ANALYTIC SOLUTION TO THE SUSCEPTIBLE-INFECTIVE DISEASE SPREAD MODEL WITH VARYING CONTACT RATE

Hamed Yarmand

Graduate Program in Operations ResearchNorth Carolina State University, Raleigh, NC 27695, USA

Abstract

A common class of epidemiological models developed for the spread of infectious diseases is the Kermack-McKendrick model and its variations. These models are represented as systems of ordinary differential equations which are strongly nonlinear and cannot be solved analytically. In this research, we consider SEIR and SI models, two variations of Kermack-McKendrick model. We let contact rate be a function of number of infectives, which is an indicator of the disease spread during the course of the outbreak. We use the Markov process approach to represent SEIR and SI models as continuous-time Markov chains. The result would be a pure death (or birth) process with state-dependent rates for the SI model, for which we find the transient as well as the steady-state solution of the associated continuous-time Markov chain by solving the Kolmogorov forward equations. Finally we use the solution to the Markov chain to find the analytic solution to the original SI model.

1. Introduction

Ordinary differential equations (ODEs) are the basis for many mathematical models in the sciences including population models used in epidemiology. Specifically, the Kermack-McKendrick model (Kermack & McKendrick, 1927) was one of the first models developed to simulate the number of infectives in case of diseases such as bubonic plague (London 1665-1666, Bombay 1906) and cholera (London 1865). In 1979, Anderson and May (1979) resurrected the Kermack-McKendrick model, as known as the SIR model, for the spread of infectious diseases. In the original model there are three compartments S for susceptibels, I for infectives, and R for the recovered. Also it is assumed that there are no vital rates, so that the population size is constant through time. This simple epidemiological model consisting of three coupled ordinary differential equations has been used for a variety of epidemics, including HIV (Doyle , Greenhalgh, & Lewis, 2001) and SARS (Ng, Turinici, & Danchin, 2003). Due to its applicability, it has become the foundation for more complex, more realistic epidemiological models.

Many variations of the original Kermack-McKendrick model have been described, typically using names based on acronyms of the involved classes. Generalizations of this model are sometimes referred to as SEIRS models. We will refer to Kermack-McKendrick model and its variations as K-M models. Some of these models incorporate a fourth class (Exposed) within the population, accounting for diseases with a latent period. Also some of these models account for nonpermanent immunity, thus allowing individuals to again become susceptible (thus the second S in the acronym). Hethcote (1976) investigated a variety of mathematical models whose classes and interactions are a subset of the SEIRS model, including SI, SIS, and SIR variations. Several investigators have focused on a single specific model, computing theoretical bifurcation points as well as some transient and steady-state solutions. This includes work on an SEI model (Pugliese, 1990), an SEIR model (Li, Graef, Wand, & Karsai, 1999), and an SEIS model (Fan, Li, Wang, Wand, & Karsai, 2001).

In what follows, we represent SEIR and SI models as ODE systems, which are the basis of the associated continuous-time Markov chains.

SEIR and SI modelsIn the SEIR model, the population is categorized into

four different compartments (or epidemiological classes) at each point of time. These compartments are susceptible, exposed, infective, and recovered. The susceptible compartment contains those individuals who do not have any type of immunity to the disease and can become infected. When there is an adequate contact of a susceptible with an infective so that transmission occurs, the susceptible enters the exposed compartment of those in the latent period, who are infected but not yet infectious. After the latent period ends, the individual enters the infective compartment of those who are infectious in the sense that they are capable of transmitting the infection. When the infectious period ends, the individual enters the recovered compartment consisting of those with permanent (at least for the time horizon under study) infection-acquired immunity. Flu is an example of diseases which spread is usually modeled using the SEIR model.

We assume that there are no vital rates. This assumption is valid if for example the time horizon is short enough, so that births or deaths could be ignored. We assume that the

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mean latent and infectious periods are equal to and ,

respectively. We denote the population at each of the compartments at time by , , , and (or

simply , , , and ), respectively, and the total

population size by . Since there are no vital rates, the total population size is constant and we have

for all .

If is the effective contact rate, that is average number of adequate contacts (i.e., contacts sufficient for

transmission) of a person per unit of time, then is the

average number of contacts with infectives per unit of

time of one susceptible, and is the number of new

cases per unit of time due to the susceptibles. As a result, if we denote the horizontal incidence, the rate at which susceptibles become infected, at time by , then we have

(1)

This nonlinear form of the horizontal incidence is called the standard incidence, because it is formulated from the basic principles above (Hethcote, 2000).

