Analysis of some Epidemic Models in complex networks and some ideas about isolation strategies
AAnalysis of some Epidemic Models in complexnetworks and some ideas about isolation strategies
Carlos Rodr´ıguez LucateroDepartamento de Tecnolog´ıas de la Informaci´onUniversidad Aut´onoma Metropolitana-CuajimalpaTorre IIIAv. Vasco de Quiroga 4871Col.Santa Fe Cuajimalpa, M´exico, D. F.C.P. 05348, M´exicoemail:[email protected]
Abstract
Many models of virus propagation in Computer Networks inspired by
SIS,SIR,SEIR , etc. epidemic disease propagation mathematical models that can be foundin the epidemiology field have been proposed in the last two decades. The purposeof these models has been to determine the conditions under which a virus becomesrapidly extinct in a network. The most common models proposed in the field of viruspropagation in networks are inspired by SIS-type models or their variants. In suchmodels, the conditions that lead to a rapid extinction of the spread of a computervirus have been calculated and its dependence on some parameters inherent to themathematical model has been observed. In this article we will try to analyze aparticular model proposed in the past and show through simulations the influencethat topology has on the dynamics of the spread of a virus in different networks. Aconsequence of knowing the impact of the topology of a network can serve to proposeeffective isolation strategies to reduce the spread of a virus through modifications tothe original network of contacts. I will talk about this subject at the final section ofthe present article.
Mathematics Subjects Classification : Keywords : Dynamical Systems; Virus Spreading; isolation strategies; Complex Net-works. a r X i v : . [ m a t h . O C ] J a n Introduction
The issue of virus spread as well as the conditions under which it is extinguished or con-taminated most of the nodes of a computer network has been studied for at least twodecades. The models that have been proposed are based on mathematical models of virusspread in the field of epidemiology [5],[6],[7]. These models are differential equations thattry to capture the dynamics of a virus spread in order to answer questions such as, Howlong will the epidemic disappear? At what point will it peak? Will it remain endemicallyat a certain level of infected? Will it generate permanent immunity to those who sufferedfrom the disease?.These mathematical models emerged from the area of epidemiology are known as com-partmental because they model the process as states in which an individual finds himselfduring an epidemic and transition probabilities between these states. In other words, anindividual can be susceptible, exposed, infected, or recovered from a virus, and these statesare modelled as compartments. From these compartments and the transition probabilitiesbetween them, relationships can be established in the form of differential equations thatdescribe the dynamics of the spread of a disease [11]. The compartments used by one ofthose models depend on the disease to be modelled, since there are diseases that producepermanent immunity such as measles, while some diseases such as seasonal influenza donot produce permanent immunity. The spread of a disease in epidemic processes sharessimilarities with the massive attacks on a computer network and therefore some ideas fromthe mathematical modelling of epidemics served as the basis for models of virus spread incomputer networks. However, it is worth mentioning that some problems that arise in thestudy of the spread of viruses in computer networks, as well as their respective solutions,may be useful for epidemiology. There have been very famous massive attacks on computernetworks. One of the first denial-of-service-type attacks turned 20 on February 7, 2020,and was conducted by a 15 years old Canadian hacker whose pseudonym was
Mafiaboy and whose real name is Michael Calce. Denial of service type attacks consist of sending ahuge number of service request packets to a target server in such a way that it exceeds thecapacity of said server to respond to so many orders, thus causing it to crash. This massiveattack revealed the vulnerability of networks such as the internet and led to a study ofthe causes of said vulnerability and and gave birth to a new type of computer virus. Thisvulnerability aroused the interest of researchers in the field of network security and theyrealized that certain topologies favoured a faster dissemination of information than othersor that a certain type of interconnection kept the nodes of a network connected despitethe fact that some lines were faulty. Some studies showed that the type of interconnec-tion structures that are formed in networks such as the Internet, Facebook, tweeter havesimilar characteristics and that they produce the appearance of giant components as wellas the phenomenon of small worlds. The formalization of the relationship of the topologyof the networks and the appearance of the aforementioned phenomena uses the theory ofrandom graphs as a mathematical tool. The type of interconnection structures that areformed in social networks such as facebook, tweeter and the internet have been charac-terized as particular graphs whose distribution of degrees of the nodes is known as the2ype of power laws due to the mathematical form that expresses said distribution whichin algebraic terms would be p k ∼ Ck − γ with 2 < γ ≤
