Stability analysis of an SIR model with immunity and modified transmission function

 
 
 
  • Abstract
  • Keywords
  • References
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  • Abstract


    This paper examines an SIR epidemic model with a non-monotonic incidence rate. We analyzed the model by considering after infection, only a fraction of transmitted part is shifted to infectious and remaining part gets recovered without becoming infectious. We also analyze the dynamical behavior of the model and derive the stability conditions for the disease-free and the endemic equilibrium. We have found a threshold condition, in terms of basic reproduction number  which is, less than one, the disease free equilibrium is globally attractive and if more than one, the endemic equilibrium exists and is globally stable. We illustrate theoretical results by carrying numerical simulation.


  • Keywords


    Basic Reproduction Number; Disease Frees Equilibrium; Endemic Equilibrium; Epidemiology; Non-Monotonic Incidence; SIR Model; Stability.

  • References


      [1] Anderson RM. Transmission dynamics and control of infectious disease agents. In: Anderson RM, May RM, eds.Population Biology of Infectious Diseases. Springer-Verlag, Berlin, 1982, pp. 149–76. http://dx.doi.org/10.1007/978-3-642-68635-1_9.

      [2] Anderson RM, May RM, Infectious Diseases of Humans. Dynamics and Control. Oxford University Press, Oxford, 1991.

      [3] Diekmann O, Heesterbeek JAP.Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation. Wiley and Sons, Chichester, 2000.

      [4] D. Xiao and S. Ruan: Math. Biosci. 208, 2007, 419-429. http://dx.doi.org/10.1016/j.mbs.2006.09.025.

      [5] H.W. Hethcote: The Mathematics of Infectious Disease, SIAM Rew. 42, 2000, 599-633. http://dx.doi.org/10.1137/S0036144500371907.

      [6] H.W. Hethcote and P. Van Den Driessche: J. math. Boil. 29, 1991, 271-287.

      [7] S Khekare et al., Global Dynamics of an Epidemic Model with a Non-monotonic Incidence Rate, IJOSR-JM, 2014, 71-77.

      [8] L.Perko,Differential Equations and Dynamical Systems, Texts in Applied Mathematics,Vol. 7,Springer-Verlag,New work, 1991

      [9] S. Ruan and W. Wang: J. Differential equations, 188, 2003, 135-163. http://dx.doi.org/10.1016/S0022-0396(02)00089-X.

      [10] V. Capasso and G. Serio: Math. Biosci. 42, 1978, 43-61. http://dx.doi.org/10.1016/0025-5564(78)90006-8.

      [11] W. M. Liu, S. A. Levin and Y. Iwasa: J. Math. Biol., 1986, 187-204. http://dx.doi.org/10.1007/BF00276956.

      [12] W. M. Liu, H. W. Hethcote and S. Levin: J. Math. Biol., 25, 1987, 359-380. http://dx.doi.org/10.1007/BF00277162.

      [13] W.M. Liu, P. van den Driessche, Math. Biosci. 128, 1995, 57-69. http://dx.doi.org/10.1016/0025-5564(94)00067-A.

      [14] W.R. DERRICK AND P.VAN DEN DRIESSCHE: J MATH. BIOL. 31, 1993, 495-512. http://dx.doi.org/10.1007/BF00173889.

      [15] W. Wang: Math.Biosci. 201, 2006, 58-71. http://dx.doi.org/10.1016/j.mbs.2005.12.022.


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    Stability analysis of an SIR model with immunity and modified transmission function

     

    Nidhi Nirwani *, V.H.Badshah, R.Khandelwal

     

    School of Studies in Mathematics Vikram University, Ujjain (M.P.) India

    *Corresponding author E-mail: nd.mathematics2009@gmail.com

     

     

    Copyright © 2015 Nidhi Nirwani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

     

    Abstract

     

    This paper examines an SIR epidemic model with a non-monotonic incidence rate. We analyzed the model by considering after infection, only a fraction of transmitted part is shifted to infectious and remaining part gets recovered without becoming infectious. We also analyze the dynamical behavior of the model and derive the stability conditions for the disease-free and the endemic equilibrium. We have found a threshold condition, in terms of basic reproduction number  which is, less than one, the disease free equilibrium is globally attractive and if more than one, the endemic equilibrium exists and is globally stable. We illustrate theoretical results by carrying numerical simulation.