With the horizontal incidence in (1), the system of differential equations which describes the dynamics of the SEIR model would be as follows.

(2-a)

(2-b)

(2-c)

(2-d)

If the latent period is so short that could be ignored and also there is no recovery after infection (at least in the time horizon under study), the exposed and recovered compartments can be eliminated, resulting in the SI model, in which any individual is either susceptible or infective. An example of such diseases is HIV. Therefore the ODE system which describes the disease spread dynamics would become as follows.

(3-a)

(3-b)

Both systems (2) and (3) are strongly nonlinear and have no generic analytic solution, although there are

numerical methods to approximate the solution. Indeed the associated ODE systems of all K-M models are strongly nonlinear, since they all have the nonlinear form of the horizontal incidence (1).

Nevertheless, finding the exact solutions to K-M models is of great importance, since knowing the exact solution would be very helpful in analyzing the model properties and characteristics of the equilibrium points, finding different thresholds, conducting sensitivity analysis on different model parameters, etc.

In addition, in K-M models, it is assumed that the contact rate, which is a reflection of people’s social behavior regarding the disease spread, is assumed to be constant during the course of the outbreak. However, people’s social behavior changes as society copes with the outbreak (Larson, 2007). Therefore we considered the contact rate as a function of number of infectives, which is an indicator of the disease spread and its threat for susceptible individuals; hence we denote the contact rate by in the rest of this paper to emphasize its dependence on the number of infectives. Naturally should be decreasing in , but this is not a requirement in our model.

In this research, we use a Markov Process (MP) approach to formulate SEIR and SI models, which are two instances of K-M models, as continuous-time Markov chains (CTMCs). Then we solve the Kolmogorov forward equations for the developed CTMC for the SI model and find the transient and steady-state probability distribution of the state of the system. Then we use the state probability distribution to find the analytic solution to the original SI model.

The rest of this paper is organized as follows. In the next section we present a review of the literature related to solutions of K-M models. In Section 3, we represent SEIR and SI models as CTMCs. In Section 4, we derive the analytic solution of the SI model by solving the Kolmogorov forward equations for the CTMC associated with the SI model. We conclude this paper in Section 5 by summarizing our findings and suggesting some future directions.

2. Literature Review

Literature related to solutions to K-M models falls in two general categories. In the first category, researchers directly deal with the epidemiological model and attempt to find the exact or a good approximate solution; while in the second category, other approaches whose results can be used to find the exact or approximate solution are followed. First we briefly review the literature in the first category, which is not as developed as the second category.

In special cases of K-M models, the exact solution can be found. For example, Newman (2002) uses a network approach to find the exact solution of the SIR model in the unrealistic case of fixed infectiveness time and fixed and

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uncorrelated probability of transmission between all pairs of individuals. Also he has solved cases in which times and probabilities are nonuniform and correlated.

Gani (1965) considers the differential-difference equations of the SIR model with constant population size and gives the partial differential equation which the associated probability generating function satisfies and outlines a mathematical method for solving it. Unfortunately, the mathematics involved are so complicated, limiting its success somewhat in the SIR model, to very small population sizes of at most 3 individuals.

Gart (1968) considers an SI model with constant population size and two kinds of susceptibles having very different infection rates. Then he gives the exact solution in an implicit form and also derives an approximation to the solution which permits simple estimates of the infection rates.

There have also been efforts to develop numerical methods to approximate the solution of K-M models. Boquet (2010) uses discrete models to qualitatively capture all the dynamics of these models. He develops two discrete models based on the Lewis-Glass hypercube projection method (Lewis & Glass, 1991) and the Laubenbacher-Stigler polynomial interpolation method (Laubenbacher & Stigler, 2004) to find discrete approximations of the dynamics of the associated K-M model which are qualitatively similar to the dynamics of the continuous model.

