Sexually transmitted diseases (STDs) need to be studied systematically to better understand their global spread. Transmission of Chlamydia trachomatis is a severe public health issue, with roughly 90 million new cases per year. Globally, Chlamydia trachomatis is the most frequent bacterial cause of STDs.
Methods:
To better understand the dynamics and transmission of Chlamydia, the susceptible-exposed-infected-recovered-susceptible (SEIRS) model was constructed. Using a system of nonlinear ordinary differential equations, a basic reproduction number has been calculated at an equilibrium point, and the system is locally and globally asymptotically stable at both disease-free and endemic equilibrium points. Numerical simulations illustrate the behavior and flow of Chlamydia infections in different compartments.
Results:
Conclude from the proposed study that 25% of individuals have been exposed to Chlamydia, of which 20% of individuals get infections due to sexual activity and 55% of individuals get recovered. Twenty percent of individuals have been exposed to Chlamydia, of which 37% of individuals have been infected due to an unhygienic environment. Of those, 43% of individuals recovered. Also, it has been found that people are more likely to get infections because of an unhygienic environment than sexually active people. The recovery rate is also much better for people who have been infected because of an unhygienic environment.
Conclusions:
Sexually transmitted infections can be reduced by up to 10%. While infection due to an unhygienic environment can be controlled up to a certain intensity. According to this research, public awareness campaigns and the improvement of personal hygiene will play a major role in reducing the spread of the epidemic in the future.
Among sexually transmitted diseases (STDs), Chlamydia trachomatis infection is the most common [1]. According to the World Health Organization (WHO), there will be 129 million cases of Chlamydia trachomatis infection in 2020 [2]. The bacteria Chlamydia trichromatic causes Chlamydial infection. It can be transmitted in two main ways: one is through sexual activity with an infected individual, while the other is due to an unhygienic environment. Once infected, a person may transmit the disease to their partners via intercourse, anal sex, or oral sex. Whereas, a non-sexual method of transmission includes direct hand-to-hand contact, sharing of bedding, clothing, or towels, and transmission by flies that have come into contact with an infected person’s discharge from the eyes or nose. In rare situations, infected vaginal fluid or semen might come into contact with a person’s eye, producing conjunctivitis. Chlamydia trachomatis is also the leading cause of blindness worldwide [3]. It is a very common STD that can happen to both men and women, affecting about 4.2% of women and 2.7% of men worldwide [4, 5], but it is more common in women. However, Chlamydia is more prevalent among young individuals who engage in sexual activity, whereas infection rates are greater in younger women aged fifteen to twenty-four [6]. In women, Chlamydial infection affects the throat, rectum, and cervix, causing serious damage to the reproductive system. Moreover, it causes pelvic inflammatory disease (PID) with a subsequent risk of infertility. As a result, pregnancy may become difficult or impossible for a woman. Aside from ectopic pregnancy, it may also cause miscarriage [7, 8]. The transmission of Chlamydia from an infected mother to her baby may occur during vaginal delivery [6]. Also, the symptoms of Chlamydia infection in women include vaginal discharge with a bad smell, vaginal bleeding, vaginal itching and burning, abdominal pain, cramping during menstruation, and fever. In men, the symptoms include pain and swelling in the testicles, pain, and burning when urinating, and turbid discharge from the penis. Chlamydia has an incubation period of seven to twenty days. It refers to the time between infection and the development of the disease in a person. Infections with Chlamydia are treatable and curable. Azithromycin is an antibiotic that is often provided in a single, high-dose prescription or doxycycline is an antibiotic that must be taken twice a day for about one week before it will be effective. It is essential to avoid any sexual activity throughout the therapy period. Even if an infected individual has successfully cured a previous infection, it is still possible to transmit and get Chlamydia if exposed to the same person again.
Literature survey
Mathematical models have proven to be one of the most useful tools for studying the spread of any infection. The mathematical epidemic model of STDs is used to study the behavior of STDs and their impacts on society. These models were originally used to demonstrate the relevance of the contact structure and dynamic characteristics of infections [9]. A compartmental mathematical model to study the Chlamydia resurgence [10]. In the absence of tools to change sexual networks, a vaccine will be necessary to stop infection transmission. Chlamydia trachomatis and gonorrhea co-dynamics models with optimum control analysis have been studied and examined to assess the impact of targeted treatment for each of the diseases on their co-infections in a population [11]. Mathematical modeling and study of the transmission dynamics of blinding trachoma with the impact of awareness programs and found a lack of competent health care systems and public awareness programs to blame for the outbreak of the blinding trachoma disease [12]. A sophisticated investigation of Chlamydia trachomatis infection sensitivity was performed using a susceptible-exposed-infected-recovered-susceptible (SEIRS) model, and they also analyzed the population turnover and the impact of screening [13]. When the asymptomatic interval is longer than the symptomatic phase, the impact of a screening program is more apparent. To understand how various forms of policy analysis are employed and how the studies have evolved with changes in the field, it is examined by published Chlamydia models [14].
