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I. Introduction

THE epidemiology of viral diseases is a discipline that deals with the study of determinants, predictions and control of factors related to health and disease, as well as the study of the dynamics and distribution of viral diseases in a population [1].

In order to establish the dynamics of these diseases and to carry out a pursuit or control to these, it is made use of mathematical tools, like the ordinary differential equations, which allows to establish relations between these behaviors by means of variables and parameters or factors that influence the development or extinction of the disease.

There are various mathematical models that explain the dynamics of their epidemiological states by assuming that each individual has the same opportunity to come into contact with any other individual in the population. For example, for diseases such as HIV, there are two types of states: those who have already contracted the disease and remain infected and infectious to others, and those who are susceptible to contracting it because they are in a risk zone. This type of model is known as the IS model and is characterized by the fact that the individual acquires the virus from an infected person without acquiring immunity [2, 3]. Similarly, for diseases such as influenza, can be described by the SIS model [4, 5], which gives way to the individual can move from an infected state to susceptible and be prone to acquire the disease again. For diseases in which the individual acquires immunity are explained by SIR models [6].

One of the major challenges is the spread of diseases for people who are associated with a limited number of individuals in a population, as millions of people now cross the borders of many countries every day, increasing the likelihood of an epidemic or pandemic, as well as the invasion of ecosystems and environmental degradation that can create opportunities for existing and new infectious diseases. Therefore, one of the solutions to this difficulty is to fit an epidemiological model, such as the case of the IS, SIS or SIR, into a network when few infected individuals enter.

Therefore, the aim of this work is to analyze the dynamics of SI, SIS and SIR models, adjusted or not in a network, and to compare the necessary conditions to locate or not the disease in a constant and closed population along the time. In order to understand this process, the following steps are developed: in the second section the classical SI, SIS and SIR models are analyzed. In the third, fourth and fifth section, the construction of the SI, SIR and SIS models in networks, respectively, is shown and an analysis of the short- and long-term dynamics is made to determine the conditions in which the disease spreads or not and to compare them with the conditions of the models described given in the second section. Similarly, construction methods are shown to approximate model solutions on the network due to the complexity of finding an analytical solution.

II. Classic epidemiological models

By assuming that the nodes and their edges in a network do not present variations over time, a qualitative analysis is then made to the main epidemiological models, represented by a system of ordinary differential equations, without considering birth or natural death rates due to disease.

A. Model SI

Consider a population with individuals, constant and closed over time, divided into two states, infected when in contact with an infected and susceptible to this. Let and the number of susceptible and infected individuals, respectively, at any one time , so that .

As seen in Fig. 1 grafo 𝒌-regular, if represents the rate of infection, and by assuming that the individual has the same opportunity to come into contact with any other individual in the population, the change in with respect to is represented by,

where is the flow of infection between healthy and infectious individuals per unit of time.

Since , then change of per unit of time is given by

Thus, the final model is

If and represent the fractions of susceptible and infected individuals at the time respectively, then and (1) is equivalent to

Due to the biological interest in analyzing the dynamics of since the beginning of the disease . Consider y the number of infected and susceptible individuals, respectively, equivalent to and as initial conditions of (2).

Since , from (2) we have

that is,

When integrating with respect to for both sides of (4), we have to

where is a constant of integration. When considering the particular solution of (3) in

An important question in any epidemic is whether or not the infection spreads over time . In the case of propagation, we must determine how it develops over time or when it will start to diminish. Indeed, as

for everything from (2) an epidemic is shown to always spread and eventually infect all susceptible individuals if . However, if then for everything , that is, the population will remain in a susceptible state. Similarly, for fixed, converge to 1 with a faster propagation speed each time as seen in Fig.2.

Therefore, the following result has been verified.

Theorem 1Let a solution of (2) with and initial condition . Then when while for , that is, when y for .

B. SIR Model

Unlike the SI model, consider that the population at the time is divided into three stationary, susceptible states , infected by the disease and recovered without acquiring the disease again

In view of Fig.3 and as stated [7], suppose that the susceptible fraction which becomes an infectious fraction is proportional to the product of its fractions, that is, the rate of loss of the susceptible fraction is , where is the infection rate. Therefore, the change in the susceptible fraction regarding time is given by,

where the negative sign represents the loss of the susceptible fraction.

Since also indicates the gain rate of the infectious fraction, and represents the rate of profit of the recovered fraction, i.e, indicates the exit of the infectious fraction, the change of with respect to time is represented by

and the equation that describes the change of the recovered fraction es,

As the total population was divided into three states, susceptible, infectious and recovered, you have to equivalent to and therefore the model is represented by

The objective is to determine the conditions in which the epidemic spreads or not from , , and , equivalent to , and .

