1. Introduction

Front. Appl. Math. Stat.

Frontiers in Applied Mathematics and Statistics

Front. Appl. Math. Stat.

2297-4687

Frontiers Media S.A.

10.3389/fams.2022.1101029

Applied Mathematics and Statistics

Original Research

COVID-19 and syphilis co-dynamic analysis using mathematical modeling approach

Teklu

Shewafera Wondimagegnhu

^* Terefe

Birhanu Baye

Department of Mathematics, College of Natural and Computational Sciences, Debre Berhan University, Debre Berhan, Ethiopia

Edited by: Yasser Aboelkassem, University of Michigan–Flint, United States

Reviewed by: Vedat Erturk, Ondokuz Mayıs University, Türkiye; Pushpendra Kumar, Central University of Punjab, India

*Correspondence: Shewafera Wondimagegnhu Teklu ✉ luelzedo2008@gmail.com; ✉ shewaferaw@dbu.edu.et

This article was submitted to Mathematical Biology, a section of the journal Frontiers in Applied Mathematics and Statistics

01 02 2023

2022

1101029

17 11 2022 30 12 2022

2023

Teklu and Terefe

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this study, we have proposed and analyzed a new COVID-19 and syphilis co-infection mathematical model with 10 distinct classes of the human population (COVID-19 protected, syphilis protected, susceptible, COVID-19 infected, COVID-19 isolated with treatment, syphilis asymptomatic infected, syphilis symptomatic infected, syphilis treated, COVID-19 and syphilis co-infected, and COVID-19 and syphilis treated) that describes COVID-19 and syphilis co-dynamics. We have calculated all the disease-free and endemic equilibrium points of single infection and co-infection models. The basic reproduction numbers of COVID-19, syphilis, and COVID-19 and syphilis co-infection models were determined. The results of the model analyses show that the COVID-19 and syphilis co-infection spread is under control whenever its basic reproduction number is less than unity. Moreover, whenever the co-infection basic reproduction number is greater than unity, COVID-19 and syphilis co-infection propagates throughout the community. The numerical simulations performed by MATLAB code using the ode45 solver justified the qualitative results of the proposed model. Moreover, both the qualitative and numerical analysis findings of the study have shown that protections and treatments have fundamental effects on COVID-19 and syphilis co-dynamic disease transmission prevention and control in the community.

syphilis COVID-19 COVID-19 and syphilis co-infection protection numerical simulation

1. Introduction

Communicable diseases are illnesses caused by pathogenic microbial agents such as bacteria, viruses, fungi, and parasites, which affect human beings throughout the world [1]. The novel coronavirus (COVID-19) infection is a lethal disease that has been a major global public health concern. The COVID-19 pandemic has affected various animals mostly infecting millions of human beings in different nations throughout the world [2–6]. It has been spreading mainly through sneezing, individuals interacting with each other in a certain time frame, or through coughing [7]. Although different species of animals are thought to be the source of COVID-19 transmission, bats have been shown to be coronavirus hosts [8]. Many nations throughout the world have started to practice various prevention and control strategies such as lockdown approach, quarantine, isolation, and closing schools [3, 9].

Syphilis is a major sexually transmitted disease and has been affecting millions of individuals both in low- and high-income countries of the world [10]. It is a chronic systemic disease caused by Treponema pallidum bacterium which is mainly transmitted through sex, blood contact, and mother-to-child during birth [4, 10–16]. Diagnosis, treatment, and using a condom are the basic control mechanisms of syphilis spreading in the community [10]. If left untreated, syphilis progresses through four stages: primary, secondary, latent, and tertiary [17–19]. The first three infection stages can transmit the disease to other susceptible groups of individuals, the transmission can occur via sexual contact, and in most cases, the tertiary stage is not transmissible through sexual contact [19]. It can be a cause of different cardiovascular and neurological diseases [17]. Approximately 90% of new syphilis substantial morbidity and mortality data are recorded in low-income countries around the world [11, 16]. Co-infection is an infection of an individual with two or more microorganisms' species [20, 21]. COVID-19 is an opportunistic infection for people with a weak immune system who were already infected by acute and chronic infections such as pneumonia, TB, and HIV/AIDS.

Mathematical modeling approach research done by scholars using a deterministic method [10, 14], a stochastic method [7, 22], or a fractional order method [23–32] has made a great contribution to linking the scientific approach with real-world physical situations and also for the decision-making process for solving real-world problems [33]. Different scholars have formulated and analyzed mathematical models on COVID-19 transmission [7, 8, 22, 24–26, 29, 30, 34–37], syphilis transmission [10, 17–19, 23], and other infectious diseases transmission [20, 21, 27, 28, 33, 38–40]; however, no one has done analysis on COVID-19 and syphilis co-infection transmission dynamics.