3. The formation of networkswith these topological characteristics are closely related to the type of connection protocolused and is known as preferential attachment . It is said that this type of network is bothrobust and vulnerable since if they were removed by randomly choosing more than 90%of the edges of the associated graph, it would continue to be a connected graph, while ifthey were removed by strategically choosing a number small number (2 . Epidemics have been with humans for a long time. One of the oldest diseases is leprosy.Another disease that devastated Europe in the Middle Ages was the Black Death. Littlewas known about these diseases and how to cure them. In these circumstances, it wasnecessary to try to understand the way in which the epidemic process was developing inorder to at least stay safe from them. In the absence of objective knowledge based onscience, there was a tendency to believe that such calamities were divine punishments.Perhaps since those times the most helpful strategy consisted of isolating oneself. Somediseases such as smallpox were treated since ancient times using the variolation method,which consisted in inoculating the scales of a sick patient in a healthy individual andobserving that this allowed said individual to acquire a certain type of immune protection.The knowledge of this empirical method of immunization was imported to Europe fromthe colonies and later served as a starting point for the development of vaccines for thisdisease [11]. In search of answers to questions that disturb humanity during the years, oneof the first mathematical models of the spread of smallpox appeared in the year 1760 andwas proposed by Daniel Bernoulli [3]. Later, deterministic mathematical models of virusspread began to be developed at the beginning of the 20th century. In 1906 a discretetime model was formulated and analyzed for the measles epidemics [10]. In 1911 somedifferential equations based models for the malaria disease were formulated in [18], [19],[20] and [21].Mathematical models of this type of phenomenon clarify which are the important pa-rameters to take into account to obtain concepts such as thresholds, basic reproductionnumbers, number of contacts and replacement numbers. This in turn allows us to makecomputer simulations. Having the information provided by epidemiological models helpsto know what data must be collected in an epidemic, identify trends, make estimates andcalculate uncertainties in these estimates. Later, in 1926, models were formulated fromwhich thresholds could be calculated from which an epidemic outbreak appears and this3appens when the number of susceptible reaches a critical value [19],[22] and [23]. Themodel presented in [23] is a reference in the modelling with non-linear dynamic systems ofthe phenomenon of virus propagation in networks. This type of model is known as com-posed by compartments since it conceives the states through which an individual passesduring an epidemic as boxes or compartments. The compartments are labelled by theletters
M, S, E, I, R and S that cover the different characteristics that one may have, suchas having generated antibodies from the mother’s womb by maternal transmission, beingsusceptible, being exposed, being infected and having recovered from the disease respec-tively. It is entered vertically either to the state M or to the state S and it is exitedvertically when passing to a state of death. There are horizontal transfers between state M to S , from S to E , from E to I and from I to R as it is shown in figure 1. Some of thesecompartments will be present in a specific model and others will not be, depending on theparticular characteristics of the disease to be modelled. For instance, if the disease to bemodelled produces permanent immunity, do not have a passive immunity transmitted bythe mother and there is no incubation period, then the compartments that will be presentare S, I R (Susceptible, Infected and Recovered) and will be called a
SIR -model.Figure 1: The general transfer diagram for the MSEIR modelFrequently in epidemic models the average number of secondary infections from oneindividual is called the reproduction number and is denoted as R and normally is biggerthan 1. When the time span of the process is quite long, the model is called endemic otherwise it is called epidemic . In the case of endemic models, factors as population growth4r population decrease must be included and it affects the calculation of R . In othermodels the the demographic age structure are taken into account and this can change thecalculation of R as well. In the next subsection I will formulate the classic mathematicalepidemic models known as SIR and