From (6)we can be seen that , that is, the susceptible fraction will decrease as long as there are infectious individuals, and therefore for everything . Similarly, , therefore , for everything , if there are infectious individuals.

On the other hand,

· If , this is , then and when , this is, the fraction of infected people decreases.

· If when , then if . Therefore, the number of infected people will increase and there will be an epidemic. So, for some there will be an epidemic outbreak if .

Since the first two equations of (6) do not depend on , then

which if and if .

By integrating (7) with respect to , we have to

where is a constant of arbitrary integration.

Considering the initial conditions y at (8)we have

and therefore the dynamics of (6) on the plane in initial condition y is given by

From (9) you must y which there is a point such that with . Note that , that is, the point corresponds to a balance of (6).

On the other hand, the maximum number of the infectious fraction, at any moment of time, satisfies when . From (6) you must if for . When replacing the value of in (9) you have to

as seen in Fig.4.

Note that, from Fig.4, if then for everything and so when . Therefore, the following result shows the dynamics of the model (6) with respect to the relationship between the parameters , and its initial conditions.

Theorem 2. Let a solution of (6) with initial condition . If , when . For , if then when and, if then first increases until a maximum value is reached and then decreases to zero when . The susceptible fraction is a decreasing function and the limit is the only root in of the equation

Fig.5 shows the dynamics of (6) for fixed and various initial conditions .

C. SIS Model

This model extends the IS model by considering that individuals can recover but do not acquire immunity to the disease. In this case, consider and as the susceptible and infected fraction, respectively, at the time and .

Looking at Fig.6, the following model is proposed

where is the rate of infection and recovery rate, with initial conditions and .

Since for everything , the change in the fraction of infected people with respect to time takes the form of

than by the method of separation of variables, with initial conditions and , the particular solution of (11) is given by

Therefore, we have the following result,

Theorem 3. Be the solution of (10) with initial condition and . If then when , which tends to disappearto . If then

and so the disease does not disappear over time.

Fig.7 shows the dynamics of (10) for various conditions on the , .

III. SI model in a network

Consider a network between individuals, represented by a network , where the population is represented by nodes and the contact between individuals is represented by edges. This network can be represented by an adjacency matrix size which represents the number of edges for each pair of nodes.

Consider that each node in the moment belongs to an infected state by a disease or a susceptible state . That is, for a node arbitrary, or , where .

If each node is connected by a neighbor , the change in probability of an infected node with respect to time is given by

where is the flow of infection between the infected node and every susceptible neighbor from in the moment[8].

Since for everything , the change in the probability of a node susceptible with respect to time is represented by .

Therefore, the model to be considered is

equivalent to

or

From (13) we can be seen tha t , that is, is decreasing and converge to zero when infectious nodes exist and , is growing and converge to one as there are susceptible nodes . Therefore, the dynamic (13) is equivalent to the dynamic in (2).

Analogous to the initial conditions of the classical SI model, suppose the disease starts with an infected node or a of nodes, chosen at random, such that and .

Since in (14) converge to one, the behavior of the system must be analyzed for a short time to determine its propagation speed. Indeed, if is close to zero, and for big enough. Thus, from (14) and ignoring the terms of quadratic order, we have

equivalent, in matrix form, to

where .

If represented as a linear combination of the own vectors , associated to the own values of the matrix , this is ,

where are constants that depend on , of (16) we have,

Then,

with particular solution

By replacing (18) in (17),

and therefore, it is expected that the solution grows exponentially from short moments of time and, unlike the conclusions in the dynamics of the classical SI model given in Theorem 1, the growth depends on and . That is, if , the disease spreads more slowly if the network is more dispersed, that is, if and, for denser networks, this is, , the disease spreads more quickly.

Fig.8 shows some simulations of (13), made in Matlab, in a network -regular [9], i.e. with neighbors for each node . There is evidence that the contagion spreads throughout the network to increasingly shorter as the number of neighbors increases.

A. Model based on the degree of a node

Since the model (13) cannot be solved analytically, a model must be fitted to approximate the solutions of the SI model in a network whose dynamics in which its states are explained by the degrees of the nodes in the network [8].

As seen in Fig.9, a network is said to have components if any subgraph of , and the largest component of is the subgraph that has the largest number of edges.