Oshinubi et al. [41] proposed and analyzed a new age-dependent compartmental model for COVID-19 transmission. The qualitative analysis of the model includes the non-negativity and boundedness of the model solutions in a given region, and the existence, uniqueness, and stability of the model solutions. Using parameter estimation from three different nations Kuwait, France, and Cameroon, they carried out numerical simulations and have shown the fundamental role of vaccination on COVID-19 transmission. Babaei et al. [34] proposed and examined a model for novel coronavirus transmission with Caputo's fractional order approach. The finding of the study shows that quarantine has a very fundamental role to control transmission. Iboi et al. [17] formulated and analyzed a new multi-stage syphilis model to examine the role of transitory immunity loss in the spreading process. The analysis shows that the disease-free and unique endemic equilibrium points are globally asymptotically stable when the corresponding basic reproduction number is less than unity and greater than unity, respectively. The results show that high treatment rates in the primary and secondary stages have a positive effect on the remaining stages of infection. Nwankwo et al. [38] formulated a mathematical model to examine the interaction between HIV/AIDS and syphilis pathogens with syphilis treatment on the co-infection of syphilis and HIV/AIDS where treatment or HIV is not accessible. High treatment in the primary stage has a fundamental role in reducing both single infections and co-infections in the population. Teklu et al. [42] formulated a six-compartmental COVID-19 transmission model to examine the impacts of intervention measures. The results show that protection, treatment, and vaccinations are fundamental to minimizing infection in the population.

Because different scholars have been mainly concerned with studying COVID-19 and syphilis single infections, no one has studied syphilis and COVID-19 co-infection using a mathematical model approach. Therefore, in this study, we are interested in filling the gap by formulating and analyzing syphilis and COVID-19 model intervention strategies.

The remaining part of the article is organized as follows. Section 2 presents COVID-19 and syphilis co-infection model construction. Section 3 describes the qualitative model analysis. Section 4 presents the numerical and sensitivity analyses. Section 5 presents the discussions and conclusions of the whole research study.

2. Model construction

We have considered COVID-19 and syphilis co-infection by separating the four syphilis infection stages (primary, secondary, latent, and tertiary) into two, the asymptomatic and symptomatic groups, and we have divided the population N(t) into 10 mutually exclusive states, which are described in Table 1 as follows:

N(t)=Pc(t)+Ps(t)+S(t)+Ic(t)+Qc(t)+Ias(t)+Iss(t) +Ts(t)+Tcs(t)+T(t).

Table 1

Variables' definitions.

State variables	Definition
S	Susceptible individuals for both COVID-19 and syphilis
P_c	COVID-19 protected individuals
P_s	Syphilis protected individuals
I_c	COVID-19 infected individuals
Q_c	COVID-19 isolated with treatment individuals
I_as	Syphilis asymptomatic infected individuals
I_ss	Syphilis symptomatic infected individuals
T_s	Syphilis treated individuals
I_cs	COVID-19 and syphilis co-infected individuals
T	Co-infected treated individuals

Assumptions and definitions of basic terms:

➢ Co-infectious humans do not transmit both infections simultaneously.

➢ COVID-19 infection is transmitted to susceptible individuals from I_c and I_cs infectious groups at the transmission rate as follows: (1)λc=β2(Ic+ϕ1Ics).

➢ Syphilis infection is transmitted to susceptible individuals from I_as, I_ss, and I_cs infectious groups at the force of infection rate as follows: (2)λs=β1(Ias+ϕ2Iss+ϕ3Ics).

Using variable and parameter definitions given in Tables 1, 2, respectively, the flowchart of the COVID-19 and syphilis co-infection model is represented in Figure 1.

Table 2

Parameter definitions.

