- Research
- Open Access
Analysis of an SIS epidemic model with treatment
- Jinghai Wang1Email author and
- Qiaohong Jiang1
https://doi.org/10.1186/1687-1847-2014-246
© Wang and Jiang; licensee Springer 2014
- Received: 11 August 2014
- Accepted: 5 September 2014
- Published: 24 September 2014
Abstract
An SIS epidemic model with saturated incidence rate and treatment is considered. According to different recovery rates, we use differential stability theory and qualitative theory to analyze the various kinds of endemic equilibria and disease-free equilibrium. Finally, we get complete configurations of different endemic equilibria and disease-free equilibrium.
Keywords
- epidemic model
- incidence rates
- treatment
- globally asymptotically stable
1 Introduction and model
Infectious diseases have tremendous influence on human life and will bring huge panic and disaster to mankind once out of control. Every year millions of human beings suffer from or die of various infectious diseases. In order to predict the spreading of infectious diseases, many epidemic models have been proposed and analyzed in recent years (see [1–13]). Some new conditions should be considered into SIS model to extend the results.
Li et al. (see [13]) studied an SIS model with bilinear incidence rate and treatment. The model takes into account the medical conditions. The recovery of the infected rate is divided into natural and unnatural recovery rates. Because of the medical conditions, when the number of infected persons reaches a certain amount , the unnatural recovery rate will be a fixed value . The study of this model should be divided into two cases to discuss with and . In this paper, we study an SIS model with saturated incidence rate and treatment, and we extend some recent results.
where () is the rate at which infected individuals are treated; A is the recruitment rate of individuals (including newborns and immigrants) into the susceptible population; is the nonlinear incidence rate; d is the natural death rate; γ is the rate at which infected individuals are recovered; ε is the disease-related death rate, A, d, γ, δ, ε, α are all positive numbers.
2 Existence of equilibria
We easily see that model (1.1) has a disease-free equilibrium .
Let . We study equation (2.5) as follows.
So this case need not be considered.
Therefore model (1.1) has a disease-free equilibrium and has an endemic equilibrium except the disease-free equilibrium when .
Let . We study equation (2.6) as follows.
So this case need not be considered.
Then .
If , it is easy to see that (2.6) has no positive root if .
At the same time, holds if and only if .
It is easy to see that , which implies that (2.6) has two positive equilibrium points , if , (2.6) has only one positive equilibrium point if , (2.6) has no positive equilibrium point if .
Therefore, if , holds.
we get or .
From the above discussion, we get the following conclusions.
Theorem 2.1 If , model (1.2) has only one disease-free equilibrium ; if , model (1.2) has a unique endemic equilibrium except the disease-free equilibrium ; if , is a unique endemic equilibrium of model (1.1).
Theorem 2.2 If , then is a unique endemic equilibrium of model (1.3) if ; is a unique endemic equilibrium of model (1.1) if and .
If , model (1.3) has two positive equilibrium points , if ; model (1.3) has only one positive equilibrium point if ; model (1.3) has no positive point if ; is an endemic equilibrium of model (1.1) if ; is an endemic equilibrium of model (1.1) if or .
If , model (1.3) has no endemic equilibrium.
3 Stability of equilibria
Theorem 3.1 The disease-free equilibrium is stable if and is a saddle point if ; the endemic equilibrium is a stable node if it exists; the endemic equilibrium is a stable node if it exists; if the endemic equilibrium points , exist, then is a stable node of model (1.1) if and is a stable node of model (1.1) if .
Thus, is a stable node if , and is a saddle point if .
So, is a stable node if it exists.
If , does not exist, then .
then is a stable node if it exists.
If , holds, then is a stable node if and . Similarly, is a stable node if and . This completes the proof. □
Theorem 3.2 If , there is no limit cycle of model (1.1).
Thus if .
Then there is no limit cycle of model (1.1) if . This completes the proof. □
Theorem 3.3 There is no limit cycle of model (1.1) if .
