On the Dynamics of Sexually Transmitted Diseases Under Awareness and Treatment

Abstract

In this paper, we develop and extend the work of Jia and Qin on sexually transmitted disease models with a novel class of non-linear incidence. Awareness plays a central role both in the susceptible and the infectious classes. The Existence, uniqueness, boundedness, and positivity of solutions are systematically established. Concavity arguments and the occurrence of a vertical asymptote are essential in the proof of the existence of a unique endemic equilibrium. Conditions for the stability of all steady states are investigated. In particular, numerical simulations are performed in order to capture the asymptotic behavior of solutions.

AMS Classification: 92D30, 34D23.

1. Introduction

Disease incidence plays a crucial role in mathematical epidemiology and it is essential in the computation of the basic reproduction number. Non-linear incidences are known to induce complex or chaotic behavior as oppose to standard incidences frequently used in classical infectious disease models [1–6]. A class of non-linear incidences particularly useful in the modeling of sexually transmitted diseases was introduced in [7] by the authors in the modeling of HIV/AIDS epidemic. The model considered however was not properly conceptualized as the density of individuals with full-blown AIDS not receiving ARV treatment did not bear any influence on the infection rate of the disease. In addition a number of inaccuracies are displayed in this paper like the unknown variable T missing in the third equation of system (2.1) and also a mistake occurred in the computation of the sign of a₃ in the proof of the stability of the endemic equilibrium. The purpose of this paper is to develop and extend the work on [7] by deriving a realistic model for sexually transmitted diseases with a proper non-linear incidence rate with a valid biological significance and perform a full analysis of the resulting model. In Section 2, the model is derived and presented. Well-posedness analysis, positivity and boundedness are considered in Section 3 followed by stability analysis of the critical points of the system in Section 4, numerical simulations in Section 5 and the conclusion.

2. The Model

In this paper, a model with five compartments is formulated with non-linear incidence Sg(t, I) incorporated into it. The incidence is presumed to be a time dependent non-linear response to the size of the infectious population.

The compartments are denoted by S(t), I(t), T(t), A(t) and R(t) which represent the number of susceptible individuals, the number of infected individuals with the potential of transmitting the disease as they are not under treatment and do not take any form of protection while engaging in sexual activities, the number of individuals under treatment, the number of infectious individuals engaging in safe sex, and heathy individuals that engage in safe sex, respectively, at time t. The model represented in Figure 1 is governed by the system of nonlinear ordinary differential equations

endowed with initial conditions

S(0) ≡ S₀ > 0, I(0) ≡ I₀ > 0, T(0) ≡ T₀ ≥ 0, A(0) ≡ A₀ ≥ 0, R(0) ≡ R₀ ≥ 0.

Figure 1

The parameters in the evolution system (1) are described in Table 1:

The total population N(t) is given by S(t) + I(t) + T(t) + A(t) + R(t). By adding all the equations of the system (1), we obtain the rate of change of N(t), which is given by

and N(t) varies over time and is nearing a stable fixed point as t → ∞. Therefore, the biologically feasible region for the system (1) is given by

It is easy to see that the set Ψ is positively invariant. Next we present a systematic analysis of our evolution equation.

3. Mathematical Analysis

We start by investigating the well-posedness of the model (1). Given the fact that the variables represent biologically densities, it is important to show that all the variables remain positive at all time.

Lemma 1. For any non-negative initial conditions (S₀, I₀, T₀, A₀, R₀), system (1) has a local solution which is unique.

Proof: Let x = (S, I, T, A, R), system (1) can be rewritten as x′(t) = f(t, x(t)), where f : ℝ⁶ → ℝ⁵ is a C¹ vector field. By the classical differential equation theory, we can confirm that system (1) has a unique local solution defined in a maximum interval [0, t_m).     □

Lemma 2. For any non-negative initial conditions (S₀, I₀, T₀, A₀, R₀), the solution of (1) is non-negative and bounded for allt ∈ [0, t_m).

Proof: We start by showing positivity of the local solution for any non-negative initial conditions. It is easy to see that S(t) ≥ 0 for all t ∈ [0, t_m). Indeed, assume the contrary and let t₁ > 0 be the first time such that S(t₁) = 0 and . From the first equation of the system (1), we have , which presents a contradiction. Therefore S(t) ≥ 0 for all t ∈ [0, t_m). Using the same argument, positivity I(t), T(t), A(t) and R(t) in the interval [0, t_m) are established. Furthermore from (2), we have that

Therefore the solution N(t) is bounded in the interval [0, t_m).     □

Theorem 1For any non-negative initial conditions (S₀, I₀, T₀, A₀, R₀), system (1) has a unique global solution. Moreover, this solution is non-negative and bounded for allt ≥ 0.

Proof: The solution does not blow up in a finite time as it is bounded, it is therefore defined at all time t ≥ 0. Other properties of the solution follow from lemma (1) and lemma (2).     □

Setting m = d + τ₁ + τ₂ and n = d + ν₁ + ν₂, system (1) transforms into a reduced system

4. Models With Time Independent Non-linear Response

In this section, we assume that the non-linear response function is not time dependent, ie g(t, I) ≡ g(I). Following [7], it is further assumed that

4.1. The Basic Reproduction Number

In this section, we use the next generation method [8] to obtain the basic reproduction number. Let z be the transpose of (I, A, T, S, R). We rewrite the system (4) in the matrix form

where

and

The disease free equilibrium of system (4) takes the form

Following [8], we compute the basic reproduction number using the formula below

where

and

and ρ is the spectral radius of the matrix FV⁻¹. Given the fact that

it follows that

4.1.1. Stability of the Disease-Free Equilibrium

The stability of the disease-free equilibrium will be investigated in this subsection.

