ABSTRACT

In this work, we provide some sufficient conditions to study the global asymptotic stability of the endemic equilibrium for certain models in mathematical epidemiology with nonlinear incidence and removal functions. We also present numerical examples in order to illustrate our results.

Keywords:
global asymptotic stability; Hurwitz matrix; epidemiology models

1 INTRODUCTION

Forecasting the evolution of an infectious disease has been the main motivation for the construction of mathematical epidemiological models. This is because knowing the evolution of the infectious disease allows the design of public health strategies to control or eradicate the disease. In epidemiology, many mathematical models descend from the classical SIR model of Kermack and McKendrick, established in 1927. When the infectious diseases confer permanent acquired immunity, these diseases can be modeled by classical susceptible-infectious-recovered (SIR) models. In epidemiological models the total population, N(t), is divided into any number of classes according to their epidemiological status. In the SIR model, S is the number of individuals in the susceptible class, I is the number of individuals who are infectious but not isolated and R is the number of individuals who are recovered.

We propose the following SIR model

with N(t) = S(t) + I(t) + R(t). In model (1.1), it has an inflow of newborns into the susceptible class at rate µN and deaths in the classes at rates µS, µI and µR. Notice that, The births balance the deaths. Therefore, the population size N is constant. Also, it is assumed that the infected population recovers at a rate γ and joins the recovered class. The interaction between susceptible and infected population will produce new infected individuals. These contagion processes are characterized by the C ¹ incidence function f (S, I). Define $g\left(I\right):=(\mu +\gamma )I+g_1\left(I\right)$, for some g ₁(I) ≥ 0, we make the following assumptions:

The asymptotic behavior of SIR models with the general nonlinear incidence rate have been studied by many researchers; see ¹1 V. den Driessche & J. Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 180 (2002), 29-48.^{), (2}2 A. Dénes & G. Rost. Global stability for SIR and SIRS models with non-linear incidence and removal terms via Dulac functions. Discrete and continuous Dynamical systems series B, 21(4) (2016), 1101-1117.^{), (6}6 A. Korobeinikov. Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol., 68 (2016), 615-626.^{), (7}7 A. Korobeinikov & P. Maini. Nonlinear incidence and stability of infectious disease models. MMB IMA, 22 (2005), 113-128. and ³3 Y. Enatsu, Y. Nakata & Y. Muroya. Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays. Discrete and continuous Dynamical systems series B , 15(1531-3492) (2011), 61.. However, the treatment rate of the infectious is assumed to be linear i.e., g ₁(I) = 0. In ⁸8 E. Rivero-Esquivel, E. Avila-Vales & G. García-Almeida. Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment. Mathematics and Computers in Simulation, 121 (2016), 109-132. doi: https://doi.org/10.1016/j.matcom.2015.09.005. URL http://www.sciencedirect.com/science/article/pii/S0378475415001937.
https://doi.org/10.1016/j.matcom.2015.09... ^{), (9}9 W. Wang. Backward bifurcation of an epidemic model with treatment. Math. Biosci. , 201 (2006), 58-71. a SIR model with a saturated treatment is studied i.e., $g_1\left(I\right):=\frac{rI}{1+\alpha I}$, nevertheless it deals only with a specific incidence rate. The general strategy in the previous works has been, first to establish the existence of an endemic point, then to prove that it is unique and later to proof the global stability in the feasibility region.

In this work, we develop criteria on global stability without the need to prove the existence and uniqueness of an endemic point, our result will also allow us to recover several results in the literature such as those above mentioned and extend them for general removal terms.

2 RESULTS

Without loss of generality, since the total population N(t) is constant, we take N(t) = 1. Then 1 = S(t) + I(t) + R(t). To analyze the endemic points, we reduce model (1.1) to a two-dimensional system as R does not appear in the first two equations of (1.1), the third equation can be ignored. This observation gives the simpler system

By hypotheses i) and iii), we have that (2.1) always admits the disease-free equilibrium state E ₀ = (S ₀ , I ₀) = (1, 0).

In the following result, we prove that all solutions of system (2.1) are eventually confined in the a compact subset, which is a positive invariant region.

Lemma 1. The region

is a positive invariant region for system (2.1).

Proof. Consider a solution of system (2.1), given by (S(t), I(t)), with initial condition (S(0), I(0)) ∈ ∆. We define w(t) = S(t) + I(t). Using equations given in (2.1), it is obtained

Multiplying expression (2.2) by e ^µt leads to

Notice that, equation (2.3) can be written as

By integrating (2.4) in [0, t] and since w(0) = S(0) + I(0) ≤ 1, then

Therefore.

That is, a solution (S(t), I(t)) of system (2.1) does not come out on the side S + I = 1.

On the other hand, consider the initial condition (S(0), I(0)) = (0, I(0)) ∈ ∆ with I(0) > 0. Then, the dynamics of system (2.1) in the vertical side of ∆ is given by

From the reduced system given by (2.7), it is shown that every solution with initial condition (S(0), I(0)) = (0, I(0)) goes into ∆. Therefore, these solutions do not come out for the vertical side of ∆.

