Asian Journal of Mathematics & Statistics

Year: 2017 | Volume: 10 | Issue: 1 | Page No.: 1-12
DOI: 10.3923/ajms.2017.1.12

Analysis of a SIRI Epidemic Model with Modified Nonlinear Incidence Rate and Latent Period

Shivram Sharma , Viquar Husain Badshah and Vandana Gupta

Abstract: Objective: This study proposed an SIRI epidemic model with modified nonlinear incidence rate and latent period. Methodology: The equilibrium point of the model was investigated and showed that it was globally stable without latent period, further this study proposed the stability and Hopf bifurcation for the model with the latent period. Results: This study showed that the infected population on sociological and psychological effects seemed to be similar and the numerical examples were also given to illustrate the theoretical results. Conclusion: If basic reproduction number is less than or equal to one, the disease free equilibrium is globally asymptotically stable and if basic reproduction number is greater than one, then the endemic equilibrium exist and globally without latent period.

Keywords: latent period, Lyapunov function, global asymptotically stable, SIRI epidemic model and Hopf bifurcation

INTRODUCTION

The SIRI model is appropriate for some diseases such as human and bovine tuberculosis and herpes¹ in which recovered individuals may revert back to the infective class due to reactivation of the latent infection or incomplete treatment. Hethcote et al.² proposed an epidemiological model with a delay and a nonlinear incidence rate. Martins et al.³ has been given a scaling analysis in the SIRI epidemiological model. Global stability of an SIR epidemic model with time delays and different incidences studied by several author’s^4-9. Moreira and Yuquan¹⁰ have considered an SIRI model with general saturated incidence rate. Blower¹¹ formulated a compartmental model for genital herpes and Sharma et al.¹² considered a delayed SIR model with nonlinear incidence rate:

as follows:

where, S(t) is the number of susceptible individual, I(t) is the number of infective individual and R(t) is the number of recovered individuals, then we have the following model, the parameter r is the logistic growth rate, K is the carrying capacity, μ>0 is the rate of natural death such that r>μ, δ>0, is the rate of disease related death, ρ>0 is the rate of recovery, is the incubation period and α is the parameters that measures infections with the inhibitory effect. Xu and Ma¹³ studied a delayed SIRS epidemic model with a nonlinear incidence rate. Enatsu et al.¹⁴ proposed Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model. Van den Driessche et al.^15,16 proposed an integro-differential equation to model a general relapse phenomenon in infectious diseases. A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse is studied by Georgescu and Zhang¹⁷.

Several author’s^17-21 have been proposed SEIRI and SIRI epidemic model with different incidence and Guo et al.²² considered the dynamical behaviors of an SIRI epidemic model with nonlinear incidence rate and latent period, namely:

as follows:

The model (1) describes the psychological effect of certain serious diseases on the community when the size of the set of infective individuals is getting larger.

MODEL DESCRIPTION AND ANALYSIS

The human population is divided into three classes such that at time t there are susceptible humans (S) infected humans (I) and recovered humans (R). Thus the size of the human population is given as N = S+I+R. The susceptible population increase by birth rate b, the size of human population is decreased by natural death rate d, infected individuals (I) become temporarily immune by the rate μ which γ is the rate at which the recovered class revert to the infective class. This study considers the general incidence rate in an epidemiological model:

where, kI represents the infection force of the disease and:

represents the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. Furthermore, assume that the latent period is a constant τ, that is, a susceptible becomes infective if he contacted the infected τ times ago. Therefore, delay can be incorporated into the non linear incidence function as follows:

In this section, this study will consider the following SIRI epidemic model with the non linear incidence rate and latent period:

where, S(t), I(t), R(t) are the number of susceptible, infective and recovered individuals respectively, α₁, α₂ are the parameters which measure the effects of sociological and psychological and the rest of the parameters are describes in above. If N(t) is the number of total population then N(t) = S(t)+I(t)+R(t), Adding the three Eq. 2 then as a result it has:

Clearly:

and:

Hence Eq. 2 can be rewritten as:

Invariant regions: Suppose the system (4) with initial condition:

where, the space of continuous function from [-τ, 0] to the set of all ordered pair whose components are non-negative real numbers with the norm:

From (4) it can be seen that on the line and on the line Hence, all orbit of equation (4) must go into the first quadrant. From (3), it can be shown:

Again, it shows:

the orbit of Eq. 4 getting at the boundary must go into the interior of . Thus the region S_Δ is an invariant set and also an absorbing set of Eq. 4 in the first quadrant. From the above discussion the following theorem is obtained.

