Stability Analysis of an SVIR Epidemic Model with Non-linear Saturated Incidence Rate

Transcription

1 Applied Mathematical Sciences, Vol. 9, 215, no. 23, HIKARI Ltd, Stability Analysis of an SVIR Epidemic Model with Non-linear Saturated Incidence Rate Muhammad Altaf Khan 1, Zulfiqar Ali 2, L. C. C. Dennis 3, Ilyas Khan 4, Saeed Islam 1, Murad Ullah 5 and Taza Gul 1 1 Department of Mathematics, Abdul Wali Khan, University Mardan Khyber Pakhtunkhwa, Pakistan 2 Department of Mathematics, ISPAR-Bacha Khan University, Charsadda Khyber Pakhtunkhwa, Pakistan 3 Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, 3175 Perak, Malaysia Corresponding author 4 College of Engineering Majmaah University, Majmaah Kingdom of Saudi Arabia 5 Department of Mathematics, Islamia College University, Peshawar, Pakistan Copyright c 214 Muhammad Altaf Khan 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 In this paper, we present an SVIR epidemic model with non-linear saturated incidence rate. Initially the basic formulation of the model is presented. Two equilibrium point exists for the system; disease free and endemic equilibrium. The stability of the disease free and endemic equilibrium exists when the basic reproduction less or greater than unity, respectively. If the value of R, less then one then the disease free equilibrium is locally as well as globally asymptotically stable, and if its exceeds, the endemic equilibrium is stable both locally and globally. The numerical results are presented for illustration. Mathematics Subject Classification: 92D25, 49J15, 93D2

2 1146 Muhammad Altaf Khan et al. Keywords: SVIR model, Reproduction number, Stability analysis, Numerical results 1 Introduction The mathematical modeling of natural phenomena or disease modeling is one of the major research area for mathematicians and biologist. The mathematical models of disease or natural phenomena often involved complexity and non-linearity. These complexities and non-linearities cannot be solved analytically. The models that present the epidemiology of a particular disease can be analyzed by studying their dynamical behavior, reproduction number, stability analysis, bifurcation analysis and their numerical results. By studying such properties of the epidemiological models, one can get a reliable and useful information about the disease control and spread. In literature, several articles are available that gives a mathematical descriptions of such non-linear phenomena [1, 2, 3]. The mathematical modes describes infectious diseases can be modeled in ODEs, PDEs or sometimes both of them. The first mathematical model in epidemicity has been presented by [4]. In [4] a simple SIR model that includes three state variables namely, susceptible, infected and recovered. After the development of this model a lot of mathematical models have been presented for different infectious diseases [5, 6]. The main purpose of such mathematical models to get insight to study the disease dynamics and control. Mathematical models that describes disease dynamics can be modeled for different purposes, for example, in [7], the author s formulated a mathematical model SVEIS with temporary immunity and saturated incidence rate. The model focuses the disease vaccination which is spread in the host population through horizontal transmission. They derived the basic properties of the model and the stability results presented, which completely co-exists with basic reproduction number, for more relevant references [8, 9, 1, 11, 12]. In this paper, we propose a mathematical model that includes vaccination and saturated incidence rate. Initially we formulate the model with the parameters therein. The model contains the state variables, susceptible -S(t), vaccinated-v (t), infected-i(t) and recovered-r(t). We assume that their is no migration and the population is constant. Presently, all the people have no immunity and get risk to be infected. The present model is constructed for the disease which is spread in the population horizontally. The structure of the paper follows is as: The problem formulation with their basic properties is discussed in detail in section 2. The local stability of disease free and endemic is discussed in section 3. In section 4, we study the global stability of disease free and endemic equilibrium. In section 5 and 6, respectively we show the numerical results and discussion.