Organization of proposed study
A model has been constructed in this paper to better understand how Chlamydia spreads. Section 2 contains the model’s formulation and description and also contains the calculation of the basic reproduction number at equilibrium points, while section 3 contains the computation of stability on a local and global scale, and section 4 contains the computation of numerical simulation in such a way that it helps in the analysis of Chlamydia infection and the effect of recovery rate on disease transmission. Section 5 concludes the model.
Formulation and description of the <italic>Chlamydia</italic> model
To analyze the transmission dynamics of Chlamydia, a compartmental model is constructed. This compartment model consists of two groups of populations with two strains of infectious stages: the transmission of Chlamydia due to sexual activity and an unhygienic environment. The total population of humans at the time t, N(t) is divided into five compartments, each with a class of susceptible individuals S(t), who are healthy but can get infected through direct and indirect contact with infectious individuals. Class of exposed individuals E(t), who have been infected due to sexual activity and an unhygienic environment but are not yet infectious. Is(t) and Iu(t) a class of infected individuals who are infected and can transmit the disease due to sexual activity and an unhygienic environment, respectively. R(t) class of recovered individuals who have been infected and then recovered from the disease. The transmission dynamics of Chlamydia are described graphically in Figure 1, and the parameters used in the model are described in Table 1.
Transmission dynamics of Chlamydia
Description of model parameters
Parameters
Description
Value
Source
B
Birth rate
0.018
Calculated
α1
Rate of transmission from S to E
0.8
Assumed
α2
Transmission rate from E to IS
0.67
Calculated
α3
Transmission rate from E to IU
0.32
Calculated
α4
Recovery rate from IS
0.92
Calculated
α5
Recovery rate from IU
0.95
Calculated
α6
Rate of transmission from R to S
0.05
Assumed
μ
Escape rate
0.01
Assumed
A dynamical system of non-linear ordinary differential equations for the model is formulated as follows:
dSdt=B−α1SE+α6R−μS,dEdt=α1SE−α2E−α3E−μE,dISdt=α2E−α4IS−μIS,dIUdt=α3E−α5IU−μIU,dRdt=α4IS+α5IU−α6R−μR.
The total human population of this model is presented as:
N(t)=S(t)+E(t)+IS(t)+IU(t)+R(t)
Adding the non-linear ordinary differential equation of system (1), we have
ddt=(S+E+IS+IU+R)≤B−μN.
This implies that limt→∞sup(S+E+IS+IU+R)≤Bμ.
Thus, the feasible region for the model is Λ, which is a positively invariant. i.e., every solution of the model, with initial conditions Λ remains there for all t > 0.
Λ={(S,E,IS,IU,R)∈R+5:S+E+IS+IU+R≤Bμ,S>0,E≥0,IS≥0,IU≥0,R≥0}
Existence of the equilibrium points
To obtain the disease-free equilibrium point E0for the system of non-linear differential equations, put the right-hand side of Eq. (1) equal to zero thus,
E0=(Bμ,0,0,0,0).
This means when there is no infection E = IS = IU = R = 0. This model has a unique disease-free equilibrium point.
Due to sexual activity or the use of unhygienic environments, there will always be an optimum number of individuals who will be sitting in each compartment. This point is referred to as an endemic point in epidemiology.