Theorem 1: The region is an invariant set and also an absorbing set of Eq. 4 in the first quadrant.

The basic reproductive number for the model (4) is defined as follows:

Equilibrium and stability analysis: The model (4) always has disease free equilibrium (DFE) E₀ = (0, 0) for any set of parameter values. The endemic equilibrium of the model (4) is the solution of:

From the first and the second equation of system (7), get:

From the Eq. 8 it can be easily shown that if ℜ₀<1, there is nopositive solution as in that case coefficient of I^h, I and constant term are all negative. Set:

Thus:

When ℜ₀>1, then g’(I)<0 since is strictly monotonic decreasing for I>0. Also:

This implies that Eq. 4 has one unique positive solution, namely E_* = (I_*, R_*), called endemic equilibrium.

Here and I_* is a positive solution of (8). Thus the following theorem is obtained.

Theorem 2:

Now, the stability of two equilibrium and Hopf bifurcation for Eq. 4 with or without latent period is investigated.

Threshold dynamics for the case τ = 0: From the Eq. 4:

Theorem 3:

Proof: The Jacobian matrix of Eq. 9 at the disease free equilibrium:

The characteristic equation is:

Where:

When ℜ₀<1, then both a₁>0, a₂>0 by Hurwitz criterion the disease free equilibrium E₀ = (0, 0) is locally asymptotically stable.

When ℜ₀ = 1, then a₂ = 0 and a₁>0 which implies that the one eigen value is 0 and the other is -a₁<0 therefore the disease free equilibrium E₀ is locally stable. Furthermore, it takes the Lyapunov function:

From ℜ₀ = 1, it can be shown:

and:

and this has a unique point E₀ = (0, 0). It follows from the Lasalle invariant principle that the disease free equilibrium E₀ is asymptotically stable when ℜ₀ = 1.

When ℜ₀>1 we have a₂<0, one eigen value will be positive which implies E₀ of Eq. 9 is an unstable. Furthermore, Eq. 9 has an unique endemic equilibrium E_* = (I_*, R_*).

The Jacobian matrix of Eq. 9 at the disease free equilibrium E_* = (I_*, R_*) is:

Let:

and:

The characteristic equation of M_* is:

Where:

and:

Since (I_*, R_*)∈S_Δ so both a₃ and a₄ are positive. Hence by the Hurwitz criterion the endemic equilibrium E_* is a locally asymptotically stable.

Furthermore, discuss the topological structure of the trajectories of Eq. 9.

Theorem 4: Equation 9 has neither a nontrivial periodic orbit nor a singular closed trajectory including a finite number of equilibria in S_Δ .

Proof: Set:

Select:

Thus, α₃ - α₂ < 0 and:

Then:

By the Bendixen-Dulac criterion²³, the Eq. 9 does not have nontrivial periodic orbits or a singular closed trajectory which contains a finite number of equilibrium.

This proof is completed.

Theorem 5: If ℜ₀>1, then the two stable manifolds of saddle E₀ are not in S_Δ.

Proof: Suppose there is a stable manifold in S_Δ, then the stay in the S_Δ or traverse through S_Δ Assuming the former case, we know that the satisfy one of the following cases (see²³):

where, A_p is negative limit set of L_p. Case (i) and Theorem 3 lead to a contradiction and case (ii) and Theorem 4 lead to a contradiction.

If the ran out of the region S_Δ it is divided into two parts to S_Δ, (if the two stable manifolds are all in S_Δ and all passed through this region S_Δ they are divided into three parts of S_Δ) since I = 0 or R = 0 is not an orbit of Eq. 9. Then the equilibrium E_* is not in one part. In this part, we arbitrarily select a point Q≠O and Q is not in any of the stable manifolds, then the positive half orbit through the point Q will run out of this part or Ω_Q is a closed orbit or a singular closed trajectory, where Ω_Q is a positive limit set of L_Q.

This is the contradiction of a saddle or Theorem 1 or Theorem 4.

This completes the proof.