3 Stability analysis of an SVIR epidemic model Model Formulation: In this section, we investigate the basic model formulation by dividing the total population into four compartments; that is, S(t) represents the susceptible, V (t) represents the vaccinated compartment, I(t) the infected and R(t) shows the recovered individuals. The model that governs a system of differential equation is presented in the following form: ds = µ(1 p)b (µ + q)s β 1SI 1 + αi + γv, dv = µpb + qs β 2 ξv I (µ + γ)v, di = β 1SI 1 + αi + β 2ξV I (µ + δ + ω)i, dr = δi µr, (1) Subject to initial conditions S() = S, V () = V, I() = I, R() = R. (2) The parameters with their description are presented in Table 1. The addition Table 1: Parameter Description Parameter Description µb Population recruitment rate. p The fraction of individuals to be vaccinated. µ Natural death rate. β 1 The disease contact rate. β 2 The interaction between vaccinated and infected. γ Wanning of vaccine. δ Recovery rate. q The individuals who needs vaccination. ω The disease induced death rate ξ Reflects the effect of vaccine reducing the infection rate α The saturation constant. of the system (1), gives dn = µb µn ωi,

4 1148 Muhammad Altaf Khan et al. where N = S + V + I + R. From above equation, we get lim t supn(t) N o, with lim t supn(t) = N o iff lim t supi(t) =. From the first equation of the system (1), it follows and the second equation gives lim t sups(t) S o, lim t supv (t) V o, So from the above we if N > N o, then dn <. We can write now } Ω = {(S, V, I, R) R + : S + V + I + R N o, S S o, V V o. 2.1 Equilibria The system (1) has always the disease free equilibrium at E = (S, V,, ), given by ( ) (µ + γ)(1 p)µb E = (S, V qµb(1 p),, ) =, µ(µ + γ + q) µ(µ + γ + q),,. Endemic Equilibria: S = Bµ(γ + (1 p)(µ + I ξβ 2 )) γ qγ + µ + I ξβ 2, V = R = δi µ. Bµ(p + q pq) γ qγ + µ + I ξβ 2, 2.2 Basic reproduction number In epidemiology, the basic reproduction number shows about the disease spread and control. If R < 1, then the disease free equilibrium is stable, the disease dies out from community. If R > 1 the endemic equilibrium exists, and the disease permanently exist in the community. In order to determine an expression for system (1), we follow [18]: Let x = (I, V ), then it follows from system (1): dx = F V,

5 Stability analysis of an SVIR epidemic model 1149 F = [ β1 SI 1+αI + β 2ξV I ] [, and V = (µ + δ + ω)i µp B + qs (µ + γ)v [ ] β1 S F = Jacobian of F at DF E = + β 2 ξv, [ ] µ + δ + ω V = Jacobian of V at DF E = µ + γ The next generation matrix for system (1) is [ ] F V 1 β 1 S +β 2 ξv = µ+δ+ω. ], The spectral radius R of the matrix F V 1 is R = ρ[f V 1 ] = β 1(µ + γ)(1 p)µb + β 2 ξqµb(1 p) µ(µ + γ + q)(µ + δ + ω) is the required basic reproduction number for the system (1). 3 Local Stability In this section, we discussed the local stability of the disease free and endemic equilibrium. In system (1), the fourth equation is independent of the rest, so we omit it, we obtain the following reduced model: ds dv di Subject to initial conditions = µ(1 p)b (µ + q)s β 1SI 1 + αi + γv, = µpb + qs β 2 ξv I (µ + γ)v, = β 1SI 1 + αi + β 2ξV I (µ + δ + ω)i. (3) S() = S, V () = V, I() = I. (4) We state and prove the following results. Theorem 3.1. At E, the disease free equilibrium of the system (3)is stable locally asymptotically, when R < 1. Proof: The Jacobian matrix at the point E given by (µ + q) γ β 1 S J = q (µ + γ) ξβ 2 V (5) R 1