Even in the absence of uncertainty, traditional numerical methods, such as Euler’s method or the Runge-Kutta schemes, only approximate the solution of the associated ODE system, since truncation errors from both function approximation and machine arithmetic are present. When there is uncertainty in the initial conditions and/or model parameters, normal use of traditional methods cannot account rigorously for the uncertainties. Therefore there has been efforts to find verified (i.e., mathematically and computationally guaranteed) solutions of such systems of ODEs. For example, Enszer and Stadtherr (2010) have considered continuous epidemiological models that are systems of ODEs and formulated as initial value problems (IVPs) and used interval methods (also called validated or verified methods) to find rigorous bounds on the solution of these systems of ODEs, especially systems that involve uncertainty in initial conditions or model parameters.1

In the second category of related literature, the most important tools are birth and death processes (BDPs), since some instances of K-M models, such as SI model, can be represented as one- or multi-dimensional BDPs, as shown in Section 3 for the SI models.

BDPs have been considered from long time ago for their applications in various fields such as population models (Bartlett, Gower, & Leslie, 1960), population genetics (Mode, 1962), epidemic theory (Gart, 1968), species interaction processes (Leslie & Gower, 1960), and medicine (see Birkhead (1986) for an application to drug resistance in cancer therapy). Therefore there has been research on finding the analytic solution to BDPs. Kendall (1948) gives the analytic solution to a generalized one-dimensional BDP (1-BDP) as a population growth model. His main idea is to replace the differential-difference equations for the population distribution by a partial differential equation for its generating function. He assumes in his model that the birth and death rates are known functions of time, which is not a valid assumption for the SI model, which can be represented as a 1-BDP.

Billard (1981) gives the analytic solution to a bounded two-dimensional birth and death process (2-BDP) where the transition rates are arbitrary time-independent functions of the population sizes. His approach is based on the combined techniques of Severo (1969a) and Severo (1969b) where he works directly with the Chapman-Kolmogorov equations.

Parthasarathy and Sudhesh (2006) present a power series expression in closed form for the transient probabilities of a state-dependent 1-BDP. They transform the underlying forward Kolmogorov differential-difference equations into a set of linear algebraic equations by employing Laplace transforms, and then use continued fractions to derive the transient probabilities.

Brandwajn (1979) develops a semi-numerical iterative approach to the solution of the balance equations of a finite 2-BDP. His method is semi-numerical in that it uses the formal knowledge of the stationary probability distribution of one variable, and the iteration is applied to the conditional probabilities of the second variable given the first one.

We see that some of the models in both categories only give approximate solutions to K-M models. In those models which give the exact solution, the complex nature of the mathematics has still prevailed, limiting the applicability of those models to small population sizes. In addition, some of these models only give the exact solution in special cases or under limiting assumptions or in an implicit form. Also most of the developed models consider a constant contact rate, which may not be a realistic assumption.

In comparison with the existing literature, our model gives the exact solution of the SI model with varying contact rate in an explicit form with the minimum level of complexity of the mathematics involved.

3. Representing SEIR and SI Models as CTMCs

1 See Nedialkov et. al. (1999) for a review of interval methods for IVPs.

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In this section we represent SEIR and SI models with varying contact rate and no vital rates (hence with constant population size ) as CTMCs.

SEIR Model as a CTMCIn K-M models, the disease natural history is

represented as the transfer diagram, which depicts the movement of one individual between different diseases progress stages. The transfer diagram of the SEIR model is shown in Figure 1.

Figure 1: Transfer diagram of the SEIR model

Let denote the state

of the system at time . Then the state space would be

with size

(4)

where denotes the number of epidemiological classes;

therefore for the SEIR model. Note that formula

(4) gives the number of ways to sum non-negative

integer numbers to get the positive number . For

instance, if and (e.g. an SEIR model with 20 individuals), then the size of the state space would be 1,771.

The associated stochastic process is actually a right-shift process in which only it is possible for one individual to move to the next compartment; i.e. move from compartment S to E, E to I, or I to R with the following transfer rates for the original state ,

(5-a)

(5-b)

(5-c)

(5-d)

where denotes the transfer rate from state to

. The associated CTMC for population size is shown

in Figure 2. Circles refer to states which are labeled from 1 to 20 in the parenthesis. The outbreak starts in any of initial states and ends in any of final states.

SI Model as a CTMCThe transfer diagram of the SI model is depicted in

Figure 3.

