Mathematical model of sexual orientations with advocacy

Introduction

Sexual orientation has been defined as the attraction one feels to men, women, or both [1,2] . It has been a challenge conceptualising the sexual orientation phenomenon since it has been characterised by shifting societal and scientific perceptions. In past years, sexual orientation has always been seen through a binary lens where heterosexuality was perceived as the standard. In contrast, other orientations were seen as a deviation from the norm. Magnus Hirschfeld and Alfred Kinsey began to change this perception by pushing for a deeper understanding of human sexuality [3] . Since then, there has been increased research work showing sexual orientations as a varied spectrum, leading to recognition of lesbians, gays and Bisexual Individuals (LGB) [4] . In 2015, the Supreme Court of the United States of America stated that a permit should be issued to same-sex couples [5] . Despite these efforts to recognize different spectrums in sexual orientations, a lot of LGB people still go for therapy to change their sexual attraction. In the United States, for instance, 60% of sexual minority people visit psychiatrists, psychologists, and religious leaders to change their sexual orientation [6] . Advocacy is the deliberate process of influencing those who make or have responsibility for implementing policy decisions. At the same time, an advocate spublicly supports or recommends a particular cause or policy, or pleads on another's behalf [7,8] . Advocacy is an action that aims to change laws, policies, practices, and attitudes. It can include public activities designed to gain media attention, such as flashmobs, demonstrations, or marches, non-public activities such as meetings with politicians, workshops, and trainings for key decision-makers and community leaders.

Lobbying in international forums; or mass campaigning through letter-writing, public meetings, or social media [9] . Mathematicians have also furthered knowledge in this study area by modelling phenomena revolving around this topic and other real-life situations [10,11] . Such studies include, but are not limited to, HIV/AIDS models considering sexual preferences [12] , analysing the impacts of bisexual behaviour in a global community [13] , a model of sexual transmission of Hepatitis B considering sexual orientations [14] among LGBT populations. This paper seeks to develop a comprehensive mathematical model of sexual orientations with advocacy, to further knowledge in the existing literature.

Model formulation

This study uses classical SIRS (susceptible, infected, recovered and susceptible) to model sexual orientations with advocacy Model assumptions  It was assumed that homosexuality (same-sex marriage) is not hereditary; instead, people are convinced to practice it.  One can only be influenced by a gay, a lesbian, a bisexual male, or a bisexual female into homosexuality.  Advocacy is only done by Gay, Lesbian, Bisexual males or Bisexual females.

Model development

The model is formulated into ten compartments, Males (M), Females (F), gays (G), lesbians (L), heterosexual males (Hm), heterosexual females (Hf), bisexual males (Bm), bisexual females (Bf), recovery centers for male (Rm) and recovery centers for females (Rf). Gays, lesbians, bisexual males and bisexual females will be referred to as homosexual populations. Below is the schematic representation of the model.

Model Equations

Invariant regions

This section studies boundedness and positivity to establish if the model is in a feasible region for consideration.

Boundedness Theorem 1

If the initial conditions of system 1 are in domain Q, then all solutions of system 1 enter and remain in the same domain Q.

Proof

It is first shown that all feasible solutions are uniformly bounded in a proper subset of Q. Splitting the system into the male component (Nm) and the female component (Nf) given as Nm = M + G + Hm + Bm + Rm and Nf = F + L + Hf + Bf + Rf. Let (Nm = M + G + Hm + Bm + Rm) be any system solution with non-negative initial conditions. Since . This implies that 0 ⩽ 𝑁 𝑚 ⩽ 𝜆 1 𝜅 1

. Similarly, the female population

𝜅 2 [15] . Hence the proof.

Positivity

𝑄 = {𝑀(𝑡), 𝐺(𝑡), 𝐿(𝑡), 𝐻 𝑚 (𝑡), 𝐻 𝑓 (𝑡), 𝐵 𝑚 (𝑡), 𝐵 𝑓 (𝑡), 𝑅 𝑚 (𝑡), 𝑅 𝑓 (𝑡)} ∈ 𝑅 + 10 and 𝑀(0) ⩾ 0, 𝐺(0) ⩾ 0, 𝐿(0) ⩾ 0, 𝐻 𝑚 (0) ⩾ 0, 𝐻 𝑓 (0) ⩾ 0, 𝐵 𝑚 (0) ⩾ 0, 𝐵 𝑓 (0) ⩾ 0, 𝑅 𝑚 (0) ⩾ 0, 𝑅 𝑓 (0) ⩾ 0 then, the region will be positively bounded.

