Abstract
The stability of the SIR epidemic model with information variable and limited medical resources was studied. When the basic reproduction ratio , there exists the disease-free equilibrium and when the basic reproduction ratio , we obtain the sufficient conditions of the existence of the endemic equilibrium. The local asymptotical stability of equilibrium is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion. We also discuss the global asymptotical stability of the endemic equilibrium by autonomous convergence theorem. A numerical analysis is given to show the effectiveness of the main results.
1. Introduction
Mathematical epidemiology, that is, the building and analysis of mathematical models describing the spread and control of infectious diseases, plays an important role in the areas of biology. Various epidemic models have been proposed and explored extensively and great progress has been achieved in the studies of disease control and prevention. In the classical epidemic models, it is usually assumed that the change of the total population size satisfies logistic model or constant. We used nonlinear incidence, standard incidence rate, general incidence rate, and saturated incidence rate to concentrate on describing the spread of disease through the population. The analysis of all of these models has also been done by some other workers; see for example [1, 2] and the references therein.
In order to control the spread of epidemic, we consider the new variable , called information variable which summarizes information about the current state of the disease that is depending on current values of state variables and also summarizes information about past values of state variables. Many authors have used this variable in their models (see, e.g., [3–5]).
In many models, it is usually assumed that the removal rate of the infective is proportional to the number of the infective, which implies that the medical resources such as drugs, vaccines, hospital beds, and isolation places are very sufficient for the infectious disease. However, in reality, every country has an appropriate or limited capacity for treatment. Recently, Kar and Mondal have introduced the continually differentiable treatment function (see in [6]), where models the maximal supply of available medical resources per unit time and describes the reverse effect of the infected individuals being delayed for treatment, which have important effects on the spread of infectious disease. and the efficiency of the supply of available medical resources are independent. In many developed countries, in addition to the limitedness of the medical resources, the efficiency of the supply of available medical resources also has an important effect on the transmission of infectious disease, which depends on many factors such as the control strategies and the production of drugs or vaccines. In [7], the SIR models with limited medical resources have been studied.
To better understand their effects on the spread of infectious diseases, in this paper, we will discuss the model with saturated incidence rate (see [7]), information variable and treatment function .
The paper is organized as follows: in Section 2, we explore the existence of disease-free equilibria point and endemic equilibrium point and investigate the effect of the limited medical resources and their supply efficiency; in Section 3, we analyze the local asymptotic stability of the disease-free equilibria and the endemic equilibrium; in Section 4, we analyze the global asymptotic stability of the endemic equilibrium and present a global analysis of the model; a numerical analysis and a simple discussion are given to conclude this paper in Section 5.
2. The Model and the Existence of Equilibrium
In paper [6], a simple example is the classical SIR epidemic model with information variable and saturated incidence rate which has been studied as follows:
In paper [7], another simple example is the classical SIR epidemic model with limited medical resources which carefully investigated the dynamics of the following SIR model:
Based on the above motivations, in this paper, we further explore the SIR epidemic model with saturated incidence rate , information variable , and a continually differentiable treatment function (see [8]) to characterize the saturation phenomenon of the limited medical resources. The model can be described by the following system of equations: where , , and and , , and denote the numbers of susceptible, infective, and recovered individuals at time , respectively. is the intrinsic growth rate of susceptibles, is the carrying capacity of susceptibles, is the saturation factor that measures the inhibitory effect, is the transmission or contact rate, , are the natural death rate of the infective and recovered individuals, is the natural recovery rate, is the disease-related mortality, is the maximal medical resources supplied per unit time, and is half-saturation constant. are all positive.
Then the nonlinear integrodifferential system (4) can be transformed into the following set of nonlinear ordinary differential questions:
Since the dynamical behavior of the last equation of system (5), that is, the dynamics of , depends only on the dynamics of , the dynamics of is the same as . So we do not consider the last equation of system (5) in our discussion. Here we will study the following nonlinear ordination differential equations:
Denote , .
Theorem 1. (1) The system (6) has a trivial equilibrium and the disease-free equilibrium .
(2) If , further if , the system (6) has one endemic equilibrium except the disease-free equilibrium.
Proof. (1) Let , we have or ; it is not easy to find that the system has a trivial equilibrium and the disease-free equilibrium and .
(2) If , from the first question of (6), we have , from the third question of (6), we also have . Then substituting them into the first question of (6) yields
where
It is clear that if and , then . We have .
The roots of (7) satisfy
Therefore, if and , it is obvious that the sign of is positive or negative. There always exists a unique positive root, that is, . So (6) has one endemic equilibrium.
3. The Local Stability Analysis of Equilibria and Bifurcation
In this section, we will examine the local stability of the equilibria by analyzing the eigenvalues of the Jacobian matrices of (6) at the equilibria and using Routh-Hurwitz criterion.
Let be the arbitrarily equilibrium of the system (6); then the Jacobian matrix at of the system (6) is
Theorem 2. The free equilibrium point is unstable for any .
Proof. The Jacobian matrix at is
The eigenvalues are , , and . Therefore is unstable.
Theorem 3. (1) If , that is, , then is unstable for all .
(2) If , that is, , then is locally asymptotically stable for all .
Proof. The Jacobian matrix at is The eigenvalues are , , and .(1)If , that is, , then ; therefore is unstable.(2)If , that is, , then ; therefore is locally asymptotically stable.
Note. In Theorem 3, we can see that the stability of disease-free equilibrium point changes from stable to unstable when increases through 1. Therefore, we use as bifurcation parameter.
For simplicity, let and .
Theorem 4. The system (6) undergoes transcritical bifurcation at the equilibrium point when bifurcation parameter .
Proof. When , the Jacobian matrix at is
Then has a geometrically simple zero eigenvalue with right eigenvector and left eigenvector .