Endemic equilibrium point
Eend*=(S*,E*,IS*,IU*,R*)
where S*=α2+α3+μα1,E*=(α5+μ)(α6+μ)(α4+μ)(−μ2+(−α2−α3)μ+Bα1)/(α1μ(μ3+(α2+α3+α4+α5+α6)μ2+((α2+α3+α5+α6)α4+(α2+α3+α6)α5+α6(α2+α3))μ+((α2+α3+α6)α5+α3α6)α4+α2α5α6)),IS*=(α5+μ)α2(α6+μ)(−μ2+(−α2−α3)μ+Bα1)/(α1μ(μ3+(α2+α3+α4+α5+α6)μ2+((α2+α3+α4+α6)α3+(α2+α3+α6)α4+α6(α2+α3))μ+((α2+α3+α6)α4+α6α2)α5+α3α4α6)),IU*=(α6+μ)(α4+μ)α3(−μ2+(−α2−α3)μ+Bα1)/(α1μ(μ3+(α2+α3+α4+α5+α6)μ2+((α2+α3+α5+α6)α4+(α2+α3+α6)α5+α6(α2+α3))μ+((α2+α3+α6)α5+α3α6)α4+α2α5α6)),R*=((α2α4+α3α5)μ+α4α5(α2+α3))(−μ2+(−α2−α3)μ+Bα1)/(α1μ(μ3+(α2+α3+α4+α5+α6)μ2+((α2+α3+α5+α6)α4+(α2+α3+α6)α5+α6(α2+α3))μ+((α2+α3+α6)α5+α3α6)α4+α2α5α6)).
Basic reproduction number
To get the threshold value for the transmission of Chlamydia, a basic reproduction number is formulated using the next-generation matrix (NGM) algorithm [15]. The basic reproduction number for the system is obtained as the spectral radius of the matrix (fv−1) around the disease-free equilibrium point.
Let X = (S, E, IS, IU, R) then model rewrite as X’ = F(X) – V(X) where F(X) represents the rate of appearance of new infections in the compartment and V(X) represents the rate of transfer individuals which are given by,
F(X)=[α1SE0000]andV(X)=[E(α2+α3+μ)−α2E+IS(α4+μ)−α3E+IU(α5+μ)−α4IS−α5IU+R(α6+μ)−B+α1SE−α6R+μS]
By calculating the Jacobian matrices at E0, we find that D(F(E0))=[f000] and D(V(E0))=[v0J1J2] where, fand v are 5 × 5 matrices defined as f=∂Fi∂Xj(E0) and v=∂vi∂Xj(E0). Finding f and v we get,
f=[α1S000α1E00000000000000000000]andv=[α2+α3+μ0000−α2α4+μ000−α30α5+μ000−α4−α5α6+μ0α1S00−α6α1E+μ].
Here, v is a non-singular matrix that we can find v–1. Now we calculate the NGM fv−1 and the largest modulus of eigenvalues of fv−1 is the basic reproduction number of the model. The formulated basic reproduction number
R0=α1Bμ(α2+α3+μ).
Stability
In this section, we will discuss the local stability and global stability for equilibrium points.
Local stabilityTheorem-1
Disease-free equilibrium point E0 of the model is locally asymptotically stable if Bμ<α2+α3+μα1.
Proof: Evaluating the Jacobian matrix for the model at point E0 (disease-free equilibrium point) gives,
J(E0)=[−μ−α1Bμ00α60α1Bμ−α2−α3−μ0000α2−α4−μ000α30−α5−μ000α4α5−α6−μ].
Thus, the eigenvalues of J(E0) are given by λ1 = –μ, λ2 = –(α6 + μ), λ3 = –(α5 + μ), λ4 = –(α4 + μ), and λ5=Bα1−μ(α2+α3+μ)μ. Clearly, λ1, λ2, λ3 and λ4 are negative. Also, if Bμ<α2+α3+μα1 then λ5< 0.
As all eigenvalues of the matrix are negative therefore disease-free equilibrium point E0 of the model is locally asymptotically stable.
Theorem-2
The endemic equilibrium point Eend* is locally asymptotically stable if it satisfies the following condition, S≤max{α5α1,μ2α1α5}.
Proof: Linearizing the system around the endemic equilibrium point Eend* gives the Jacobian matrix
J(E*)=[−α1E*−μ−α1S*00α6α1E*α1S*−α2−α3−μ0000α2−α4−μ000α30−α5−μ000α4α5−α6−μ].