Parameters	Biological definitions
Λ	The annual recruitment number of population in the community
τ₁	Portion of recruitment rate protected from COVID-19
τ₂	Portion of recruitment rate susceptible to both COVID-19 and syphilis
τ₃	Portion of recruitment rate protected from syphilis
β	COVID-19 protection loss rate
π	Syphilis protection loss rate
μ	Natural death rate of individuals
θ₁	Modification parameter
θ₂	Modification parameter
θ₃	Modification parameter
ρ	Treatment rate of COVID-19 infectious
ϵ	Progression rate of asymptomatic syphilis infectious to symptomatic syphilis infectious
ε	Treatment rate of COVID-19 and syphilis co-infections
γ	Treatment rate of symptomatic syphilis infectious
δ	Immunity lose rate against syphilis treatment
β₁	Syphilis transmission rate
β₂	COVID-19 transmission rate
ω₁	COVID-19 infection induced death rate
ω₂	Syphilis infection induced death rate
ω₃	COVID-19 and syphilis co-infection induced death rate
θ	Immunity lose rate against syphilis after treated from co-infection

Figure 1

Flowchart of the model (3) with forces of infections λ_C and λ_s as in (1) and (2), respectively.

Using the flowchart represented in Figure 1, the corresponding system of differential equations of the complete co-infection model (3) is written as follows:

(3)dPcdt=τ1Λ-(β+λs+ μ)Pc,dPsdt=τ3Λ-(π+λc+ μ)Ps,dSdt=τ2Λ+βPc+πPs+δTs+θT-(λs+λc+ μ)S,dIcdt=λcS+λcPs-(θ1λs+ρ+μ+ω1)Ic,dQcdt=ρIc-μQc,dIasdt=λsS+λsPc-(θ2λc+ϵ+ μ)Ias,dIssdt=ϵIas-(θ3λc+γ+μ+ω2)Iss,dIcsdt=θ2λcIas+θ3λcIss+θ1λsIc-(ε+μ+ ω3)Ics,dTsdt=γIss-(δ+ μ)Ts,dTdt=εIcs-(θ+ μ)T.

2.1. Qualitative properties of the model (3)

System (3) represents the human population; we want to prove that all the solutions of the model are non-negative and bounded, respectively, in the following region:

(4)Ω={(Pc, Ps, S, Ic, Qc, Ias, Iss, Ts, Tcs, T)∈ℝ+10,N≤Λμ}

Theorem 1: Let P_c(0) > 0, S(0) > 0, P_s(0), I_c(0) > 0, Q_c(0) > 0, I_as(0) > 0, I_ss(0) > 0, T_s(0) > 0, I_cs(0) > 0, T(0) > 0 be the initial solutions of the system (3), then P_c(t), P_s(t), S(t), I_c(t), Q_c(t), I_as(t), I_ss(t), T_s(t), T_cs(t), and T(t) are positive in the region ℝ+10 for any time t > 0.

Proof: Let τ = sup{t > 0:P_c(t) > 0, S(t) > 0, P_s(t), I_c(t) > 0, Q_c(t) > 0, I_as(t) > 0, I_ss(t) > 0, T_s(t) > 0, I_cs(t) > 0, T(t) > 0}.

Since P_c(t), P_s(t), S(t), I_c(t), Q_c(t), I_as(t), I_ss(t), T_s(t), T_cs(t), and T(t) are continuous, and we deduce that τ > 0. If τ = +∞, then positivity holds, but, if 0 < τ < +∞, P_c(τ) = 0 or P_s(τ) = 0 or S (τ) = 0 or I_c(τ) = 0 or Q_c(τ) = 0 or I_as(τ) = 0 I_ss(τ) = 0 or T_s(τ) = 0 or T_cs(τ) = 0 or T(τ) = 0.

From model (3) first equation, we do have

dPcdt+(β+λs+μ)Pc= τ1Λ.

After some calculations of integration, we got

Pc(τ)=a1Pc(0)+a1∫0τe∫(β+λs+μ)dtτ1Λdt>0, where a1=e-∫(β+λs+μ)dt>0, Pc(0)>0,Pc(τ)>0,so that Pc(τ)≠ 0.

From model (3) second equation, we have

dPsdt=τ3Λ-(π+λc+ μ)Ps.

After some calculations of integration, we have

Ps(τ)=b1Ps(0)+b1∫0τe∫(π+λc+μ)dtτ3Λdt>0, where b1=e-∫(π+λc+μ)dt>0, Ps(0)>0,Ps(τ)>0,so that Ps(τ)≠ 0.

From model (3) third equation, we have

dSdt=τ2Λ+βPc+πPs+δγTs-S(λs+λc+ μ).

After some calculations, we have

S(τ)=c1S(0)+c1∫0τe∫(λs+λc+μ)dt(τ2Λ+βPc+πPs+δγTs)dt>0, where

c1=e-∫(λs+λc+μ)dt>0, S(0)>0, and by the definition of τ we have P_c(t) > 0, P_s(t), T_s(t) > 0, S(τ) > 0, so that S(τ) ≠ 0.