SIS and show how they work with some simplesimulations that I have implemented in
MATLAB . The first epidemic model that I will describe is the classic
SIR model that models diseaseswhere some individuals start out being susceptible. These individuals can transition toa state of infection by contagion and after a time they transition to a state of recoveryobtaining permanent immunity. The arrow labelled as horizontal incidence in figure 1represent the infection rate with which a susceptible individual is in contact with infectednodes and make them transit to an infected state. S ( t ) represent the number of susceptiblein time t , I ( t ) the number of infected in time t , N the total size of the population, s ( t ) = S ( t ) N the fraction of susceptible of the total population in time t , i ( t ) = I ( t ) N the fraction ofinfected of the total population in time t , N the population size, s ( t ) = S ( t ) N the fraction ofsusceptible, i ( t ) = I ( t ) N the fraction of infected of the total population in time t and β theaverage number of adequate contact or sufficient for transmission. The average numberof contacts with infected per unit time of one susceptible is expressed as βIN = βi and βIN S = βN is the number of new cases per unit time. The transitions from the boxes M, E and I in figure 1 are calculated δM, (cid:15)E and γI . These terms represent the exponentiallydistributed waiting time in each box that in the cases of the I compartment in figure 1 thetransfer rate γI P ( t ) = e − γt is the fraction that is still in the infective class t units afterentering this class and γ is the mean waiting time.The parameters R , σ and the replacement number R are related with the threshold.For more details about the classical SIR model consult the article [11]After having defined the relevant elements of the phenomenon of the spread of a disease,we are able to define the following
SIR epidemic model: dSdt = − β ISN , S (0) = S ≥ dIdt = β ISN − γI, I (0) = I ≥ dRdt = γI, R (0) = R ≥ S ( t ) + I ( t ) + R ( t ) = N . If we divide by N (the total population) the equations1 we get dsdt = − βis, s (0) = s ≥ didt = βis − γi, i (0) = i ≥ r ( t ) = 1 − s ( t ) − i ( t ) where s ( t ) , i ( t ) and r ( t ) are de fractions in the classes.We are also able to define the following SIR endemic model5
Sdt = µN − µS − β ISN , S (0) = S ≥ dIdt = β ISN − γI − µI, I (0) = I ≥ dRdt = γI − µR, R (0) = R ≥ S ( t ) + I ( t ) + R ( t ) = N . The SIR model 3 is almost the same as the epidemicversion 1 except that it has an inflow of newborns into the susceptible class at rate µN and deaths in the classes at rates µS, µI and µR . If we divide by N (the total population)the equations 3 we get dsdt = − βis + µ − µs, s (0) = s ≥ didt = βis − ( γ + µ ) i, i (0) = i ≥ r ( t ) = 1 − s ( t ) − i ( t ) where s ( t ) , i ( t ) and r ( t ) are de fractions in the classes.For a deep exposition of more sophisticated models and analysis of their respectivethresholds consult [11].The next figure correspond to a simulation of the SIR modelFigure 2: Simulation of classical epidemic SIR model with β = 0 . , γ = 0 . SIR model, the behavior in time of the infected curve reachesits peak or acme and then decreases until it is extinguished as expected, since the type ofepidemic processes that the model describes is related to infections that produce permanentimmunity, which is also observed in the corresponding growth in behavior over time of therecovered curve. It is also important to note that the curve of the proportion of Infectedvs. Susceptibles converges to a fixed point in the coordinates (0 , SIS model can be defined as follows dSdt = − β ISN + γI, S (0) = S ≥ dIdt = β ISN − γI, I (0) = I ≥ SIS modelFigure 3: Simulation of classical epidemic SIS model with β = 1 , γ = 0 . SIS model, the behavior in time of the infected decreases untilit converges to some given level and simultaneously the number of susceptible grow untilsome level and stays there as expected, since the type of epidemic processes that the modeldescribes is related to infections that produce temporary immunity. It is also importantto note that the curve of the proportion of Infected vs. Susceptible converges to a fixedpoint in the coordinates (0 . , . Mathematical epidemic models assume that each individual has on average the same num-ber of contacts. In my opinion, this hypothesis is not always fulfilled because it does notcorrespond to the type of structures that arise in networks such as the internet in mediasuch as Facebook or tweeter. In addition, the type