The characteristic equation of the Jacobian matrix is λ5+α4λ4+α3λ3+α2λ2+α1λ+α0 where α4=Eα1+α2+α3+α4+α6+5μ+(α5−Sα1),
α3=Eα1(α2+α3+α4+α5+α6+4μ)+α2(α4+α5+α6+4μ)+α3(α4+α5+α6+4μ)+α4(α6+4μ)4μα6+9μ2(α4α5−Sα1α4)+(α5α6−Sα1α6)+(4α5μ−4Sα1μ)+(μ2−Sα1α5),α2=7μ3+(6Eα1+6α2+6α3+5α4+5α6)μ2+(3E(α2+α3+α4+α5+α6)α1+(α2+α3+α6)3α4+3(α5+α6)(α2+α3))μ+E((α2+α3+α5+α6)α4+(α2+α3+α5)α6+α5(α2+α3))α1+((α5+α6)α4+α5α6)(α2+α3)(α4α5α6−Sα1α4α6)(3α4α5μ−3Sα1α4μ)(3α5α6μ−3Sα1α6μ)(6α5μ2−6Sα1μ2)(α4μ2−Sα1α4α5)(α6μ2−Sα1α5α6)(3μ3−3Sα1α5μ),α1=2μ4+(4Eα1+4α2+4α3+2α4+2α6)μ3+(3E(α2+α3+α4+α5+α6)α1+(3α2+3α3+2α6)α4+3(α5+α6)(α2+α3))μ2+2E((α2+α3+α5+α6)α4+(α2+α3+α5)α6+α5(α2+α3))α1+2((α5+α6)α4+α5α6)(α2+α3)μ+E(((α3+α5)α6+α5(α2+α3))α4+α2α5α6)α1+α4α5α6(α2+α3)(4α5μ3−4Sα1μ3)+(3α5α6μ2−3Sα1α6μ2)+(3α5α4μ2−3Sα1α4μ2)+(2α4α6μ−2Sα1α4α6μ)+(α4α6μ2−Sα1α4α5α6)+(2α4μ3−2Sα1α4α5μ)+(2α6μ3−2Sα1α5α6μ)+(3μ4−3Sα1α5μ2),α0=(Eα1+α2+α3)μ3+(E(α2+α3+α4+α5+α6)α1+(α4+α5+α6)(α2+α3))μ2+(E((α2+α3+α6)α4+(α2+α3+α6)α5+α6(α2+α3))α1+((α5+α6)α4+α5α6)(α2+α3)μ+E(((α2+α3+α6)α4+α2α5α6)α1+α4α5α6(α2+α3))μ+(α5μ4−Sα1μ4)+(α4α5μ3−Sα1α4μ3)+(α5α6μ3−Sα1α6μ3)+(α4α5α6μ2−Sα1α4α6μ2)+(α4α6μ3−Sα1α4α5α6μ)+(α6μ4−Sα1α5α6μ2)+(α4μ2−Sα1α4α5μ2)(μ5−Sα1α5μ3).
The endemic equilibrium point Eend* is locally asymptotically stable [16] if it satisfies, Sα1 < α5 and Sα1α5 < μ2 which implies that S<α5α1 and S<μ2α1α5. Hence, the endemic equilibrium point Eend* is locally asymptotically stable if S≤max{α5α1,μ2α1α5}.
Global stability
Here, we discuss the global stability behavior of the equilibrium point E0 and Eend* by Lyapunov’s function [17].
Theorem-3
The disease-free equilibrium point E0 is globally asymptotically stable.
Proof: The disease-free equilibrium point E0 is global asymptotically stable in the feasible region Λ.
Consider the Lyapunov function L1(t) = S(t) + E(t) + IS(t) + IU(t) + R(t)L1'(t)=S′(t)+E′(t)+IS'(t)+IU'(t)+R′(t)L1'(t)=B−μ(S+E+IS+IU+R)L1'(t)=0
Since E0 belongs to the feasible region Λ, S is bounded above by Bμ, this implies dL1dt≤0. Moreover, dL1dt=0. Therefore, the only trajectory of the system on which dL1dt=0 is E0 . Hence, by LaSalle’s Invariant Principle [18], E0 is globally asymptotically stable.
Theorem-4
The endemic equilibrium point Eend* is globally asymptomatically stable in the feasible region Λ.