Similarly, by proving the remaining state variable, we have

I_c(τ) > 0, hence I_c(τ) ≠ 0, Q_c(τ) > 0 hence Q_c(τ) ≠ 0, I_as(τ) > 0 hence I_as(τ) ≠ 0, I_ss(τ) > 0 hence I_ss(τ) ≠ 0,T_s(τ) > 0 hence T_s(τ) ≠ 0, T_cs(τ) > 0 hence T_cs(τ) ≠ 0, and T(τ) > 0 hence T(τ) ≠ 0.

Thus, τ is not finite, and hence = +∞, which means all the model solutions are non-negative.

Theorem 2: The model feasible region Ω stated in (4) is bounded in ℝ+10.

Proof: The total human being of the model (3) is as follows:

N(t)=Pc(t)+Ps(t)+S(t)+Ic(t)+Qc(t)+Ias(t) +Iss(t)+Ts(t)+Tcs(t)+T(t).

Differentiating both sides gives the following result

dNdt=dPcdt+dPsdt+dSdt+dIcdt+dQcdt+dIasdt+dIssdt+dTsdt +dTcsdt+ dTdt. =Λ-μN-ω1Ic-ω2Iss-ω3Ics,where τ1+τ3+τ2= 1. ⇒dNdt≤Λ-μN.

After some steps, we have 0≤N (t) ≤ Δμ, and hence, the model solutions with positive initial solutions are bounded in Ω.

3. Model analysis in qualitative approach

The complete COVID-19 and syphilis co-infection model (3) depends on the results of the two sub-models analysis.

3.1. COVID-19 mono-infection model analysis

From the complete model (3), we have the COVID-19 mono-infection model taking values P_s = I_as = I_ss = T_s = I_cs = T = 0 as follows:

(5)dPcdt=τ1Λ-(β+ μ)Pc,dSdt=τ2Λ+βPc-(λc+ μ)S,dIcdt=λcS-(ρ+μ+ω1)Ic, dQcdt=ρIc-μQc.

with N₁(t) = P_c(t) + S(t) + I_c(t) + Q_c(t) as a total population and λ_c = β₂I_c.

3.1.1. COVID-19 infection-free equilibrium

The COVID-19 infection-free equilibrium of the model (5) at I_c = 0 is Ec0=(Pc0,S00,0,0)=(τ1Λβ+μ, τ2Λ(β+μ)+βτ1Λμ(β+μ),0,0 ).

3.1.2. COVID-19 mono-infection reproduction number

This mono-infection model has one infectious class, I_c, and we can obtain basic reproduction numbers without a method of the next-generation matrix as follows:

dIcdt=λcS−(ρ+μ+ ω1)Ic, =β2IcS−(ρ+μ+ ω1)Ic, =(β2S−(ρ+μ+ ω1))Ic, =(β2Λ(τ2(β+μ)+βτ1)μ(β+μ)−(ρ+μ+ ω1))Ic, =(ρ+μ+ω1)(β2Λ(τ2(β+μ)+βτ1)μ(β+μ)(ρ+μ+ω1))− 1)Ic, =(ρ+μ+ω1)(ℜ0c−1)Ic, whereℜ0c=β2τ2Λ(β+μ)+β2βτ1Λμ(β+μ)(ρ+μ+ ω1).

3.1.3. COVID-19 incidence equilibrium point

The COVID-19 incidence equilibrium point of the system (5) is Ec*=(Pc*,S*, Ic*, Qc*), where

Pc*=τ1Λβ+μ, S*=τ3Λ(β+μ)+τ1βΛ,(λc*+μ)(β+μ),Ic*=(τ3Λ(β+μ)+τ1βΛ)λc*(λc*+μ)(β+μ)(ρ+μ+ω1),Qc*=(τ3Λ(β+μ)+τ1βΛ)ρλc*μ(λc*+μ)(β+μ)(ρ+μ+ ω1).

Theorem 3: The COVID-19 mono-infection model (5) has a unique COVID-19 incidence (endemic) equilibrium point whenever ℜ0c> 1.

Proof: Using equation (1), we have the following:

λc*=β1Ic*=β1(τ3Λ(β+μ)+τ1βΛ)λc*(λc*+μ)(β+μ)(ρ+μ+ ω1).

Then the non-zero value of λc* after a simple simplification is as follows:

λc*=μ(R0c-1)>0, if and only if R0c> 1.

Hence, the COVID-19 mono-infection model (5) has a unique incidence equilibrium point if and only if R0c> 1.