Where, 𝐶 1 , it is a constant of integration. Putting initial conditions;

Hence, the region Q is positively invariant, and the model is well-posed mathematically. Since the model monitors human populations, all the variables and parameters are assumed to be non-negative for t ⩾ 0.

Equilibrium states, reproductive number, and stability 4.1 Homosexuality-free equilibrium point (HFE). Theorem 3

System 1 has an HFE obtained by setting all the homosexual and recovered classes to zero. [16] Proof E 0 = (M 0 ,G 0 ,Hm 0, Bm 0, Rm 0, F 0 ,L 0 ,Hf 0 ,Bf 0 ,Rf 0 ) = (ξ1,0, ξ2,0,0, ξ3,0,ξ4,0,0)

The Basic Reproduction Number

The next-generation method [17,18] was used to obtain reproduction numbers. Using the notation f for a matrix of new infection terms and v for the matrix of the remaining transfer terms in system 1, yields

It follows that basic reproduction numbers for model 1, denoted by 𝑅 0 1 , 𝑅 0 2 , 𝑅 0 3 and 𝑅 0 4 are given by evaluating the eigenvalues of F.V -1 as follows

Thus, the basic reproduction numbers are:

Local Stability Analysis of Homosexuality-Free Equilibrium Point (HFE) Theorem 4

The HFE of model system 1 is locally asymptotically stable (LAS) if 𝑅 0 1 < 1, 𝑅 0 2 < 1, 𝑅 0 3 < 1, and 𝑅 0 4 < 1 and unstable otherwise [17] .

Proof

To establish the local stability, the Jacobian of the model was evaluated at E 0 . The stability of this steady state is then determined based on the signs of the eigenvalues of the corresponding Jacobian, which are functions of the model parameters.

All eigenvalues of (3) were negative under the condition,

< 1. Thus, the homosexuality Free Equilibrium (HFE) of the system is locally asymptotically stable, i.e.𝑅 0 1 < 1, 𝑅 0 2 < 1, 𝑅 0 3 < 1, and 𝑅 0 4 < 1 and unstable otherwise.

Numerical simulations support analytical results that in the absence of homosexuality, the number of individuals in homosexual classes will tend to zero, as shown below. ) is an M-matrix (the off-diagonal elements of A are non-negative) and Ω is the region where the model makes biological sense [16, [17][19]. Then we can conclude that 𝐸 0 is a globally asymptotically stable (GAS) equilibrium of model system 1, provided that 𝑅 0 1 < 1, 𝑅 0 2 < 1, 𝑅 Since 𝐺 ˆ(𝑋, 𝑍) ≥ 0 by theorem 5, 𝐸 0 Globally asymptotically stable.

Existence of Homosexuality Endemic Equilibrium point (HEEP)

Conditions for the existence of equilibrium are sought for which the model is endemic in the population [17] . For HEEP to exist, the infectious classes of the system 1 must be greater than zero, that is, G > 0, Bm > 0, L > 0, Bf > 0. Setting infectious classes to zero and substituting them with each other, and letting them be greater than zero, results in

Thus, HEEP has existed since 𝑅 0 1 > 1, 𝑅 0 2 > 1, 𝑅 0 3 > 1, and 𝑅 0 4 > 1

Global stability of the Homosexuality Endemic Equilibrium point (HEEP) Theorem 6

The HEEP is globally asymptotically stable whenever 𝑅 0 1 < 1, 𝑅 0 2 < 1, 𝑅 0 3 < 1, and 𝑅 0 4 < 1.

Proof

Where, ∇1 = (κ1 +β1F +π1), ∇2 = (κ1 +ψ1 +µ1F +µ1θ1Bf), ∇3 = (κ1 +γ1G), ∇4 = (κ1 + φ1), ∇5 = (κ1 + η1 + χ1 + Γ1), ∇6 = (κ2 + β2M + π2), ∇7 = (κ2 + ψ2 + µ2M + µ2θ2Bm), ∇8 = (κ2 + γ2), ∇9 = (κ2 + φ2), ∇10 = (κ2 + η2 + χ2 + Γ2).