Theorem 2The disease free equilibriumE⁰is globally asymptotically stable if, and unstable if.

Proof: The Jacobian matrix , evaluated at E⁰, is given by

The characteristic equation that results from the Jacobian matrix is given by det = 0. Thus, we get

The characteristic equation (7) has three negative real roots, which are

and the other 2 roots, λ₄ and λ₅, are roots of the equation

where

We now need to consider the signs of λ₄ and λ₅. Note that

Assuming

we have that

Moreover

It implies that

It follows that a₁ > 0. As a result the roots λ₄ and λ₅ are strictly negative. We can conclude that all roots of (7) have negative real parts, therefore, the disease free equilibrium is locally asymptotically stable [8–10]. Furthermore assuming that

we have that a₂ < 0, it follows that the characteristic equation f(λ) = 0 has a least a strictly positive root. Therefore, the disease free equilibrium E⁰ is unstable.

4.2. Existence of an Endemic Equilibrium

In this subsection, we investigate the existence of an endemic equilibrium for the system (4).

Proposition 1. The system of differential equations (4) admits a unique endemic equilibrium if and only if.

Proof: Let E^* = (S^*, I^*, T^*, A^*, R^*) be an equilibrium point. Then the components of E^* satisfy the following set of equations

From the last three equations of the system (9), we have that

Substituting T^*, A^* and R^* into the first two equations, we obtain

It follows that

and

Next we set

It is enough to show that there exists a point I^* ∈ ℝ⁺ such that h(I^*) = g(I^*). In other words, we will show that the curves of the functions h and g intersect at a point I^*.

Note that

is a vertical asymptote of the function h(I). For all

we have that

and

It follows that the function h is increasing and concave upward in the interval

with a vertical asymptote at the right end of the interval. Note that the function g is increasing and concave downward in the closed interval

As a result if

which is equivalent to the condition , then Equation (12) has a unique root I^* in the interval

Furthermore if

then h(I) < 0. There is no intersection point with g(I) since g is a positive function. Therefore there exists a unique endemic equilibrium point E^* = (S^*, I^*, T^*, A^*, R^*) provided that . In addition if

equivalent to the condition , there is no endemic equilibrium for the system (4).

4.2.1. Stability of the Endemic Equilibrium

Lemma 3. Letg(I) be a positive smooth function defined on the interval [0, ∞). Suppose that assumptions H₁and H₂hold, then the following inequality is satisfied

Proof:

We have that

as g″(I) ≤ 0. This implies that the function g(I) − Ig′(I) is increasing on the interval [0, ∞). Given the fact that g(0) − 0g′(0) = 0, it follows that g(I) − Ig′(I) ≥ 0.     □

Theorem 3If, then the endemic equilibriumE^*is locally asymptotically stable.

Proof: The Jacobian matrix of the endemic equilibrium is given by

The characteristic equation that results from the Jacobian matrix is given by det = 0. Thus, we get

where

The characteristic Equation (15) has a negative real double root

and three other roots, λ₃, λ₄ and λ₅, which are the roots of the equation

From Lemma (3), we have that

It follows that

Hence,

and

Moreover we have that

It follows that

As a result, by the Routh−Hurwitz stability criterion [11], all the roots of the characteristic polynomial (15) have strictly negative real parts. Therefore the endemic equilibrium is locally asymptotically stable.

5. Numerical Simulations

In this section, we provide numerical simulations for the evolution system of ordinary differential equations (1) to support the theoretical findings. Without loss of generality we set

Note that conditions of assumptions (H₁) and (H₂) are satisfied. Furthermore we let

Next we explore two scenarios involving static simulations and time-dependent simulations respectively.

5.1. Static Simulations

Picking and β = 1 and substituting in the expression of the basic reproduction number, we get that . According to Theorem 2 the disease-free equilibrium, E⁰ = (50, 0, 0, 0, 25), is globally asymptotically stable. In Figure 2, it clearly shows that the disease eventually dies out.

Figure 2

Picking and β = 0.1 and substituting in the expression of the basic reproduction number, we get that . According to Theorem 3 the endemic equilibrium, E^* = (42.5, 3.6, 2.7, 4.9, 21.2), is locally asymptotically stable. In Figure 3, all the graphs converge to the endemic equilibrium.

Figure 3

5.2. Time-Dependent Simulations

Picking , it can be observed in Figure 4 that graphs converge to the disease free equilibrium as time increases. It therefore suggests the global asymptotical stability of the disease free equilibrium and the extension of the disease in time.

Figure 4

Picking , Figure 5 clearly shows that the susceptible population vanishes in a short span of time and the disease essentially affects all people in the population.

Figure 5

6. Concluding Remarks

In this paper, we formulated and investigated a mathematical model describing the dynamics on sexually transmitted disease models with a novel class of non-linear incidence. We showed that the derived non-autonomous system of differential equations governing the evolution of the process was well-posed and the solution happened to be positive and bounded. The role of awareness in the susceptible and infectious classes was explored and investigated. A vertical asymptote and concavity arguments were critical in the proof of existence of an endemic equilibrium for the system and its asymptotical stability. In particular, numerical simulations were performed in order to predict the asymptotic behavior of solutions and support the theoretical findings.

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