6 115 Muhammad Altaf Khan et al. The characteristics equation of the Jacobian matrix J is given by (λ 3 + a 1 λ 2 + a 2 λ + a 3 ) =, (6) where a 1 ) = (µ + γ)(µ + q) + (µ + δ + ω) (1 R >, a 2 [ )] = (µ + γ)(µ + q) 1 + (µ + δ + ω) (1 R >, a 3 ) = (µ + γ)(µ + q)(µ + δ + ω) (1 R >. (7) We have ( a 1 a 2 a 3 = (µ + γ)(µ + q) (µ + δ + ω) 2 + (µ + γ)(µ + q) ) [(µ + δ + ω) + 1] )(1 R >. (8) The Routh-Hurtwiz criteria is satisfied as a 1 >, a 2 >, a 3 > and a 1 a 2 a 3 > if R < 1. Thus, all the eigenvalues of the system (3) has negative real part. The disease free equilibrium of the system (3) at E is locally asymptotically stable. Next, we show under some sufficient conditions the system (3) around E is locally asymptotically stable if R > 1. Theorem 3.2. The system (3) is locally asymptotically stable at E if R > 1, otherwise unstable. Proof: At the endemic equilibrium E the Jacobian matrix of the system (3)is given by: J 1 = (µ + q) β 1I 1+αI γ β 1 S (1+αI ) 2 q β 2 ξi (µ + γ) β 2 ξv β 1 I 1+αI β 2 ξi β 1S + β (1+αI ) 2 2 ξv (µ + δ + ω) (9) By elementary row operation, we obtain the matrix J 1 as follows: µ µ (µ + δ + ω) J 1 = µ(q + β 2 ξi + (µ + γ)) µβ 2 ξv q(µ + δ + ω) µβ 2 ξi µβ 1I µ(µβ 1+αI 2 ξv + q(µ + δ + ω)) (µ+δ+ω)β 1I 1+αI..

7 Stability analysis of an SVIR epidemic model 1151 The first eigenvalue of Jacobian matrix J 1 is clearly negative, i.e. µ <. The remaining eigenvalues are obtained by showing tracej 1 < and detj 1 >. We calculate [ ] tracej 1 = µ q + β 2 ξ(i + µv ) + q(µ + δ + ω) (µ + δ + ω)β 1I <, 1 + αi and [ ][ detj 1 = µ q + β 2 ξ(i + µ + γ) µq(µ + δ + ω) + µ 2 β 2 ξv β 1 I ] + (µ + δ + ω) 1 + αi +(µβ 2 ξv + q(µ + δ + ω))(µi (β 2 ξ β 1 ) + µβ 2 ξαi 2 ) >. The determinant detj 1 > if (β 2 ξ + β 1 ) >. The endemic equilibrium of the system (3) at E has negative real part. Thus, we conclude that the endemic equilibrium E of the system (3) is locally asymptotically stable, if R < 1. 4 Global Stability In this section, we study the global stability of the disease free and endemic equilibrium by lyapunov function. The disease free stability is presented in the following form. Theorem 4.1. The disease free equilibrium of the model (3) is globally asymptotically stable if R < 1. Proof: To prove this result, we construct the following lyapunove function: L = u 1 (S S ) + u 2 (V V ) + u 3 I, (1) where u 1, u 2, u 3 are positive constants to be determined later. Differentiating equation (1) with respect to time t, we obtain L = u 1 [µ(1 p)b (µ + q)s β 1SI 1 + αi + γv ] + u 2[µpB + qs β 2 ξv I (µ + γ)v ] +u 3 [ β 1SI 1 + αi + β 2ξV I (µ + δ + ω)i], After some arrangements, we get L = β 1SI 1 + αi (u 3 u 1 ) + β 2 ξv I(u 3 u 2 ) + S(u 2 q (µ + q)u 1 ) + V (u 1 γ u 2 (µ + γ)) +u 1 µ(1 p)b + u 2 µp B u 3 (µ + δ + ω)i, Let us chose the constants u 1 = u 2 = u 3 = 1. Finally, we obtain L = (µn µb) (δ + ω)i <. Thus, the disease free equilibrium of the system (3) is stable globally asymptotically, if R < 1.