Theorem 4: COVID-19 infection-free equilibrium point of the model (5) is locally asymptotically stable if ℜ0c<1; otherwise, it is unstable.

Proof: The Jacobean matrix of the model (5) at the COVID-19 infection-free equilibrium point is

J(Ec0)= (-(β+μ)000β-μ-β2(τ2Λ(β+μ)+βτ1Λ)μ(β+μ)000β2(τ2Λ(β+μ)+βτ1Λ)-μ(β+μ)(ρ+μ+ω1)μ(β+μ)000ρ-μ).

Further, the characteristics equation after a certain calculation gives us as follows:

λ1=-(β+μ)<0 or λ2=-μ<0,orλ3=(ρ+μ+ω1)(R0c- 1).

Thus, each eigenvalue of the Jacobian matrix is negative if R0c<1 implies the COVID-19 infection-free equilibrium point is locally asymptotically stable whenever R0c< 1.

Theorem 5: The COVID-19 infection-free equilibrium point denoted by Ec* of the COVID-19 mono-infection model is globally stable if R0c<1; otherwise, it is unstable.

Proof: Take the representative Lyapunov function l(I_c ) = aI_c, where a=1(ρ+μ+ ω1)),

l(Ic)=aIc=1(ρ+μ+ ω1)Ic, dldt=1(ρ+μ+ω1)((β2S−(ρ+μ+ ω1))Ic), ≤1(ρ+μ+ω1) (β2Λ(τ2(β+μ)+βτ1)−μ(β+μ)(ρ+μ+ω1)μ(β+ μ))Ic, ≤(β2Λ(τ2(β+μ)+βτ1)−μ(β+μ)(ρ+μ+ω1)μ(β+μ)(ρ+μ+ ω1))Ic, ≤μ(β+μ)(ρ+μ+ω1)(β2Λ(τ2(β+μ)+βτ1)μ(β+μ)(ρ+μ+ω1)−1μ(β+μ)(ρ+μ+ ω1))Ic, ≤μ(β+μ)(ρ+μ+ω1)(R0c−1μ(β+μ)(ρ+μ+ ω1))Ic, ≤(R0c− 1)Ic.

Thus, dldt<0, if R0c<1, and the equality dldt=0 holds if R0c=1, and hence the COVID-19 infection-free equilibrium point is globally asymptotically stable if R0c< 1.

Theorem 6: The COVID-19 incidence denoted by EC* of the COVID-19 mono-infection model (5) is locally asymptotically stable whenever R0c>1; otherwise, it is unstable.

Proof: The Jacobean of the system (5) at EC*

J(EC∗)=(−(β+μ)000β−(β2Ic *+μ)−β2S∗00β2Ic*β2S*−(ρ+μ+ω1)000ρ−μ).

From the Jacobean matrix, the characteristics equation, after simplification, gives as follows:

(−(β+μ)−λ)(−μ−λ)[(−(β2Ic∗+μ)−λ)(β2S*−(ρ+μ+ ω1)−λ)+β2S∗β2Ic∗]= 0.

Then we do have the eigenvalues λ₁ = −μ < 0 or λ₂ = −(β + μ) < 0 or

a0λ2 + a1λ+a2=0 where a0= 1, a1= (β2(τ3Λ(β+μ)+τ1βΛ)μ(R0c−1)+μ(λc∗+μ)(β+μ)(ρ+μ+ω1)(λc∗+μ)(β+μ)(ρ+μ+ω1)) +(β2τ2Λ(β+μ)+τ1βΛ)[(λc∗+μ)μR0c−1](λc∗+μ)(β+ μ), a2=β2(τ3Λ(β+μ)+τ1βΛ)μ(R0c−1)(λc∗+μ)(β+μ)(ρ+μ+ω1) + (β2τ2Λ(β+μ)+τ1βΛ)[(λc∗+μ)μR0c−1](λc∗+μ)(β+μ) + β2(τ2Λ(β+μ)+τ1βΛ)μ(R0c−1)(λc∗+μ)(β+μ)(ρ+μ+ ω1).

Hence, all the coefficients of the characteristics equations are positive when R0c>1; thus, the COVID-19 incidence equilibrium point has local asymptotic stability when R0c> 1.

3.2. Analysis of syphilis sub-model

The syphilis sub-model is obtained from the system (3) by making P_c = I_c = Q_c = I_cs = T = 0 and is as follows:

(6)dPsdt=τ3Λ-(π+ μ)Ps,dSdt=τ2Λ+πPs+δγTs-S(λs+ μ),dIasdt=λsS-Ias(ϵ+μ),dIssdt=ϵIas-Iss(γ+μ+ω2 ),dTsdt=γIss-Ts(δγ+ μ).