Figure 3: Transfer diagram of the SI model

If denote the state of the system at time

, the state space would be

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Initial state

Final state

Isolated state

S E I R

3000(1)

2100(2)

2010(3)

2001(4)

1200(5)

1110(6)

1101(7)

1020(8)

1011(9)

1002(10)

0300(11)

0210(12)

0201(13)

0120(14)

0111(15)

0102(16)

0030(17)

0021(18)

0012(19)

0003(20)

Figure 2: Continuous-time Markov chain associated with the SEIR model for population size N=3

For instance, in an SI model with 20 individuals, there would be 21 states in according to formula (4) which

checks with above.The associated stochastic process is also a right-shift

process in which the only state transition occurs if one individual moves from compartment S to compartment I with the following transfer rate for the original state

,

(6-a)

(6-b)

If we define the stochastic process , we

see that is a pure death process, which is a BDP. Alternatively, we may define the stochastic process

to obtain a pure birth process. The associated CTMC, which we will refer to as CTMC-SI, for population size is shown in Figure 4. Note that we

have eliminated the state in CTMC-SI, since without any infectives there would be no dynamics.

4. Analytic Solution of the SI model

In this section, we derive the Kolmogorov forward equations for CTMC-SI. The fact that the SI model can be represented as a pure death (or pure birth) process results in a triangular structure in the linear ODE system obtained from the Kolmogorov forward equations allowing for solving the equations recursively, and finally finding the analytic solution of the original SI model. The details are as follows.

We refer to state as state . For simplicity of notation, we denote the set of integer numbers such that by for any two

integers , i.e. .

Let be a row vector

denoting the state probability distribution, i.e. denotes the probability that CTMC-SI is in state at time

, with the initial probability distribution

, where

and . Also let

be the transition

rate matrix so that denotes the rate at which

CTMC-SI makes a transition from state into state . Then from (6) we have

(7)

for .Now we derive the Kolmogorov forward equations for

CTMC-SI using (7). For simplicity, we drop which indicates that the probabilities and the transition rates are in general a function of time. If we denote the column of by , Kolmogorov forward equations can be represented as

(8)

Therefore for CTMC-SI we obtain from (8) the following triangular linear ODE system.

(9)

where . Note that the

right side of equations in (9) sum to zero, verifying

(10)

Assuming is so that for , solution to

system (9) is obtained recursively using the following formula.

(11)

By applying formula (11) to system (9) recursively, we find the solution to the Kolmogorov forward equations as follows.

(12)

where

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N-1,1 N-2,2 1,N-1... 0,NFigure 4: Continuous-time Markov chain associated

with the SI model for population size N=3

(13)The values of are found in this order:

. As an example, assuming

, we find using (12) and (13).

These relationships for are easily verified by noting that they satisfy equations in system (9) for

if .Note that there is only one observing state in

CTMC-SI, which is state , and all other states are transient, since from (12) we have

(14-a)

(14-b) where (14-b) can be shown directly or obtained from (14-a) and (10) by letting .

Now we can find the analytic solution to the original SI model as follows.

(15-a)

(15-b)

For any population size , the recursive equations (13) may be easily programmed in any programming language to obtain coefficients , which are used in (12) to calculate

. Finally the exact solution to the SI model can be calculated using (15).

5. Conclusions and Future Directions

We represented SEIR and SI models with varying contact rates as continuous-time Markov chains. Then for the SI model, we derived and solved the associated Kolmogorov forward equations. Finally we used the state probability distribution of the Markov chain to find the analytic solution of the original SI model.

Our method has the minimum of complexity regarding the mathematics involved and can be easily coded to give the exact solution of the SI model. In addition, our methodology can be used for other instances of Kermack-McKendrick models, and in fact for many instances of right-shift processes. Currently we are following this approach to develop an efficient algorithm to find the exact solution of the SEIR model.

AcknowledgmentsThe author would like to thank his advisor, Dr. Julie S.

Ivy for her invaluable assistance, support and guidance in this research.

This research was carried out by the North Carolina Preparedness and Emergency Response Research Center (NCPERRC) which is part of the UNC Center for Public Health Preparedness at the University of North Carolina at Chapel Hill’s Gillings School of Global Public Health and was supported by the Centers for Disease Control and Prevention (CDC) Grant 1PO1 TP 000296.

ReferencesBiographical SketchHAMED YARMAND is a PhD student in Operations

Research in the Department of Industrial and Systems Engineering at North Carolina State University. He received his Bachelor of Science degree in Industrial Engineering in 2008 at Sharif University of Technology in Tehran, Iran. He received his Master of Science degree in Operations Research in 2010 at North Carolina State University. His research interests include applications of computer simulation, control theory, and Markov processes in disease modeling, and applied game theory. His email address is hyarman@ncsu.edu

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