Now
According to [9], the system (6) undergoes transcritical bifurcation at the disease-free equilibrium point , hence the theorem.
Theorem 5. If and , then the endemic equilibrium is local asymptotical stability for , where
Proof. The Jacobian matrix at is
The characteristic equation is
That is,
where
Let , , and . We have
Let . Then and has an equine positive root:
Therefore, from the condition of Theorem 5, we have , that is, . If , we have , that is, . Hence, the theorem is trivially proved by Routh-Hurwitz criterion.
Note. If , the equilibrium may be unstable (see Figure 3).
4. The Global Stability Analysis of the Equilibrium Point
Here we will shortly describe the general method by which the global stability analysis for the endemic equilibrium will be performed through the approach due to Li and Muldowney [10]. Consider the autonomous dynamical system: where , is an open set and simply connected and , , .
Let be an equilibrium of (23). We recall that is said to be globally stable in , if it is locally stable and all trajectories in converge to . Assume that the following hypotheses hold.There exists a compact absorbing set .Equation (23) has a unique equilibrium in .
The basic idea of this method is that if the equilibrium is locally stable, then the stability is assured provided that and hold and no nonconstant periodic solution of (23) exists. Therefore, sufficient conditions on capable of precluding the existence of such solutions have to be detected.
Li and Muldowney showed that if and hold and (23) satisfies a Bendixson criterion, that is, robust under local -perturbations of at all nonequilibrium nonwandering points for (23), then is globally stable robust under local -perturbation and based on the introduced LozinskiǏ measure.
Let be a matrix-valued function, that is, on , and consider where the matrix is and the matrix is the second additive compound matrix of the Jacobian matrix , that is, . Generally speaking, for a matrix , is a matrix (for a survey on compound matrices and their relations to differential equations see [11]) and in the special case , one has Consider the LozinskiǏ measure of with respect to a vector norm in , , (see [12]) It is proved in [10] that, if and hold, condition guarantees that there are no orbits giving rise to a simple closed rectifiable curve in which is invariant for (23), that is, periodic orbits, homoclinic orbits, and heteroclinic cycles. In particular, condition (28) is proved to be a robust Bendixson criterion for (23). Besides, it is remarked that, under the assumptions and , (28) also implies the local stability of .
The analysis of the global stability of the endemic equilibrium may be usefully approached by means of the Poincare-Bendixson trichotomy. If the endemic equilibrium is globally asymptotically stable, then the disease will permanently be present in the population in case of infinitesimal initial prevalence. Here we will provide an analytical proof of global stability of by giving sufficient conditions. Global stability analysis for the endemic equilibrium will be performed through the approach due to Li and Muldowney. The instability of implies the uniform persistence; that is, there exists a constant such that any solution with in the orbit of the system satisfies
Consider the following assumptions: where
Lemma 6 (Li and Muldowney [10]). Assume that conditions and hold; then is globally asymptotically stable in provided that a function and a measure exist such that (28) is satisfied.
Theorem 7. Under the assumption , , (29), and (30), the endemic equilibrium of the system (6) is globally asymptotically stable.
Proof. The Jacobian matrix at of system (6) isThe second additive compound matrix of isConsider the function .
Then
Therefore
Also
Therefore
where
Consider the norm in as
where denotes vector and denotes the LozinskiǏ measure with respect to the norm
where and . Here and are matrix norms with respect to the vector norm and denotes the LozinskiǏ measure with respect to the norm.
Therefore
Therefore ,
From the system (6), ; therefore
Hence
That is,
Hence
By integrating both sides at the same time, we obtain
The proof is completed by Lemma 6 and (30).
5. Numerical Simulations
To demonstrate the theoretical results obtained in this paper, we will give some numerical simulations. We consider the hypothetical set of parameter values as follows.(1)Consider ; ; ; ; ; ; ; ; ; ; . By directly computing, we obtain . According to Theorem 3, we know the free disease equilibrium of system (6) is locally asymptotically stable for this case (see Figures 1(a)–1(c) and 1(d)).(2)Consider ; ; ; ; ; ; ; ; ; ; . Through calculation, we know , . According to Theorem 7, we know the positive equilibrium of system (6) is locally asymptotically stable for this case (see Figures 2(a)–2(c) and 2(d)).(3)Consider ; ; ; ; ; ; ; ; ; ; . Through calculation, we know , . According to Theorem 5, we know the positive equilibrium of system (6) is unstable for this case (see Figures 3(a)–3(c) and 3(d)).
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
(a)
(b)
(c)
(d)
6. Conclusion
In this paper, the stability of the SIR epidemic model with information variable and limited medical resources has been revisited. By analyzing the model, we have found the disease-free equilibria and exist when the basic reproduction ratio . At the same time we have proved the local asymptotic stability of the disease-free equilibrium. The conclusion reveals that the disease dies out, when ; then disease becomes endemic. changes the stability of the disease-free equilibrium and delay parameter and change the stability of the endemic equilibrium. It is shown that the disease-free equilibrium is unstable and the unique endemic equilibrium is globally asymptotically stable under some conditions. Lastly, a numerical simulation provided that, when is less than 1, the disease-free equilibrium is stable and while is more than 1, the disease-free equilibrium is unstable; that is, the endemic equilibrium exists (see Figure 1). We found that if , the equilibrium is globally asymptotically stable (see Figure 2).
If , it is concluded that the instability of the equilibrium has not been studied. It is worthwhile for us to study this case in the future work from the theorematic idea. Here we only illustrate the equilibrium is unstable if by use of the numerical simulation (see Figure 3).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the anonymous referees for their careful reading of the original paper and their many valuable comments and suggestions that greatly improve the presentation of this work. This work is supported by Natural Science of Shanxi Province (2013011002-2).