Since the degree of a node is given by the number of its neighbors, consider as the distribution in degrees of a network , where is the fraction of nodes that have degree and suppose that the infected nodes make contacts independently of each other, i.e, . Consider the probability generating function given by

The average grade is given by

More generally, we define the moments

As reference [10], when a node in a network is infected, the infection is transmitted to every connected individual except the edge from which it came. Therefore, the excess grade of a node that is one less than the grade is used. The probability of reaching a node of degree or excess degree , when following a random edge, it is proportional to , and therefore the probability that a node at the end of a random edge has a degree of excess is a constant multiple of with the constant chosen to make the sum over of the probabilities is equal to . So the probability of a node having an excess of degree in

where

This leads to a generating function for excess grade,

and the excess of medium grade, denoted by ,

Note that in the largest component of if , that is, the largest component of has fewer grade 1 connections.

If a node is infected and belongs to a small component with few or no connections in , the probability that the infected node will spread throughout the network is zero. Therefore, an SI model will be adjusted for a large component of such that . Consider as the probability of a grade neighbor is infected, and the degree of excess is distributed according to . Then, the average probability that the neighbor is infected is

where .

If the node's neighbor is infected, the probability of the disease being transmitted to the in the given time interval is . Then the total probability of transmission from a single neighbor during the time interval is and the probability of transmission from any neighbor is where is the number of neighbors of . Furthermore, 𝑖 is required to be susceptible, which occurs with probability ), so the final probability that get infected is [8]. Therefore, the change of is given by

with particular solution in ,

If we consider

(23) takes the form

To calculate an equation must be constructed in terms of without being dependent on as observed in (24). In effect, from (22) and (25) we have

equivalent to

When we consider that , (21) is equivalent to

Therefore,

is used to determine the values of .

Finally, to calculate the total fraction of infected individuals in the network, is averaged over in this way

However, the solution of (27) cannot be explicitly calculated, but an analysis can be made for both short- and long-term times. Indeed, when , of (24) must be . Then, is, by definition, positive and not decreasing, of (24), when . By assuming that the infection starts with only one or a handful of individuals, so for some constant , we have when , this implies that, in the long term, (27) it becomes

with general solution,

Then, the long-term behavior of This is determined by because grade one nodes are the last to be infected.

On the other hand, as and is decreasing, consider for short periods of time. Then, from (27),

and considering that

we have, ignoring the terms of higher order, which

where is the initial value of . Since (29) is a first-order linear equation, using the integral factor , that is,

and by integrating with respect to , the general solution for (29) is

where is a constant of arbitrary integration. When considering , the particular solution of (29) is given by

Therefore,

with given in equation (20).

Finally, for short periods of time, this is, ,

where it is considered and as stated in (20). Therefore, as expected, the initial growth of the infection is more or less exponential. Similarly, it is expected that increase rapidly if , that is, represents how quickly the network branches out as it moves away from the node where the disease first begins.

IV. SIR model in a network

Consider , and . As it is proposed [8] and in an equivalent way to the construction of the classical SIR model and the SI model with network, the changes of , , and per unit of time obey the system

where is the probability per unit of time that an infected individual will recover. In addition, consider in the instant , , and [8].

From (30) it can be seen that decreases and grows when infectious nodes exist . Therefore, and equivalent to the SI model in a given network in section 3, the dynamics must be analyzed for short time steps. If , and when , and so by ignoring the infected of quadratic order, of (30) can be approximated as

where

In matrix form, (31) takes the form,

where

If is an eigenvector associated with the eigenvalue of the matrix , then

that is, the value of associated to the own vector in . When considering as given in (17), we must

with particular solution,

Then

If , decreases exponentially to zero. On the other hand, if the main eigenvalue is small, the probability of infection must be large, or the recovery rate to make the disease start to spread, equivalent to verifying that . Therefore, and unlike the conclusions of the dynamics of the classical SIR model given in Theorem 2, the epidemic threshold occurs in , that is,

Fig.10 shows that, when fixing for a network -regular, decreases to zero as , where in a network -regular [11], and tends to grow to a certain for big enough.

A. Model based on the degree of a node

Consider , and the odds that a neighbor with a degree is susceptible, infected or recovered, respectively, in a time . When considering a node who is a neighbor of a susceptible , we have to contracts the disease from one of his neighbors other than . Then the probability that infection is given by with the degree of excess. So that recovery depends only on the probability that we have been previously infected, which is given by , where is the degree of excess, and the probability if susceptible can be derived from [8].