of structures that appear in these so-cial networks partly reflects the way in which people build networks of collaboration andinteraction in real life. For this reason, I believe that just as the mathematical models ofepidemics have contributed to the development of discrete mathematical models to study7he phenomena of virus propagation in computer networks, also the mathematical modelsof epidemics can benefit from the advances that take place in the study of virus spreadin computer networks. Since computer networks have grown rapidly, security problemshave also increased in the same proportion. In the same way, the number of types ofservices offered on networks such as the Internet has grown. One service that appearedhas been the distribution of content over the internet, which requires ensuring that theinformation reaches its destination quickly and with a good level of quality. In this typeof services, P2P-type networks have been studied. Another type of service that arises withthe appearance of networks is that of sensor networks. When trying to solve both securityand quality of service problems, it is necessary to resort to mathematical tools that allowmodeling the phenomena inherent in computer networks and answering questions such aswhich network structure is the most appropriate to ensure that the network stay connectedin the presence of faults in the lines or what type of topology ensures that the distancetraveled by an information packet is short enough for it to reach its destination withoutdelays. On the other hand, the security of a network must be guaranteed and therefore itis important to know which network topology facilitates or inhibits the spread of a virus init. These topics have aroused the interest of researchers from fields as varied as statisticalphysics or experts in the field of random graphs.The type of graphs that appear most frequently in the study of virus propagation incomplex networks and that we will use to illustrate the operation of discrete SIS models arethose of binomial distribution of degrees, distribution of degrees in power laws, exponentialtype and lattices . For the sake of clarity I give below the definitions of these types of graphs.
Definition 3.1 Power law or scale-free degree distribution graph
Is a graph whosedegree distribution of nodes follows asymptotically a power law. More formally let P ( k ) thefraction of the total number of nodes in a given graph that have k connections with othernodes. This fraction of nodes have the following behaviour P ( k ) (cid:118) k − γ (6) where γ is a parameter in the interval < γ < Definition 3.2 Binomial degree distribution graph (Erd˝os–R´enyi model, Barabasi-Albert model, etc.)
Is a random graph whose degree distribution of nodes follows abinomial probability distribution law of degrees k that can be formally defined as follows.Each of the n nodes of the graph is independently connected with other node with probability p or not connected with probability (1 − p ) . Let P ( k ) the fraction of the total number ofnodes in a given graph that have k connections with other nodes. This fraction of nodeshave the following behaviour P ( k ) = (cid:18) n − k (cid:19) p k (1 − p ) n − − k (7) Definition 3.3 Exponential degree distribution graph
Is a random graph whose de-gree distribution of nodes follows a binomial probability distribution law of degrees k that an be formally defined as follows. Let P ( k ) the fraction of the total number of nodes ina given graph that have k connections with other nodes. This fraction of nodes have thefollowing behaviour P ( k, λ ) = (cid:26) λe − λk k ≥ k < where λ > is a parameter of the distribution called rate parameter . Definition 3.4 Lattice 4 connected graph (grid graph, mesh graph, etc)
Is agraph that each node is connected to four other nodes for all the n nodes belonging to thegraph. Next I will show some examples of these graphs.Figure 4: Binomial degree distribution graphFigure 5: Exponencial degree distribution graph9igure 6: Power law degree distribution graphFigure 7: Lattice 4 graphThe understanding of the emergence of giant components or phenomena of small worlds that occurs on the Internet requires mathematical tools such as the theory of contactprocesses as well as the theory of random graphs to be able to analyze these phenomena.Many research articles have been written about such subjects [8],[1, 2, 14], [15, 16, 17].Some papers about rumours spreading on networks under the approach of contact processeshave also been written [13]. The problem of vaccine distribution on networks can beconsulted in [4].Finally the use of tools such as non-linear dynamical systems, fix-point theorems forobtaining fast extinction conditions of a virus in a network combined with a discrete versionof a
SIS epidemic model can be found in [12]. This is the model that I will describe in thefollowing subsection.