Letting P * denote all the positive values and Q * denote all the negative values of the solution after substituting evaluated constants, results in = 0 is the singleton 𝐸 * using Lassalle's invariance principle [18,19] ,

𝑑𝑘(𝑀,𝐺,𝐻 𝑚 ,𝐵 𝑚 ,𝑅 𝑚 ,𝐹,𝐿,𝐻 𝑓 ,𝐵 𝑓 ,𝑅 𝑓 )

Bifurcation Analysis

The Centre Manifold Theorem [21] was used to investigate the nature of the bifurcation of Model 1. Letting 𝑃 𝑘 be the 𝑘 th component of p and

Considering a case where 𝑅 0 * 1 = 1, 𝑅 0 * 2 = 1, 𝑅 0 * 3 = 1 and 𝑅 0 * 4 = 1, 𝑑 1 = 𝑑 1 * , 𝑑 2 = 𝑑 2 * , 𝑑 5 = 𝑑 5 * and 𝑑 8 = 𝑑 8 * were chosen as bifurcation parameters as follows:

Jacobian J * was obtained: Solving for the right and left eigenvector at HFE of the Jacobian J * resulted to u =[u1,u2,u3,u4,u5,u6,u7,u8,u9,u10] T =0 , and left eigenvectors as v = [v1, v2, v3, v4, v5, v6, v7, v8, v9, v10]=0.

Since the right and left eigenvectors result in zero, i.e., (a = 0 and b = 0), then the Centre Manifold theorem cannot be used to study the bifurcation of this system, or this system doesn't have bifurcation.

Normalized Sensitivity analysis of the parameters

The technique employed in the sensitivity analysis was based on the reproduction numbers. This helps determine each parameter's contribution level to the reproduction number. This technique uses partial derivatives of the parameters [22] . These indices are given by

Partial rank correlation coefficient (PRCC)

To identify parameters that were more influential to the model, PRCC analysis was carried out, where the strength of the outputs and inputs of the model was determined using the Latin hypercube sampling (LHS) method. [23] Figure 7 shows that in the presence of advocacy, the lesbian population is increased by 15.6% compared to the model without advocacy. Figure 8 shows that in the presence of advocacy, the bisexual female population is increased by 15.3% in comparison to the model without advocacy. The numerical results above showed that, while advocacy increases the spread of homosexuality in the population, its effects are more pronounced in bisexual female and lesbian populations.

Results and Discussion

This research developed a deterministic mathematical model to analyze sexual orientations with advocacy. The positivity, boundedness, equilibria points, and control reproduction numbers were studied. The Routh-Hurwitz criteria were used to assess the local stability of the disease-free equilibrium, the Metzler Matrix method was used to determine global stability, while Lyapunov functions were used to study the global stability of the endemic equilibrium point. Bifurcation analysis was also conducted. Lastly, a sensitivity analysis and numerical simulations were performed using Wolfram Mathematica and MATLAB software.

The model was found to be bounded, lay in the positive region, and had Homosexuality-free equilibrium points given as E 0 = (ξ1,0, ξ2,0,0, ξ3,0, ξ4,0,0). Control reproduction numbers were obtained as 𝑅 0 1 = 0.280315, 𝑅 0 2 = 4.66546, 𝑅 0 3 = 0.0203, and 𝑅 0 4 = 0.144999. The stability of HFE was found to be locally asymptotically stable (LAS) and globally asymptotically stable (GAS) whenever 𝑅 0 1 < 1, 𝑅 0 2 < 1, 𝑅 0 3 < 1, and 𝑅 0 4 < 1. The global stability of HEEP was found to be asymptotically stable (GAS) whenever 𝑅 0 1 > 1, 𝑅 0 2 > 1, 𝑅 0 3 > 1, and 𝑅 0 4 > 1. Upon carrying out bifurcation analysis, it was observed that the Center Manifold theorem could not be used to get the bifurcation of this system since the right and left eigenvectors were zero. The sensitivity analysis based on the control reproduction number showed that d1, d2, d5, d8,e1, e2,e3, and φ2 were the most influential parameters, and they would significantly influence the spread of homosexuality within the community. The sensitivity analysis was performed based on the PRCC method and LHS to identify the highly sensitive parameters in the model. It showed e2 and e3 as the most influential parameters, in that order, and ψ1 as the least influential. Numerical results showed that the model with advocacy increases the gay population, the lesbian population, the bisexual male population, and the bisexual female population by 5.7%, 15.6%, 2.6%, and 15.3%, respectively, in comparison with the model without advocacy. Thus, this study shows that while incorporating advocacy that promotes homosexuality in the community significantly increases homosexual populations in the society, Lesbian and bisexual female populations will be improved the most.

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