Theorem 6:

Proof: For any point P in S_Δ according to Zhifen et al.²³, its Omega-limit set satisfies Ω_P = {E₀} or Ω_P is a closed orbit or Ω_P is a singular closed trajectory. Theorem 4 essentially means that there is no closed orbit or singular closed trajectory in S_Δ. So it can be shown that Ω_P = {E₀}. This proves that E₀ is a global attractor in S_Δ.

From Theorem 3 and Theorem 5, E₀ is an unstable saddle in S_Δ and its two stable manifolds are not in S_Δ if ℜ₀>1 By a similar proof, it can be shown that Ω_P = {E_*} where P is an arbitrary point in S_Δ-{E₀} This proves that E_* is a global attractor in S_Δ.

It follows from Theorem 1 that S_Δ is a absorbing set in the first quadrant. Thus, E₀ in case (i) and case (ii) and E_* in case (iii) is a global attractor in the first quadrant.

This completes the proof.

Dynamical behaviors for the case τ>0: In this section, the stability and Hopf bifurcation of the equilibrium for Eq. 4 with the latent period τ>0 are studied.

Theorem 7: The disease free equilibrium E₀ = (0, 0) of Eq. 4 is globally asymptotically stable if ℜ₀<1 and the disease free equilibrium is unstable if ℜ₀>1.

Proof: The characteristic equation at E₀ = (0, 0):

This implies:

The roots (11) of have negative real parts if 0<τ<1. Suppose (11) has a purely imaginary root λ = ωi, if τ = τ₀ where ω is a positive real number. Then separating real and imaginary parts:

Hence:

Where:

If ℜ₀<1, then D₁>0, D₂>0. Accordingly, (14) does not have any real root and all roots of Eq. 11 have negative real parts for any τ>0 if ℜ₀<1. By a similar analysis, we derive that Eq. 11 has a unique zero root and all other roots with negative real parts for any τ>0 if ℜ₀ = 1. Thus, E₀ is locally asymptotically stable if ℜ₀<1 for any τ>0.

If ℜ₀>1 Eq. 11 has a root with the positive real part for any τ>0. Thus, the disease free equilibrium is unstable.

Next, it is shown that the disease free equilibrium is globally asymptotically stable if ℜ₀<1. Define the Lyapunov function:

Clearly V₁ (0, 0) = 0 and V₁ (I(t), R(t))>0 in the interior of Since ℜ₀<1 so:

and:

has a unique point E₀ = (0, 0) It follows from the Lyapunov-Lasalle invariant principle^2,4,24 that the disease free equilibrium E₀ = (0, 0) is globally asymptotically stable. This completes the proof.

When ℜ₀>1 the system has a unique endemic equilibrium E_* = (I_*, R_*) In the following, it is analyzed that the stability of E_* = (I_*, R_*), which is dependent on the parameters h and τ. To do this, the characteristic equation:

This implies:

Where:

Suppose that there is a positive τ = τ₀ such that Eq. 16 has a purely imaginary root λ = ω_i, ω>0 Then separating the real and imaginary parts, we have:

Hence:

When τ = 0 then all roots of Eq. 18 have negative real parts, which implies that:

Furthermore, we have:

and:

Where:

When ℜ₀>1 and 0<h<2 we have and are positive. In this case, Eq. 19 has no real root that is the real parts of all roots of Eq. 16 are negative. So, we obtain the following.

Theorem 8: The endemic equilibrium E_* = (I_*, R_*) of Eq. 4 is a locally asymptotically stable if ℜ₀>1,0<h<2 and τ>0.

Now, consider, h>2 if ℜ₀>1 and τ>0. Firstly, the following lemma is given.

Lemma 1: If ℜ₀>1, h>2 and H<0 where:

then there is a positive τ₀ such that Eq. 16 has a pair of purely imaginary eigenvalues ±ω₀i as t = t₀ and all eigen values with negative real parts 0<t<t₀.

Proof: From Eq. 8 it can be shown:

This implies that:

Thus, we have:

Where:

Let z = ω² of Eq. 19 it can be shown:

From H<0 there exists a unique positive root z₀ of Eq. 23. Then .

Denote:

now, select:

Then Eq. 16 if has a pair of purely imaginary roots ±i ω⁰ if τ = τ⁰. Note that any root of Eq. 16 has a negative real part for τ = 0 and Eq. 16 has no zero root for any τ>0. Then all roots of Eq. 16 has a negative real part if 0<τ<τ⁰ from the continuity of the roots on parameter τ It can be seen that:

Theorem 9: Assume that ℜ₀>1, h>2 and H<0 then there is a positive real number τ₀ such that the following conclusions hold.

