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.

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.

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 Γd 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.

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 all t ∈ [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 S′(t1)<0. From the first equation of the system (1), we have S′(t1)=Γ>0, 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 1 For 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 all t ≥ 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

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

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.

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.

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.

All authors contributed to all critical aspects of the analysis and simulations. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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