀(t), we have for t > 0

Discarding all but linear terms in Eq. (3.25), we obtain the linear integral equation

where

The behavior of solutions of (3.26) can be studied in terms of its associated characteristic roots l Î . More precisely, assuming X(t) = e^lt and substituting in (3.26) we obtain the characteristic equation

Lemma 3.9. We assume that r(t) satisfies (3.22) with r_T> 1. Then the characteristic equation (3.27) has a unique real solution

satisfying:

Proof. Let f: ® be the function defined by

If I_¥ = 0, then (3.28) is given by

and it is easily seen that f(l) is positive, monotone decreasing and satisfies:

Since f(l) is a continuous function, there exists a unique satisfying equation (3.27). Since f(0) = r_T > 1, we conclude that > 0 and a) is proved.

If I_¥ = 1 - 1/r_T, then (3.28) is given by

and the limits in (3.29) hold. Moreover, f(l) is positive and monotone decreasing in the interval

and negative elsewhere. Since it is continuous, there exists a unique

such that f() = 1. In order to prove that < 0, we distinguish two cases.

If r_T> 2, then ln(1/(r_T - 1))^1/t0< 0 and the conclusion follows from (3.30).

If 1 < r_T < 2, then 0 < f(0) = 2 - r_T < 1 = f() and the conclusion follows from the monotonicity of f.

Remark 3.10. The condition a) in Lemma 3.9 is sufficient to assure the instability of the stationary solution I_¥ = 0 in the case r_T > 1. On the other hand, although the condition b) is necessary to have the local asymptotic stability of I_¥ = 1 - 1/r_T, we do not have a precise characterization of the possible complex roots of Eq. (3.27) (see [3, 8, 7]).

The next theorem concerns the continuity of solutions of Eq. (2.13) with respect to the parameter t₀.

Theorem 3.11. Assume a and P satisfying (3.3) and (2.2) respectively. Let

be a sequence of positive real numbers such that t_n ® t₀ (t₀ > 0) as n ® +¥. Let I₀Î L^¥(-T - d,0) be a function satisfying 0 < I₀(t) < 1, where d = sup{t_n ; n Î }. If

_nÎ E(I₀) is the solution of

generated by I₀, then

_n ®

uniformly on the compacts of [0,+¥[, where

is the solution of (2.13) generated by I₀.

Proof. We have from the previous results that, for each n Î , there exists a unique solution _n of Eq. (3.31) generated by I₀, such that 0 < _n(t) < 1 a.e. in the interval ]-T -d, +¥[. In particular, it follows that is a bounded subset of the Banach space C([0,R];), for each R > 0.

Moreover, for t > 0 and h > 0 we have

and the absolute continuity of the Lebesgue integral implies that is an equicontinuous subset of C([0,R];).

Fixing R > 0, we have from the Arzelà-Ascoli theorem that there exist a subsequence and a function I_R Î C([0,R];) such that

Defining I_R(t) = I₀(t) for t < 0, we can write for t > 0

Passing to the limit as k ® ¥ in both sides of (3.32) we have

and we conclude from the uniqueness of solutions (see Lemma 3.1) that the full sequence {

_n} converges uniformly to I_R. Since R > 0 is fixed arbitrarily, the proof is finished.

Remark 3.12. Although we are tacitly assuming that t₀ > 0 in Theorem 3.11, it is interesting to point out that the conclusion is also true in the case t_n ® 0⁺. Indeed, the Eq. (2.13) with t₀ = 0 has a unique global solution generated by 0 < I₀< 1 and the same arguments in the proof Theorem 3.11 hold.

4 Numerical results and conclusions

In order to perform numerical calculations in the absence of previous statistical data, we have to propose a certain infection-rate function g(t) for the interval -T - t₀< t < 0. It has not to be a realistic function, since our purpose in this section is to illustrate the salient features of the time evolution process and not to reproduce or to predict actual situations in a realistic fashion. For this purpose, we consider the simplest case of a constant infection-rate function g(t) = I₀/T for t Î [-T - t₀,0], where I₀ represents the density of infective computers at t = 0. Furthermore, the contact-rate factor a(t) was also considered as a constant and the probability of a computer infected at t to remain infectious at t was chosen as P(t,t) = f(t-t), where f is the characteristic function of the interval [0,T[, or equivalently, P(t,t) = H(t-t) - H(t-t-T), where H(x) is the Heaviside step function.

Figure 1 shows a typical "slow infective rate case" for which, under our oversimplified considerations, aT < 1. The graph presents singularities in the interval [0,t₀ + T] due to the arbitrary choice of initial data but, as it can be seen, the infection process is self controlled, tending to disappear for increasing t values. On the other hand, Fig. 2, 3 and 4 show the more complex situation of typical "fast infective rate cases" aT > 1. In each case, strong (but damped) oscillations would appear for aT large enough. Nevertheless, an equilibrium point I_¥ = (aT-1)/aT is attained for t ® +¥, characterizing an endemic situation.

As it was shown in the previous section, the stationary solution I_¥ = 0 is an asymptotically stable solution for r_T = aT < 1. This means that when such a condition holds, the introduction of a few infected computers into an infective-free population won't give rise to an epidemic outbreak and also no endemic situations will be developed, i.e., the infection will vanish along the time.

We consider that, in spite of its inherent limitations because of the strong hypothesis included, the present work constitutes a first attempt to explain the virtual virus propagation process in a system of interacting computers and reveals some features. These result, we hope, could stimulate future researches in such a current problem. At present, we are engaged in the developing of more sophisticated models in order to take into account immunity factors (the existence of vaccines) as well as to include the spatial dependence of the infectious process. Results in this direction will be published elsewhere.

5 Acknowledgments

The authors are grateful to Dr. Felipe Acker (Instituto de Matemática, UFRJ) for suggesting the subject of this paper and the referees for corrections.

Received: 14/I/02.

#540/02.

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Publication Dates

• Publication in this collection
  20 July 2004
• Date of issue
  2003

History

• Received
  14 Jan 2002