Thus if .
Then there is no limit cycle of model (1.1) if . This completes the proof. □
4 Numerical simulation and conclusion
With different A, d, γ, δ, ε, α, it is easy to test and verify the above results, so numerical simulation is omitted. In this paper, we study an SIS model with saturated incidence rate and treatment. We get some relatively complex conclusions by stability theory and qualitative theory of differential equations. These conclusions will help policy makers to make decisions.
Declarations
Acknowledgements
The research was supported by the Fujian Nature Science Foundation under Grant No. 2014J01008.
Authors’ Affiliations
References
- Jin Y, Wang W, Xiao S: An SIRS model with a nonlinear incidence rate. Chaos Solitons Fractals 2007, 34: 1482-1497. 10.1016/j.chaos.2006.04.022MathSciNetView ArticleMATHGoogle Scholar
- Cai L-M, Li X-Z: Analysis of a SEIV epidemic model with a nonlinear incidence rate. Appl. Math. Model. 2009, 33: 2919-2926. 10.1016/j.apm.2008.01.005MathSciNetView ArticleMATHGoogle Scholar
- Xu R, Ma Z: Stability of a delayed SIRS epidemic model with a nonlinear incidence rate. Chaos Solitons Fractals 2009, 41: 2319-2325. 10.1016/j.chaos.2008.09.007MathSciNetView ArticleMATHGoogle Scholar
- Wang X, Tao Y, Song X: Pulse vaccination on SEIR epidemic model with nonlinear incidence rate. Appl. Math. Comput. 2009, 210: 398-404. 10.1016/j.amc.2009.01.004MathSciNetView ArticleMATHGoogle Scholar
- Jiang Q, Wang J: Qualitative analysis of a harvested predator-prey system with Holling type III functional response. Adv. Differ. Equ. 2013., 2013: Article ID 249Google Scholar
- Wang J, Pan L: Qualitative analysis of a harvested predator-prey system with Holling-type III functional response incorporating a prey refuge. Adv. Differ. Equ. 2012., 2012: Article ID 96Google Scholar
- Wang J: Analysis of an SEIS epidemic model with a changing delitescence. Abstr. Appl. Anal. 2012., 2012: Article ID 318150 10.1155/2012/318150Google Scholar
- Zhang T, Teng Z: Global behavior and permanence of SIRS epidemic model with t time delay. Nonlinear Anal., Real World Appl. 2008, 9: 1409-1424. 10.1016/j.nonrwa.2007.03.010MathSciNetView ArticleMATHGoogle Scholar
- Mukhopadhyay B, Bhattacharyya R: Analysis of a spatially extended non-linear SEIS epidemic model with distinct incidence for exposed and infectives. Nonlinear Anal., Real World Appl. 2008, 9(2):585-598. 10.1016/j.nonrwa.2006.12.003MathSciNetView ArticleMATHGoogle Scholar
- Zhang J, Ma Z: Global dynamics of an SEIRS epidemic model with saturating contact rate. Math. Biosci. 2003, 185: 15-32. 10.1016/S0025-5564(03)00087-7MathSciNetView ArticleMATHGoogle Scholar
- Hethcote H, Ma Z, Liao S: Effects of quarantine in six endemic models for infectious diseases. Math. Biosci. 2002, 180: 141-160. 10.1016/S0025-5564(02)00111-6MathSciNetView ArticleMATHGoogle Scholar
- Martcheva M, Castillo-Chavez C: Diseases with chronic stage in a population with varying size. Math. Biosci. 2003, 182: 1-25. 10.1016/S0025-5564(02)00184-0MathSciNetView ArticleMATHGoogle Scholar
- Li X-Z, Li W-S, Ghosh M: Stability and bifurcation for an SIS epidemic model with treatment. Chaos Solitons Fractals 2009, 42: 2822-2832. 10.1016/j.chaos.2009.04.024MathSciNetView ArticleMATHGoogle Scholar
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