key: cord-0953328-an96u4k2
authors: Shuang-de, Zhang; Hai, Hao
title: Analysis on stability of an autonomous dynamics system for sars epidemic
date: 2005
journal: Appl Math Mech
DOI: 10.1007/bf02464241
sha: 5ad706c4e7e6b39db57baf4f155165b71eb8317a
doc_id: 953328
cord_uid: an96u4k2

An extended dynamic model for SARS epidemic was deduced on the basis of the K-M infection model with taking the density constraint of susceptible population and the cure and death rates of patients into consideration. It is shown that the infection-free equilibrium is the global asymptotic stability under given conditions, and endemic equilibrium is not the asymptotic stability. It comes to the conclusion that the epidemic system is the permanent persistence existence under appropriate conditions.

Severe acute respiratory syndrome (SARS) spreads most rapidly through the 23 areas and countries in the world since the first SARS case was reported in Guangdong in November, 2002 and reached the its climax in April-May 2003. Although the period of the epidemic was over now, it was not clear about SARS origin, the mechanism of transmission, and the regularity of SARS emerging except that SARS as a novel coronavirus was known. The studies of SARS were made by a variety of ways and in many fields in order that SARS would be well known and effective measures of its prevention and treatment would be found as early as possible.

Two lectures on the dynamic model of SARS were published in Science Express of USA on May 23, 2003 E~' 2J , in which the epidemiologic data from Hong Kong and Singapore were analyzed and epidemiologic parameters were estimated and predicted by both statistical and simulating methods. The result of study proved that the isolation of the people who were contacted by the patients would stop and prevent SARS from transmissing and extending. Wang Duo and Zhao Xiaofei found that the change and development of epidemic situation would be nearly described with Kermarck-Mckendric model after they made a verifiable Autonomous Dynamics System for SARS Epidemic 915 analysis and prediction of epidemic situation of SARS on the basis of the original K-M dynamic model TM . The model was too simple to take the birth rate and density constraint of susceptible population and the cure and death rates of patients into consideration. We build a more general dynamic model of SARS including all the factors and make a further mathematical analysis on the stability of the system for SARS epidemic on the basis of the qualitative theory.

The structure of the paper is arranged as follows. In Section 1, an autonomous dynamics system for SARS epidemic is built. In Second 2, the nonnegativity and boundedness of solution of the dynamic system are discussed. Both local stability of the equilibria and global asymptotic stability of the system are separately discussed in Sections 3 and 4. Finally, we study the uniform permanence of the system in Section 5.

For easy discussion, the assumption is made as follows: A1 During the prevalence of SARS we have three populations: the susceptible whose total population density is denoted by S; the isolating whose total population density is denoted by E; the patient with the symptom of SARS whose total population density is denoted by I, where S, E, I are all the function of time t.

A2 We assume that the susceptible population density grows according to a logistic equation with carrying intrinsic growth rate a and the density confining coefficient b.

A3 If the susceptible have a touch with the patient of SARS, then they can be separated and sent to do the medical observation. We assume that the contact rate is k of the susceptible with the patient of SARS.

A4 Under the medical observation we assume that the confirmed diagnosis rate is A. Thus the 1/A = T is the delitescence of SARS.

A5 The death rate of S ARS is d and the cure rate is/x. Based on the above assumption, we have the extended K-M [4] differential model, namely, 

1 Non-Negativity and Boundedness of Solutions

Let the right side of the system ( 1 ) be equal to zero. We obtain the equation set and solve it. We have the extinction equilibrium point E 0 = (0,0,0) and infection-free equilibrium point Ef = ( a/b ,0,0) of the system ( 1 ). There is a unique endemic equilibrium point E+ = (S* ,E* ,I*) forR 0 = ak/(b(d +/x)) > 1, where 916 ZHANG Shuang-de and HAt Hai Lemma 2 R3 § is the positive invariable set of the system ( 1 ) . Proof For any solution of the system (1), that is,

This proves the Lemma 2.

1.3 Dissipativity of system Proposition 1 The system ( 1 ) is the dissipative system, namely, the solutions of the system ( 1 ) for the all initiate conditions are the boundedness in the end.

Proof Let ( S( t) ,E( t) ,I( t) ) denote the solution of the system (1) satisfying the initiate conditions ( 2), then S, E, I >I 0 for any t > 0. From the first equation of the system (1) we can have limS(t) <~ a/b. In brief, we introduce the notation Z(t) = S(t) + E(t)

where 8 = min( 1 ,A -1, (d + be)/A). Applying the comparison theorem, we have

According to the non-negativity of solution in the system ( 1 ) we know that Proposition 1 holds.

In the following we define (4) bA Applying Proposition 1 we know that/2 is the bounded set of the system ( 1 ) in the end.

The Jacbian matrix of the system ( 1 ) is 

Hence Eq. (7) has two negative real roots forR 0 = ak/( b(d +be) ) < 1 which implies that E: = (a/b ,0,0) is the stable equilibrium point. Equation (7) has a negative root and a zero root forR 0 = ak/(b(d +be) ) = 1 which implies that E: = (a/b,O,O) is the quasi-stable equilibrium point. Equation (7) has a positive real roots for R 0 = ak/( b ( d +be) ) > 1 which implies that E: = (a/b ,0,0) is the instable equilibrium point.

Based on the analysis above we come to Theorem 1. 

For convenience, we introduce the definition as follows.

Definition 2(Strong uniform permanent existence) If there ismin { liminfX(t) } > 0 for any solution X of system, then the system is called the strong uniform permanent existence.

On the basis of Definitions 1 and 2, we verify system ( 1 ) On the basis of Theorem 5 and Proposition 1, namely, the solution of the system is bounded, we can know that system (1) is the permanent persistence existence C4] .

Because we have got little knowledge about SARS and it was supposed that many accidental factors would contribute to the outbreak of SARS, we shall make a further study of SARS from many parts of its pathogeny, pathology and epidemic. We have drawn two conclusions from our mathematical model that for the system (1)the infection-free equilibrium is global asymptotically stable for Ro ~ 1 and strong uniform permanently existent for R 0 > 1. That is to say that the system ( 1 ) is a controllable and stable system for R 0 ~< 1 and it is permanently existent for R 0 > 1 (namely, the infection will become an endemic).

The stability and persistence of the system entirely depended on the threshold R 0. According to expression R o = ak/( b ( d + Ix) ), we knew that R o was decided by parameters k, d,/z, a and b. For some regions, the intrinsic growth rate a and death rate d and the density confined coefficient b can be thought as constants, but k,/z changed greatly during some period. For example, if k decreases and br increases, then R 0 will become smaller. The system (1) will become stable. Similarly, whereas k increases and/z decreases, R o will become bigger. The system ( 1 ) will become instable and permanently existent. Thus we can foresee that the SARS will be eliminated if the contact rate k is rigorously controlled and the cure rate/z is improved. The result coincides with the real situation.

The model, however, may be simple and limited. For example, the rates of exposure, death and cure were assumed to be constants, but these parameters, in fact, were related to the particular distribution of the population. So the rate of the exposure to SARS varied among the people in the different area. These factors were lost or simplified in consideration of mathematical discussion. However, our study has shown that the model is useful for the description of changing condition of the SARS. A lot of work will be done to build an effective and exact model.

Transmission Dynamics of the Etiological Agent of SARS in Hong Kong: Impact of Public Health Interventions

Transmission Dynamics and Control of Severe Acute Respiratory Syndrome

Empirical analysis and forecasting for SARS epidemic situation J]

Non-Linear Dynamic System of Biology

Persistence under relaxed point-dissipativity (with application to an endemic model