     

    Keywords: Basic Reproduction Number; Disease Frees Equilibrium; Endemic Equilibrium; Epidemiology; Non-Monotonic Incidence; SIR Model; Stability.

     

    1.         Introduction

    The concealed and apparently unpredictable nature of infectious diseases has been a source of fear and superstition since the first ages of human civilization. The asymptotic behavior of solution of an infectious disease transmission model depends not only on epidemiological formation, but also on the demographic process incorporated into the model. Many authors have proposed various kinds of epidemic models to understand the mechanism of disease transmission. Anderson R M [1] proposed transmission and control phenomena of infectious diseases. Diekmann O. Heesterbeek [3] proposed and analyzed the evidence of the increasing diversification of infectious diseases. Several studies are there for the treatment of epidemics with different kind of incidence rates which measures the transfer rate of susceptible to get infected [4,5,6,9,11,15].Thus the incidence is number of new infectious per day or per other time unit.

    We have several different incidence rates which have been proposed by many researchers in epidemic model. Capasso and Serio [10] introduced a saturated incidence rate  into epidemic models. Where  tends to a saturation level when  gets large.

    Nonlinear incidence rates of form  were investigated by Lui et. al.[12,13]. A very general form of non-linear incidence rate was considered by Derrick and Driessche [14].One of the most fundamental quantities used by epidemiologists is certainty the basic reproduction number consider and analyzed by Anderson R M, May R M[2].In this paper the result is written in terms of basic reproduction number and stability of the equilibriums are investigated. 

    2.         The mathematical model

    The model we analyze in this paper is considered within the framework of the following nonlinear ordinary differential equations.

     


                                                                                                                                                      

     

    Table 1: Variables and Parameters in the Model

    Symbol

    Description

    Number of Susceptible individuals at time t

    Number of Infective individuals at time t

    Number of Recovered  individuals at time t

    The recruitment rate of the Population

    The natural death rate of the population

    A positive constant where.

    The natural recovery rate of the infective Individuals

    The rate at which recovered Individuals lose immunity

    The proportionality constant

    The parameter which measure the affects of medical infrastructure

    The parameter which measure awareness of people through education.

     

    Existence of Equilibria

    For the system (1), for any values of parameters, it always has a disease free- equilibrium.

     

     

    Define the basic reproduction number as follows:


                                                                                                                                                                                                (2)

    Then we have the following:

     

    Theorem 2.1:

    1)           If, then there is no positive equilibrium.

    2)           If , then there is a unique positive equilibrium  called the “endemic equilibrium”. Given by

     

     

     

     

     

     

    Now we check the dynamical behavior of disease free equilibrium point  and endemic equilibrium point  . The variation matrix of  is

     

    Its characteristics equation is given by

     

     

     

    Clearly,   is stable is

     

     

    Its characteristic equation is given by

     

                                                                                                                                                 

     

     

     

    Here  provided

     

     

     

    Performing simple calculations it can be easily be verified that   under the above conditions.

    Thus by Routh Hurwitz criterion, all Eigen values of (3) will have negative real part. Hence   is asymptotically stable.

    3.         Global analysis

    In this section, we study the properties of the equilibriums and derive the stability condition for the disease- free and the endemic equilibrium of model (1).

     

    Lemma 3.1: The plane    is an invariant manifold of system (1) which is attracting in the first octant.

     

    Proof: Summing up the three equations in (1) and denoting

    , we have

     


                                                                                                                                                                        

     

    For the equilibrium point, set

    From the above equation, it is clear that

     

     

    Is one solution of (4) and for any  ,the general solution of equation (4) is

    Also,

     

     

    This completes the proof.

     

    Clearly limit set of a system (1) is on the plane.

     

     

    Thus we focus on the reduced system.

     


                                                                                                               

     

     

    Theorem3.2: System (5) does not have nontrivial periodic orbits.

     

    Proof: Consider system (5) for  and  . Take Dulac function [8] as

     

     

    Then we have

     


                                                                                       

     

    Thus the expression (6) is negative for. Hence, the conclusion follows. 

    In order to study the properties of the disease-free equilibrium  and the endemic equilibrium  we rescale (5) by taking

     

     

    Then we obtain,

     


                                                                                                                                         

     

     

    Where

     

     

    The trivial equilibrium (0, 0) of the system (7) is the disease-free equilibrium  of the model (1) and the unique positive equilibrium  of a system (7) is the endemic equilibrium  of a model (1) where

     

     

    Which is positive if 

    And  is the positive solution of the following quadratic equation


                                                                                                                                                         

    Where

     

     

     

     

     

     

     

    Obviously, equation (8) has a positive root if    .   Obviously Thus, the equilibrium points     exists.