Proof: Consider the Lyapunov function,
L2(t)=12[S(t)−S*)+(E(t)−E*)+(IS(t)−IS*)+(IU(t)−IU*)+(R(t)−R*)]2∴dL2dt=[(S(t)−S*)+(E(t)−E*)+(IS(t)−IS*)+(IU(t)−IU*)+(R(t)−R*)](S′(t)+E′(t)+I′S(t)+I′U(t)+R′(t)=[(S(t)−S*)+(E(t)−E*)+(IS(t)−IS*)+(IU(t)−IU*)+(R(t)−R*)](B−μ(S+E+IS+IU+R))=[(S(t)−S*)(E(t)−E*)+(IS(t)−IS*)+(IU(t)−IU*)+(R(t)−R*)](−μ[(S(t)−S*)(E(t)−E*)+(IS(t)−IS*)+(IU(t)−IU*)+R(t)−R*)])=−μ[S(t)−S*)+(E(t)−E*)+(IS(t)−IS*+(IU(t)−IU*)+R(t)−R*)]2≤0.
By putting B=μS*+μE*+μIS*+μIU*+μR*, we get dL2dt=−μ[(S(t)−S*)+(E(t)−E*)+(IS(t)−IS*)+(IU(t)−IU*+(R(t)−R*)]2≤0. Hence, Eend* is globally asymptomatically stable.
Numerical simulation
We simulated the transmission dynamics of Chlamydia infection, using the parametric values given in Table 1. We carry out a simulation and interpret the behavior of Chlamydia.
The variation in the population of the respective compartment concerning time in months shown in Figure 2. This represents that a large population of exposed individuals becomes infected within one month and the intensity of infected individuals decreases after around 33 days. It can be observed from the graph that infected individuals due to an unhygienic environment can be cured within five months. While infected individuals due to sexual activity can be reduced by up to 10% through proper medication and awareness.
Density in the compartment with time
The change in infected individuals due to being sexually active for different values of α2 (the rate at which exposed individuals are infected due to sexual activity) indicates Figure 3. It is observed that the number of infected individuals due to sexual activity initially increases by 5% as we increase the value of α2 but this situation reverses after about 2.5 months, and infected individuals decrease by around 10%.
Impact on infected individuals due to sexual activity due to change in α2
In Figure 4a and 4b, the trajectory field shows the intensity of infected individuals due to sexual activity and an unhygienic environment in the class of recovered individuals by Chlamydia, respectively. It can be observed from the Figure 4, that more infected individuals move toward the recovered individuals, but after some point of time it gets stable, and recovered cases decrease up to a certain level.
Intensity of infected individuals in the class of recovered individuals
The movement of susceptible individuals toward the exposed individuals show in Figure 5a. Moreover, it shows the movement of exposed individuals towards the susceptible class at a lower intensity, which illustrates the awareness of exposed individuals to disease transmission and Figure 5b depicts the movement of recovered individuals towards the susceptible class with greater intensity and vice versa.
Trajectory field and solution curve
In Figure 6 a pie chart shows the proportion of the Chlamydia model’s compartments. Chlamydia is a disease that affects 10% of the population, 8% of whom are infected by sexual activity and another 18% by an unhygienic environment, from which 21% of these people can recover from the disease.
Percentage of all compartment
According to the pie chart in Figure 7a, 25% of people have been exposed to Chlamydia, and 20% of people get infections from sexual activity, with 55% of people recovering and Figure 7b shows that 20% of people have been exposed to Chlamydia, 37% have been infected as a result of an unhygienic environment, and 43% have recovered. It is also observed that individuals are more infected due to an unhygienic environment as compared to sexually active individuals and that the rate of recovery is much higher for those who get an infection because of an unhygienic environment.
Percentage of exposed individuals and recovery rate due to infected individuals
Conclusions
A mathematical model is formulated to study the transmission of Chlamydia infection due to sexual activity and an unhygienic environment. Using a system of nonlinear ordinary differential equations, a basic reproduction number has been calculated at an equilibrium point, and the system is locally and globally asymptotically stable at both disease-free and endemic equilibrium points. Numerical simulations have illustrated the behavior and flow of Chlamydia infections in different compartments, which shows how exposed individuals are infected due to sexual activity and unhygienic environments, which demonstrates that sexually transmitted infections can be reduced by up to 10%. While the infection spreads due to an unhygienic environment, it can be controlled in five months. According to the conclusions of this research, public awareness campaigns and the improvement of personal hygiene will both play a major role in reducing the spread of the epidemic in the future.
AbbreviationSTDs:
sexually transmitted diseases
DeclarationsAuthor contributions
NHS gave the idea of the diseases and modeling. YNS detailed the conceptual facts. JNV and PMP constructed a model and did the simulation of the proposed system. JNV wrote the research paper and the remaining authors contributed to the write-up.
Conflicts of interest
The authors declare that they have no conflicts of interest.
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