Analogous to the construction of the SI model based on the degree of a node, the change of , and per unit of time is given by

where the average probability of a neighbor being infected is given by

If all the degrees of the nodes on the network have degree one, the dynamics of (34)can be expressed in an equivalent way as observed in Fig.5.

To find an analytical solution to the model (34), consider

that is, the average probability of the neighbors are recovered.

From (34) and (35), we have to

that is,

that when used in (34), the change of with respect to time is written as

equivalent to

and whose solution, with respect to the initial condition for , is

When considering

equivalent to

(38) is rewritten as

The goal is to find an equation to calculate without depending on the unknown variables . In fact, when using (40) and (41), we have to

and so on,

that is,

obtained from (34) and (36). Therefore, the total of the susceptible fraction is of the form,

To find the total fraction of , note that of (44),

than by integrating,

and use (41) and (42), you must

To calculate the total recovery fraction, it is enough to see that .

Because many times the solution of (43) cannot be explicitly calculated, certain properties are used to analyze the behavior of . Indeed, by assuming that , and , we have to and for . Therefore, from (44) we have, for ,

If , from (39) we have to and . Then, for moments of short time and given that is decreasing, considering ,

and ignoring the terms of of a higher order, it is known that (43)

Therefore

and

Thus the epidemic threshold is given by

with given in (20), equivalent to the epidemic threshold given in (33) when replacing by .

V. SIS model in a network

In a way equivalent to the construction of the SI model in a network and the classic SIS model, the changes of and per unit of time are given by

Since , the change of per unit of time, given in (48), can be written as

Assuming that for everything and constant For and ignoring the terms of of higher order, of (49) must

which is equivalent to (31). Therefore, the epidemiological threshold is given by

that is, if It is expected that decreases exponentially to zero, while if , begins to grow for short moments of time as observed in Fig.11.

A. Model based on the degree of a node

Equivalent to the construction of the SI and SIR models based on the degree of a node, the change of y per unit of time is given by

where

Since , then

As in the SIR model based on the degree of a node, if all the degrees of the nodes on the network has degree one, the dynamics of (34) can be expressed in an equivalent way as observed in Fig.7.

Because no explicit solution is known for (52), its behavior is analyzed when and . By assuming that for constant and, for moments of small time when and ignoring the terms of of higher order, of (51) and (52) that

which corresponds to a first-order linear equation. When using the integral factor, we have to

with particular solution in given by

When considering

with

it is known from (51) that

The objective is to calculate without being determined by values . Indeed, since

from (54) we have

with particular solution in

Therefore, for short term time we have

whose epidemic threshold is

which corresponds to the same threshold (47) given in the SIR model based on the degree of a node.

When it is expected that tends to the balance of (52), that is,

VI. Conclusions

The models presented depend on the hypothesis made about the population and the states of the disease. If each individual has the same opportunity to interact with another, a classical epidemiological model could be adjusted, otherwise a network model must be adjusted. Similarly, considering that the disease can have as susceptible, infected and at most recovered state, SI, SIR or SIS models were used. On the other hand, as a comparison between classic and network models was made, a constant and closed population was considered over time, so it was not considered a birth rate or natural or disease mortality.

Since the classical SI model is built by a single parameter representing the infection rate, it is determined that the disease spreads and manages to infect the entire population as long as there is at least one infected, with the difference that the speed with which it spreads decreases when its rate approaches zero. On the other hand, by fitting an SI model into a network, it was determined that the speed of propagation depends both on the infection rate and the largest eigenvalue associated with the adjacency matrix, as a way of represent a dense or dispersed network. Unlike the classical SI model, if the infection rate approaches zero, the disease spreads more rapidly if the network is denser.

For a classical SIR model, regardless of the infection and recovery rate, the infection tends to be eradicated for a long enough period of time. However, if the ratio between the recovery and infection rate is higher than the initial susceptible fraction, the disease spreads until a certain time and then tends to disappear. This dynamic is similar to the SIR model in a network, with the difference that the ratio between the recovery and transmission rate must be greater than the largest own value associated with the adjacency matrix.

On the other hand, for a classic or networked SIS model, the disease does not disappear over time when the ratio between recovery rate and contagion is greater than one or the highest eigenvalue associated with the adjacency matrix, respectively.

Since it is not possible to find an explicit solution for SI, SIS and SIR models in networks, their states are adjusted by the degree of the nodes in the network. Under this methodology and for short term time, the epidemic threshold for the SIR and SIS models are equivalent to those previously explained by replacing the own value by the excess of the average degree, and for the SI model, the speed of disease propagation is determined by the excess of the average degree above one.

References

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Notas de autor

IR