Finally, we consider a solution of system (2.1) on the horizontal side of ∆. That is, (S(0), I(0)) = (S(0), 0) with S(0) ≥ 0. Thus, system (2.1) constrained to this type of initial condition is given by

From (2.8), it is concluded that every solution of the model with an initial condition (S(0), I(0)) = (S(0), 0) remains on the horizontal side of ∆. Therefore, these solutions do not come out for the horizontal side of ∆.

Therefore, all solutions in ∆ remain in this subset when t → ∞. In conclusion, ∆ is a positive invariant region. □

From Lemma 1, it can be concluded that every solution of system (2.1) in the first quadrant will eventually enter or remain in ∆.

Let ∆^o denote the interior of ∆.

Recall that a real matrix is Hurwitz if its eigenvalues have negative real parts. In dimension 2, the condition that a matrix A is Hurwitz is equivalent to requiring Trace(A) < 0 and Det(A) > 0.

The basic reproduction number ℛ₀ has been defined as the average number of secondary infections that occur when one infective is introduced into a completely susceptible host population ¹1 V. den Driessche & J. Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 180 (2002), 29-48.. To calculate the basic reproduction number, we use the method proposes in ¹1 V. den Driessche & J. Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci., 180 (2002), 29-48.. Then, for system (2.1) , the basic reproduction number is

We denote by X the vector field formed by the right hand side of system (2.1). We get

Lemma 2.Let f (S, I) and g(I) be functions satisfying i)-iii). If ℛ₀ > 1, then E ₀ is saddle for X.

Proof. In effect, we have that E ₀ is the equilibrium point, by i) we get $\frac{\partial f\left(E_0\right)}{\partial S}=0$ and since R ₀ > 1 then $\frac{\partial f\left(E_0\right)}{\partial I}-g\text{'}\left(I_0\right)>0$. Hence the derivative of X in E ₀ is

which has eigenvalues of different signs, this concludes the proof. □

Let us introduce the following notation.

Definition 1.For C¹−functions on an open set U ⊂ ℝ² , f ₁ , f ₂: U → ℝ, with independent variables x, y, we consider the partial Wronskian with respect to y, as

Theorem 3.Let f (S, I) and g(I) be functions satisfying i)-iii). If ℛ₀ > 1 and

then (2.1) admits a unique endemic equilibrium, which is globally asymptotically stable relative to ∆^o .

Proof. We denote by X the vector field formed by the right hand side of system (2.1) and consider the vector field $\frac{1}{g}X$. Notice that $\frac{1}{g}X$ and X have the same phase portraits in ∆^o . By a straightforward computation and (2.9) we obtain

Therefore by Bendixson-Dulac’s criterion, there are no periodic solutions or polycycles in ∆^{o (4}4 A. Gasull & H. Giacomini. Some Applications of the Extended Bendixson-Dulac Theorem. In S., Ibáñez, J.P. del Río, A. Pumariño & J. Rodríguez (editors), “Progress and Challenges in Dynamical Systems.”, volume 54. Springer, Berlin, Heidelberg (2013)..

Since ∆^o is a positive invariant region for $\frac{1}{g}X$, the Poincaré-Bendixson Theorem implies that for any x ∈ ∆^o its ω-limit set is a fixed point which cannot be E ₀ because this is saddle i.e, ω(x) = {p _λ }, p _λ ∈ ∆^o , for some index, λ ∈ Λ. Notice that, Λ is the set of indexes that records the number of equilibria points of the system. This implies the existence of at least one point of endemic equilibrium.

By direct computation and (2.9) we have

Thus by (2.10) and (2.11) the Jacobian matrix of $\frac{1}{g}X\left(p\right)$ is Hurwitz for all p ∈ ∆^o . Therefore, the fixed points p _λ are isolated and asymptotically stable.

Now let

be the basin of attraction for each p _λ . These sets are open and not empty. As ${\Delta }^o={\cup }_{\lambda \in \Lambda }{W}_{p_{\lambda }}^s$ and ∆^o is connected , then it is obtained that the cardinality of Λ is one, which is denoted by #(Λ) = 1. Therefore there is a unique endemic equilibrium E ^∗, which is asymptotically stable and all solutions with initial data in ∆o converge to E ^∗. This finishes the proof. □

Recall that a function k(S, I) on ∆^o is uniformly sublinear function if

As was observed by ²2 A. Dénes & G. Rost. Global stability for SIR and SIRS models with non-linear incidence and removal terms via Dulac functions. Discrete and continuous Dynamical systems series B, 21(4) (2016), 1101-1117., if a C ²-function k(S, I) is concave with respect to the variable I i.e., $\frac{{\partial }^{2}k}{{\partial }^{2}I}\le 0$, then it is uniformly sublinear.

As a direct consequence of Theorem 2.2, we have the following statement.