With N₂(t) = P_s(t) + S(t) + I_as(t) + I_ss(t) + T_s(t), and λ_s = β₁(I_as + ϕ₂I_ss ).

3.2.1. Syphilis infection-free equilibrium

The syphilis infection-free equilibrium point of the model (6) was obtained by making I_as = I_ss = 0 and is given by Es0=(Ps0, S0, Ias0, Iss0, Ts0)=(τ3Λπ+μ, τ2Λ(π+μ)+τ3πΛμ(π+μ), 0, 0, 0 ).

3.2.2. Syphilis sub-model reproduction number

The syphilis sub-model (6) has two infectious classes, which are I_as, and I_ss, then applying the next-generation matrix method stated in [43, 44] to obtain the basic reproduction number of the system (6) by computing FV⁻¹ as follows:

F=(β1S0β1ϕ2S000),⇒F=(β1Λ(τ2(π+μ)+τ3π)μ(π+μ)β1ϕ2Λ(τ2(π+μ)+τ3π)μ(π+μ)00),

and

V=(ϵ+μ0-ϵγ+μ+ω2).

Then, we applied Mathematica coding; we have

V−1=[1ϵ+μ0ϵ(γ+μ+ω2)(ϵ+μ)1(γ+μ+ω2)], andFV−1=[β1Λ(τ2(π+μ)+τ3π)μ(π+μ)(ϵ+μ) +ϵβ1ϕ2Λ(τ2(π+μ)+τ3π)μ(π+μ)(γ+μ+ω2)(ϵ+μ)β1ϕ2Λ(τ2(π+μ)+τ3π)μ(π+μ)(γ+μ+ω2)00].

Thus, the reproduction number of the syphilis sub-model (6) is given by R0s=β1Λ[(γ+μ+ω2)+ϕ2ϵ](τ2(π+μ)+τ3π)μ(π+μ)(γ+μ+ω2)(ϵ+μ ).

3.2.3. Syphilis incidence equilibrium point of the system (6)

Making the model (6) equation to zero, we have the syphilis incidence equilibrium point given by Es*=(Ps*, S*, Ias*, Iss*, Ts*), where

Ps*=τ3Λ(π+μ),S*=(γ+μ+ω2)(ϵ+μ)(δγ+μ)(τ2Λ(π+μ)+τ3Λπ)(π+μ)((γ+μ+ω2)(ϵ+μ)(λs+μ)(δγ+μ)- γδγϵλs),Ias*=λs(γ+μ+ω2)(δγ+μ)(τ2Λ(π+μ)+τ3Λπ)(π+μ)((γ+μ+ω2)(ϵ+μ)(λs+μ)(δγ+μ)- γδγϵλs),Iss*=ϵλs(δγ+μ)(τ2Λ(π+μ)+τ3Λπ)(π+μ)((γ+μ+ω2)(ϵ+μ)(λs+μ)(δγ+μ)- γδγϵλs),Ts*=ϵλsγ(τ2Λ(π+μ)+τ3Λπ)(π+μ)((γ+μ+ω2)(ϵ+μ)(λs+μ)(δγ+μ)- γδγϵλs).

Theorem 7: The syphilis incidence equilibrium point of syphilis in the model (6) is unique whenever Rs0> 1.

Proof: From the syphilis infection rate, we have

λs∗=β1(Ias∗+ ϕ2Iss∗).λs∗=β1(λs(γ+μ+ω2)(δγ+μ)(τ2Λ(π+μ)+τ3Λπ)(π+μ)((γ+μ+ω2)(ϵ+μ)(λs∗+μ)(δγ+μ)−ϵγδγλs∗) +ϵλs∗ϕ2(δγ+μ)(τ2Λ(π+μ)+τ3Λπ)(π+μ)((γ+μ+ω2)(ϵ+μ)(λs∗+μ)(δγ+μ)− γδγϵλs∗)). ⇒λs∗=(δγ+μ) β1Λ[(γ+μ+ω2)(τ2(π+μ)+τ3π)+ϵϕ2(τ2(π+μ)+τ3π)]−μ(π+μ)[(γ+μ+ω2)(ϵ+μ)](π+μ)((γ+μ+ω2)(ϵ+μ)(δγ+μ)−γδγϵ ). ⇒λs∗=(δγ+μ)μ(γ+μ+ω2)(ϵ+μ)(R0s−1)((γ+μ+ω2)(ϵ+μ)(δγ+μ)−γδγϵ ). ⇒λs∗=k1(R0s−1),wherek1=(δγ+μ)μ(π+μ)(γ+μ+ω2)(ϵ+μ)(π+μ)((γ+μ+ω2)(ϵ+μ)(δγ+μ)−γδγϵ ).