Proof: From lemma 1, the conclusion can be given as: in case (i). To obtain the Hopf bifurcation, let λ(τ) = ξ (τ)+iω(τ) be the eigenvalue of Eq.16 such that for some initial value of the bifurcation parameter τ₀ we have ξ(τ₀) = 0 and ω(τ₀) = ω₀.

Differentiate Eq. 16 w.r.t. τ we get:

Since ξ(τ₀) = 0 and ω(τ₀) = ω₀.

Where:

From Eq. 23 and <0 it can be shown:

Thus:

which is positive when ℜ₀>1
Thus:

by continuity the real part of λ(τ) becomes positive when τ>τ₀ and the endemic equilibrium becomes unstable. A Hopf bifurcation occurs when τ passes through the critical value τ₀.

EXAMPLES

Guo et al.²² discussed on the SIRI epidemic model with nonlinear incidence rate, latent period and saturated parameter while this study discuss a SIRI model with modified nonlinear incidence rate and also see the effect of sociological and psychological rate on the infected population. In this section, some examples and their simulations to illustrate the effectiveness of the results are given.

Example 3.1: Consider the following parameters b = 0.4, d = 0.4, h = 1, k = 0.4, γ = 0.5, μ = 0.5, α₁ = 8, α₂ = 10, (S(0), I(0), R(0)) = (0.3, 0.5, 0.1). Then E₀ (0, 0), ℜ₀ = 0.7530<1. Therefore by Theorem 6 and Theorem 7, E₀ = (0, 0) is globally asymptotically stable for any latent period. S(t) approaches to its study state value while I(t) and R(t) approach zero as time increases and τ = 0 τ = 30, respectively, the disease dies out as shown in Fig. 1 and 2.

Example 3.2: Take b = 2, d = 0.4, h = 1, k = 0.4, γ = 0.5, μ = 0.5, α₁ = 8, α₂ = 10, (S(0), I(0), R(0)) = (5, 1, 4). Since ℜ₀ = 4.074>1 and 0<h<2, it follows from Theorem 5 (Theorem 8) that the endemic equilibrium E_* is globally (locally) asymptotically stable τ = 0(τ>0) The asymptotically stability of the equilibrium for the model (2) with τ = 0, τ = 40, respectively as shown in Fig. 3 and 4.

Example 3.3: Take b = 2.8, d = 0.4, h = 3, k = 0.9, γ = 0.5, μ = 0.5, α₁ = 5, α₂ = 1000, then we have ℜ₀ = 6.892>1. H = -0.549<0. Therefore by Theorem 9, the endemic equilibrium E_* is locally asymptotically stable for small latent period, while the SIRI model can go a Hopf bifurcation and produce a periodic orbit for a large latent period.

The asymptotically stability of the equilibrium for the model (2) with τ = 5 as shown in Fig. 5.

Example 3.4: Keeping other parameters fixed, if change the values of α₁ and α₂ it is seen that I* decreases as α₁, α₂ are increases and shows that the infected population on sociological and psychological effect rate seems to be similar as shown in Fig. 6 and 7.

CONCLUSION

This study proposed an SIRI epidemic model with modified nonlinear incidence rate and latent period. If ℜ₀<1 then the model (2) has one unique disease free equilibrium and if ℜ₀>1 then the model (2) has a disease free equilibrium and a unique endemic equilibrium. This study showed that the equilibriums are globally stable without latent period. Further it propose the stability of equilibriums for the model (2) with the latent period, if ℜ₀<1 then the disease free equilibrium is globally asymptotically stable for any latent period and when 0<h<2 and ℜ₀>1 the endemic equilibrium is locally asymptotically stable for any latent period. When h>2 and ℜ₀>1 the SIRI model may undergo a Hopf bifurcation and produce a periodic orbit for a large period.

SIGNIFICANCE STATEMENTS

This study discovers the global stability of disease free and endemic equilibrium of SIRI model with modified nonlinear incidence rate and latent period. This study will help the researcher to uncover the critical diseases of SIRI model with latent period. For the future research, researcher can be considering SIRI model with a logistic recruitment and latent period.

REFERENCES