    We first determine the stability and topological type of (0, 0). The jacobian matrix of a system (7) at (0, 0) is.

     

     

     

    If  = 0, then there exists a small neighborhood of (0, 0) such that the dynamic of a system (7) are equivalent to that of

     

     

     

    Theorem 3.3: The disease-free equilibrium (0, 0) of a system (7) is

     

    i)             a stable hyperbolic node if  ;

    ii)           a saddle-node if   ;

    iii)          a hyperbolic saddle if 

     

    When   , we discuss the stability and topological type of the endemic equilibrium  is

    The jacobian of the system (7) at  is

     

     


     Thus, if the above condition holds then
     is a node or a focus or a centre.

    The trace of matrix M, is given by

     

     

    Thus,   if

     


                                                                                                                                               

    Where

     

     

    Thus, if both the conditions (9) and (10) are satisfied, we get a unique equilibrium  of model (7), which is stable.

    4.         Numerical results

    Consider following values of parameters

     

     

    Note that at  we get  ,when  the value of and when  the value of  and therefore these exists unique equilibrium.

     

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.1899

    0.4980

    0.7470

    1

    1.2450

    1.5192

    5.         Conclusion

    In this paper we studied global quantative analysis of a realistic SIR model. In terms of the basic reproduction  our main result indicate that when  the disease free equilibrium is globally attractive. When  , the endemic equilibrium exists and is globally stable. Though the basic reproduction number  does not depend on a and b.This implies that the spread of disease decreases as the social or psychological protective measures for the infective increases.This implies for  the model coincides with that of Khekare et al. [7]

    References

    [1]         Anderson RM. Transmission dynamics and control of infectious disease agents. In: Anderson RM, May RM, eds.Population Biology of Infectious Diseases. Springer-Verlag, Berlin, 1982, pp. 149–76. http://dx.doi.org/10.1007/978-3-642-68635-1_9.

    [2]         Anderson RM, May RM, Infectious Diseases of Humans. Dynamics and Control. Oxford University Press, Oxford, 1991.

    [3]         Diekmann O, Heesterbeek JAP.Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation. Wiley and Sons, Chichester, 2000.

    [4]         D. Xiao and S. Ruan: Math. Biosci. 208, 2007, 419-429. http://dx.doi.org/10.1016/j.mbs.2006.09.025.

    [5]         H.W. Hethcote: The Mathematics of Infectious Disease, SIAM Rew. 42, 2000, 599-633. http://dx.doi.org/10.1137/S0036144500371907.

    [6]         H.W. Hethcote and P. Van Den Driessche: J. math. Boil. 29, 1991, 271-287.

    [7]         S Khekare et al., Global Dynamics of an Epidemic Model with a Non-monotonic Incidence Rate, IJOSR-JM, 2014, 71-77.

    [8]         L.Perko,Differential Equations and Dynamical Systems, Texts in Applied Mathematics,Vol. 7,Springer-Verlag,New work, 1991

    [9]         S. Ruan and W. Wang: J. Differential equations, 188, 2003, 135-163. http://dx.doi.org/10.1016/S0022-0396(02)00089-X.

    [10]      V. Capasso and G. Serio: Math. Biosci. 42, 1978, 43-61. http://dx.doi.org/10.1016/0025-5564(78)90006-8.

    [11]      W. M. Liu, S. A. Levin and Y. Iwasa: J. Math. Biol., 1986, 187-204. http://dx.doi.org/10.1007/BF00276956.

    [12]      W. M. Liu, H. W. Hethcote and S. Levin: J. Math. Biol., 25, 1987, 359-380. http://dx.doi.org/10.1007/BF00277162.

    [13]      W.M. Liu, P. van den Driessche, Math. Biosci. 128, 1995, 57-69. http://dx.doi.org/10.1016/0025-5564(94)00067-A.

    [14]      W.R. DERRICK AND P.VAN DEN DRIESSCHE: J MATH. BIOL. 31, 1993, 495-512. http://dx.doi.org/10.1007/BF00173889.

    [15]      W. Wang: Math.Biosci. 201, 2006, 58-71. http://dx.doi.org/10.1016/j.mbs.2005.12.022.

 

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Article ID: 4257
 
DOI: 10.14419/ijamr.v4i2.4257




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