In order to understand the spread of viruses on a network, the model proposed in [12]assumes that the nodes behave according to a SIS-type model and that they are inter-10onnected by a network. They also assume that we take very small discrete timesteps ofsize ∆ t where ∆ t →
0. The survivability results in [12] apply equally well to continuoussystems. Within a ∆ t time interval, each node i has probability r i of trying to broadcastits information every time step, and each link i → j has a probability β i,j of being up , andthus correctly propagating the information to node j . Each node i also has a node failureprobability δ i >
0. Every dead node j has a rate γ j of returning to the up state, but with-out any information in its memory. The details about the parameters of this model as wellas the fast extinction conditions and stability results can be consulted in [12]. the authorsof [12] chose to use the non-linear dynamic systems approach and fixed point theorems.The state transitions at each node are shown in the figure 8.Figure 8: Chakrabarti SIS modelThe node state has info can be think as being infected. The authors of [12] proposed toobtain an approximation of the threshold by describing the problem as non-linear dynamicsystem with N variables representing the nodes and assumed that the state of two differentnodes are independent. The independence condition can be formally expressed ad follows: ζ i ( t ) = N (cid:89) j =1 (1 − r j β ji p j ( t − p i ( t ) = p i ( t − − δ i ) + q i ( t − − ζ i ( t )) (10) q i ( t ) = q i ( t − ζ i ( t ) − δ i ) + (1 − p i ( t − − q i ( t − γ i (11)11 include below the theoretical results related to the fast extinction condition withoutdemonstration in order to have a clear description of the SIS model [26] Definition 3.5
Define S to be the N × N system matrix: S ij = (cid:26) − δ i if i = jr j β ji γ i γ i + δ i otherwiseLet | λ ,S | be the magnitude of the largest eigenvalue and (cid:98) C ( t ) = (cid:80) Ni =1 p i ( t ) the expectednumber of carriers at t of the dynamical system. Theorem 3.6 (Condition for fast extinction) . Define s = | λ ,S | to be the survivabil-ity score for the system. If s = | λ ,S | < , then we have fast extinction in the dynamicalsystem, that is, (cid:98) C ( t ) decays exponentially quickly over time. Where | λ ,S | is the magnitude of the largest eigenvalue of S , being S an N × N sys-tem matrix defined as S ij = 1 − δ i if i = j and S ij = r j β ji γ i γ i + δ i otherwise, and being (cid:98) C ( t ) = (cid:80) Ni =1 p i ( t ) the expected number of carriers at time t of the dynamical system. Twoadditional results that appears in [12] are the following Corollary 3.7 (Condition for fast extinction homogeneous case for ChakrabatiSIS model) If δ i = δ, r i = r, γ i = γ , for all i , and B = [ β ij ] is a symmetric binarymatrix (links are undirected, and are always up or always down), then the condition forfast extinction is γδ ( γ + δ ) λ ,B < δ, γ and β as well as on the largest eigen-value λ ,S of the matrix of the dynamicalsystem and therefore on the topology of the net.Based on the equations 10 and 11 of the SIS model, implement a simulation in MATLABwhose results I show below. 12igure 9: Simulation of Chakrabarti SIS model with δ = 0 . γ = 0 . , β = 0 . γ and β in incrementsof 0 .
05. Each process converges to states where the number of infected is higher than thenumber of susceptible.Figure 10: Simulation of Chakrabarti SIS model with δ = 0 . γ = 0 . , β = 0 . δ = 0 . γ = 0 . , β = 0 . γ in increments of 0 . δ = 0 . γ = 0 . , β = 0 . γ in increments of0 .