Hence, the syphilis sub-model (6) has a unique incidence equilibrium if R0s> 1.

Theorem 8: Syphilis infection-free equilibrium point of the model (3) has local asymptotic stability if R0s<1; otherwise, it is unstable.

Proof: Jacobean of the model (6) at the syphilis infection-free equilibrium point is as follows:

J( E0s)= (-(π+μ)0000π-μ-β1Λ(τ2(π+μ)+τ3π)μ(π+μ)-β1ϕ2Λ(τ2(π+μ)+τ3π)μ(π+μ)δγ00β1Λ(τ2(π+μ)+τ3π)-μ(π+μ)(ϵ+μ)μ(π+μ)β1ϕ2Λ(τ2(π+μ)+τ3π)μ(π+μ)000ϵ-(γ+μ+ω2)0000γ-(δγ+μ)).

From the Jacobian matrix, the characteristics equation after simplification is as follows:

(−(π+μ)λ)(−μ−λ)(−(δγ+μ)λ)[(β1Λ(τ2(π+μ)+τ3π)−μ(π+μ)(ϵ+μ)μ(π+μ)λ)(−(γ+μ+ω2)λ)−(β1ϕ2Λϵ(τ2(π+μ)+τ3π)μ(π+μ))]= 0.

Then the eigenvalues are λ₁ = −(π + μ) < 0 or λ₂ = −μ < 0 or λ₃ = −(δγ + μ) < 0 or a0λ2+a1λ+a2= 0.

where,

a0= 1,a1=(γ+μ+ω2)+(ϵ+μ)+β1Λ(τ2(π+μ)+τ3π)μ(π+μ ),a2=(ϵ+μ)(γ+μ+ω2)(1- R0s).

Applying Routh–Hurwitz criteria stated in [33], each eigenvalue of the matrix is negative whenever R0s<1; thus, the syphilis infection-free equilibrium point has local asymptotic stability if R0s< 1.

Theorem 9: Syphilis infection-free equilibrium point Es0 of the model (6) has global stability if R0s<1; otherwise, it is unstable.

Proof: Let the Lyapunov representative function be given as l(I_as, I_ss ) = aI_as + bI_ss, where =[(γ+μ)+ϕ2ϵ](ϵ+μ)(γ+μ+ω2), b=ϕ2(γ+μ+ ω2).

⇒l(Ias, Iss )=[(γ+μ+ω2)+ϕ2ϵ](ϵ+μ)(γ+μ+ω2)Ias+ϕ2(γ+μ+ ω2)Iss.⇒dldt=[(γ+μ+ω2)+ϕ2ϵ](ϵ+μ)(γ+μ+ω2)(λsS-Ias(ϵ+μ))+ϕ2(γ+μ)(ϵIas-Iss(γ+μ+ ω2)),≤[(γ+μ+ω2)+ϕ2ϵ](ϵ+μ)(γ+μ+ω2)(β1(Ias+ϕ2Iss)S*-Ias(ϵ+μ))+ϕ2(γ+μ)(ϵIas-Iss(γ+μ+ ω2)),≤[(γ+μ+ω2)+ϕ2ϵ](ϵ+μ)(γ+μ+ω2)(β1IasS*+β1ϕ2IssS*-Ias(ϵ+μ))+ϕ2(γ+μ)(ϵIas-Iss(γ+μ+ ω2)),≤([β1S*-(ϵ+μ)][(γ+μ+ω2)+ϕ2ϵ](ϵ+μ)(γ+μ+ω2)+ϕ2ϵ(γ+μ+ω2))Ias+(β1ϕ2[(γ+μ+ω2)+ϕ2ϵ]S*(ϵ+μ)(γ+μ+ω2)-ϕ2)Iss,≤([[(γ+μ)+ϕ2ϵ]β1S*(ϵ+μ)(γ+μ+ω2)-[(γ+μ+ω2)+ϕ2ϵ](γ+μ+ω2)])Ias+(β1ϕ2[(γ+μ+ω2)+ϕ2ϵ]S*(ϵ+μ)(γ+μ+ω2)-ϕ2)Iss,≤[(γ+μ+ω2)+ϕ2ϵ]β1Λ(τ2(π+μ)+τ3π)μ(π+μ)(ϵ+μ)(γ+μ+ω2)Ias-[(γ+μ+ω2)+ϕ2ϵ](γ+μ+ ω2)Ias+(β1ϕ2[(γ+μ+ω2)+ϕ2ϵ]Λ(τ2(π+μ)+τ3π)μ(π+μ)(ϵ+μ)(γ+μ+ω2)-ϕ2)Iss.≤([[(γ+μ+ω2)+ϕ2ϵ]β1Λ(τ2(π+μ)+τ3π)μ(π+μ)(ϵ+μ)(γ+μ+ω2)-m])Ias+(β1ϕ2[(γ+μ+ω2)+ϕ2ϵ]Λ(τ2(π+μ)+τ3π)μ(π+μ)(ϵ+μ)(γ+μ+ω2)-ϕ2),wherem=[(γ+μ+ω2)+ϕ2ϵ](γ+μ+ω2)> 1.≤m([R0sm-1])Ias+ϕ2(R0sϕ2-1) Iss,R0sm<1,Rs0ϕ2< 1,R0sm<1,R0sϕ2<1 implies R0s<1 since m=[(γ+μ+ω2)+ϕ2ϵ](γ+μ+ω2)>1, and ϕ2> 1.