Corollary 4.Let f (S, I) and g(I) be functions satisfying i)-iii). If ℛ₀ > 1 and if −g, f, are uniformly sublinear functions, then (2.1) admits a unique endemic equilibrium, which is globally asymptotically stable relative to ∆^o .

Proof. By the sublinearity of f and −g we have $\frac{\partial f}{\partial I}\le \frac{f}{I}$ and $\frac{\partial g}{\partial I}\ge \frac{g}{I}$ respectively, combining these inequalities we get

this yields the condition (2.9) as desired. The result follows from Theorem 2.2. □

An immediate consequence of Corollary 2.3 and the remarks above is the following result.

Corollary 5.Let f (S, I) and g(I) be functions satisfying i)-iii). If ℛ₀ > 1, g is convex and f concave with respect to I, then (2.1) admits a unique endemic equilibrium, which is globally asymptotically stable relative to ∆^o . The Theorem 2.2 extends and/or refines certain previously established results. It is shown in ²2 A. Dénes & G. Rost. Global stability for SIR and SIRS models with non-linear incidence and removal terms via Dulac functions. Discrete and continuous Dynamical systems series B, 21(4) (2016), 1101-1117., Theorem 2.6 that if f grows monotonically with respect to both variables, is uniformly sublinear. If in addition ℛ₀ > 1 and g(I) = (µ + γ)I, then system (2.1) has a unique positive endemic equilibrium state which is globally asymptotically stable. Notice that inequality (2.9) is valid if f is uniformly sublinear and if g(I) = (µ + γ)I. Therefore our Theorem 2.2 recovers that result of global stability.

On the other hand, for f concave with respect to I and g(I) = (µ + γ)I a similar result on global stability was established in ⁶6 A. Korobeinikov. Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol., 68 (2016), 615-626., Theorem 2.1. As we mentioned if f is concave respect to I, then f is uniformly sublinear. So Theorem 2.2 also recovers that result of global stability.

2.1 Examples

In this section, we explore epidemiological models with nonlinear incidence and removal terms. Using our previous results, we analyze the global stability of the endemic equilibrium. To understand our results more intuitively, some numerical simulations are also carried out.

Example 2.1. Consider the SIR model determined by

where$r_2,s_2\ge 1;\gamma ,\alpha ,r_1,s_1\ge 0$and β, µ > 0. Since$g\left(I\right)=(\mu +\gamma )I+r_1{I}^{r_2}+s_1{I}^{s_2}$is convex and f (S, I) concave with respect to I, then by Corollary 2.4 there is a unique endemic equilibrium which is globally asymptotically stable whenever ℛ₀ > 1. Taking the SIR model (2.14) with specific parameters µ = 0.3, γ = 0.1, r ₁ = 0.2, r ₂ = 2, s ₁ = 0.3, s ₂ = 3, α = 1 and β = 3, we get ℛ₀ = 7.5. Hence system (2.14) admits a unique endemic equilibrium which is globally stable. SeeFigure 1.

Example 2.2. Take for example the SIR model given by

where all parameters are positive. If α ≥ k and ℛ₀ > 1, then there is a unique endemic equilibrium which is globally asymptotically stable. Indeed, a straightforward computation yields W _I (f , g) ≥ 0 on ∆^o . Therefore, we can apply Theorem 2.2. In particular, for the system (2.15) with specific parameters µ = 0.25, γ = 0.6, r = 0.7, k = 0.1, α = 0.4 and β = 2.5, we get ℛ₀ ≈ 1.6129. So (2.15) admits a unique endemic equilibrium which is globally stable. SeeFigure 2. Notice that in this example g is not convex.

Example 2.3.We consider the following SIR model with incidence rate proposed in⁵5 Y.S. J. Cui & H. Zhu. The impact of media on the control of infectious diseases. J. Dyn. Differ. Equ., 20(4) (2008), 31-52.and a similar removal function g(I),

with all parameters are positive. If m ≥ k and ℛ₀ > 1, then there is a unique endemic equilibrium which is globally asymptotically stable. Indeed a direct computation gives

On the other hand, the monotonic growing condition iii), g^′(I) ≥ 0, is equivalent to (µ + γ)e ^kI + rkI ≥ r, which is true for 0 ≤ I ≤ 1, due to µ + γ > r. The last inequality is implied by ℛ₀ > 1. Therefore, we can apply Theorem 2.2. In particular, for the system (2.16) with parameters µ = 0.4, γ = 0.6, r = 0.2, m = 3, k = 1 and β = 20, we get ℛ₀ ≈ 16.6667. Thus, system (2.16) admits a unique endemic equilibrium which is globally stable. SeeFigure 3. Notice that in this example f is not monotone with respect to I.

3 DISCUSSION

In this paper, a new coordinate-free approach was proposed to establish the existence, the uniqueness and the global stability of an endemic equilibrium point for SIR models. These criteria recover several results in the literature. We also provided numerical examples in order to illustrate our results. As future work, we intend to study new sufficient conditions of type (2.9) that cover new situations or other models.

Acknowledgments

We would like to thank the anonymous referee for providing us with constructive comments, which helped to improve the manuscript.

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