05. It can be seen that the process achieves fast extinction condition given that thetopology ia lattice. So with this example we can observe that the topology can make adifference between having a fast extinction of the epidemic or converging to a state wherethe number of infected is bigger the number of suceptible and stays stable in such state.In [27] it was formulated a similar discrete epidemic model proposed by the authors of[12] having one additional states in order to let the nodes to be warned by a message or toreceive a vaccine.The idea behind this additional states was to explore prevention alternatives as well asthe possible eradication of a virus in a computer network either through warning messagesor by distribution of a vaccine [4]. The results regarding the fast extinction condition ofthe virus as well as the fixed point results were very similar to those of the [12] model.Each node can be in one of three states:
Infected , Warn Info , No Info or Dead , withtransitions between them as shown in Diagram 13. The next graph represent the transi-tions that take place in each node for this model.15igure 13: The SIRS modelMaking the same node independence probability assumption that is stated in equation(9) and taking into account the new states and transition probabilities shown in the figure13, the equations (10) and (11) as well as the new equation corresponding to w i can beexpressed as follows: p i ( t ) = p i ( t − − δ i ) + q i ( t − − ζ i ( t )) ν i (12) q i ( t ) = q i ( t − ζ i ( t ) − δ i ) + (13)(1 − p i ( t − − q i ( t − − w i ( t − γ i + χ i w i ( t − w i ( t ) = (1 − ζ i ( t ))(1 − ν i ) q i ( t −
1) (14)+(1 − χ i − δ i ) w i ( t − δ = 0 . γ = 0 . , ν = 1 , χ = 1 , β = 0 . γ and β in increments of 0 .
05. Eachprocess converges to states where the number of infected is higher than the number ofsusceptible.Figure 15: Simulation of SIRS model with δ = 0 . γ = 0 . , ν = 1 , χ = 1 , β = 0 . As can be seen in the results obtained in the simulations of both the Chakrabarti SIS modeland the SIRS model, the network topology has a significant impact on the rapid extinctionof a computer virus in a network or its permanence in it. Given that the topology of thenetwork is related to its adjacency matrix, a possible isolation strategy to minimize thefirst eigenvalue of said matrix [24]. The question is that this strategy would be reducedto the search for a Hamiltonian circuit, which in general is an NP-hard problem. Onepossibility is to look for algorithms that approximate the obtaining of the minimum valueHamiltonian path by algorithmic techniques of the closest neighbors type that seek toapproximate the minimum value Hamiltonian cycle with some guarantee of approximationand eliminate from the original interconnection graph those edges that do not belong tosaid cycle. This is what is proposed in [25].Given that the problem of minimization of eigenvalues of the adjacency matrix is NP-hard and the algorithms of approximation to the problem of calculation of Hamiltoniancycles that guarantee a certain quality of approximation, I would propose to try to trans-form any power law graph to a graph of type lattice 4 to reduce the mentioned eigenvaluein some way, even if it is not the minimum. An application to contain the spread of avirus in a network, be it computerized or of people, could be to detect the value of theparameters that characterize a process of diffusion of a virus taking into account the inter-connection medium where the epidemic process will take place and from there determinewhich network nodes to isolate by modifying the adyascences of the associated graph insuch a way as to obtain a notable reduction in the spread of a virus or even achieve itsrapid extinction.
In this article, he makes a historical account of the classic mathematical models for thestudy of epidemic processes that made it possible to develop models to address problemsof virus spread in networks. We were also able to observe that the concerns that arisewhen trying to solve problems that arise in the field of virus spread can contribute ideasto the development of mathematical models in epidemiology by incorporating aspects ofthe interconnection networks on which epidemics take place. By introducing the struc-tures of interconnection networks to epidemiological models, we can obtain elements thatguide epidemiologists in making decisions about isolation strategies to try to contain andeventually eradicate a disease through the development of strategies based on knowledge.of the dissemination of information in interconnection networks.18 eferences [1] R. Albert and A.L. Barab´asi. Error and attack tolerance of complex networks.
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