8 1152 Muhammad Altaf Khan et al. In the next, theorem we present the global stability of the endemic equilibrium of the system (3) at E. Theorem 4.2. The endemic equilibrium E asymptotically stable if R > 1, γ = (µ+γ)(µ+q), q ξ = (µ+q) q, β 1 (µ + γ)(µ + q)p > β 2 ξq(1 p), of the system (3) is globally are satisfied. Proof: In order to show this result, we construct a lyapunove function in the following: W = (µ + γ)(s S ) + The time derivative of (11) along with system (3) gives (µ + γ)(µ + q) V + (µ + γ)i, (11) q W = (µ + γ)[µ(1 p)b (µ + q)s β 1SI 1 + αi + γv ] (µ + γ)(µ + q) + [µpb + qs β 2 ξv I (µ + γ)v ] q +(µ + γ)[ β 1SI 1 + αi + β 2ξV I (µ + δ + ω)i]. (12) Equation (13) becomes after some arrangements W = µ(δ + µ + γ)(µ + q + γ)r µb(β 1 (µ + γ)(µ + q)p β 2 ξq(1 p)) β 1 (µ + γ)(µ + δ + ω)i <. (13) Thus, W <, the endemic equilibrium E of the model (3) is globally asymptotically stable, provided that R > 1. 5 Numerical Results In this section, we investigate the numerical solution of the system (1) by using the Runge-Kutta order four scheme. The state variables are chosen with different initial conditions. The numerical results are shown in Figures 1 to 3. Figure 1-2, illustrate the fact, when the basic reproduction number less than unity, when the value of basic reproduction number exceeds than unity, Figure 3 illustrate this fact.

9 Stability analysis of an SVIR epidemic model S 25 v I R Figure 1: The dynamical behavior of system (1), for different initials conditions and the parameters: when R =.194 < 1, µ =.9, B =.2, q =.9, β 1 =.2, α =.3, γ =.1, β 2 =.9, ξ =.5, δ =.1, ω =.2, p =

10 1154 Muhammad Altaf Khan et al S 6 5 v I R Figure 2: The dynamical behavior of system (1), for different initials conditions and parameters: µ =.9, B = 1, q =.8, β 1 =.1, α =.3, γ =.1, β 2 =.9, ξ =.1, δ =.1, ω =.1, p =.5

11 Stability analysis of an SVIR epidemic model S 8 v I R Figure 3: The dynamical behavior of system (1), for different initials conditions and parameters is unstable when R = > 1 : µ =.9, B = 2, q =.1, β 1 =.2, α =.3, γ =.1, β 2 =.9, ξ =.5, δ =.1, ω =.2, p =.9

12 1156 Muhammad Altaf Khan et al. 6 Discussion The mathematical analysis of SVIR epidemic model with non-linear incidence has been presented. First, we investigated the basic reproduction number R for this system (1) which completely characterized the stability of the disease free and endemic equilibrium. We observed that, when R < 1, the disease free state is stable at E locally as well as globally, Figure (1) illustrate this fact. From epidemiological point of view the disease dies out from the population. In Figure (1) we discuss the case, when all the individual is susceptible i.e., the vaccinated ratio is zero. In Figure (2) we discussed the case, when some part of the population is vaccinated. In Figure 3, we observed when the threshold exceeds, the system then becomes unstable. At E, we found that the under some sufficient conditions, the system is stable both globally and locally. If the basic reproduction number R > 1, then, the disease will persist in the population and hence from biological point of view, we can say that the unique endemic equilibrium exist and is stable locally as well as globally stable at E. Numerical results for the model carried out, in order to illustrate the theoretical results. Acknowledgements. Authors would like to thank Universiti Teknologi PETRONAS for the financial support. References [1] Govind Prasad Sahua, Joydip Dhara, Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity, Journal of Mathematical Analysis and Applications, Volume 421, Issue 2, 15 January 215, Pages [2] Yoshiaki Muroya, Toshikazu Kuniya, Further stability analysis for a multi-group SIRS epidemic model with varying total population size, Applied Mathematics Letters, Volume 38, December 214, Pages [3] J. Mena-Lorcat, H. W. Hethcote, Dynamic models of infectious diseases as regulator of population sizes, J. Math. Biol. 3 (1992) [4] Kermack, W. O. and McKendrick, A. G. A Contribution to the Mathematical Theory of Epidemicsm Proc. Roy. Soc. Lond. A 115, 7-721,

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