Hence, the syphilis-free equilibrium point is globally stable if R0s< 1.

3.3. COVID-19 and syphilis co-infection model analysis 3.3.1. The model (3) disease-free equilibrium

Making all the equations of (3) zero with I_c = I_as = I_ss = I_cs = 0, the disease-free equilibrium point of (3) is given by E0=(Pc0,Ps0,S0,Ic0,Qc0,Ias0,Iss0,Ts0,Tcs0,T0)= (τ1Λ(β+μ),τ3Λ(π+μ),Λ(τ2(β+μ)(π+μ)+τ1β(π+μ)+τ3π(β+μ))μ(β+μ)(π+μ),0,0,0,0,0,0,0 ).

3.3.2. The model (3) reproduction number

The COVID-19 and syphilis co-infection model (3) reproduction number denoted by ℜ0cs is calculated using next-generation matrix criteria, as stated in [44]. Since we have four infectious groups, those are I_c, I_as, I_ss, and I_cs, and we have

fi=[β2(Ic +ϕ1Ics) (S+Ps)β1(Ias+ϕ2Iss+ϕ3Ics)(S+Pc)0β2(θ2Ias+θ3Iss)(Ic +ϕ1Ics) +θ1β1(Ias+ϕ2Iss+ϕ3Ics)Ic].⇒f=[β2(S0+Ps0)00ϕ1(S0+Ps0)0β1(S0+Pc0)β1ϕ2(S0+Pc0)β1ϕ3(S0+Pc0)00000000],

and

vi=vi-(x)-vi+(x)=[Ic(θ1β1(Ias+ϕ2Iss+ϕ3Ics)+ρ+μ+ω1)Ias(θ2β2(Ic +ϕ1Ics) +ϵ+μ)Iss(θ3β2(Ic +ϕ1Ics) +γ+μ+ω2)-ϵIasIcs(ε+μ+ω3) ].⇒v=[(ρ+μ+ω1)0000(ϵ+μ)000-ϵ(γ+μ+ω2)0000(ε+μ+ω3)].

Then applying Mathematica, we have got

v-1=[1(ρ+μ+ω1)00001(ε+μ)00001(γ+μ+ω2)00001(ε+μ+ω3)],

and

fv-1=[β2(S0+P0s)(ρ+μ+ω1)00ϕ1(S0+P0s)(ε+μ+ω3)0β1(S0+P0c)(ε+μ)β1ϕ2(S0+P0c)(γ+μ+ω2)β1ϕ3(S0+P0c)(ε+μ+ω3)00000000].⇒|β2(S0+P0s)(ρ+μ+ω1)-λ00ϕ1(S0+P0s)(ε+μ+ω3)0β1(S0+P0c)(ε+μ)-λβ1ϕ2(S0+P0c)(γ+μ+ω2)β1ϕ3(S0+P0c)(ε+μ+ω3)000-λ00000-λ|= 0.

Then, the corresponding eigenvalues are λ₁ = 0 or λ₂ = 0 or λ3=β2(S0+P0s)(ρ+μ+ω1)=β2Λτ2μ(ρ+μ+ω1)+β2Λτ1βμ(β+μ)(ρ+μ+ω1)+β2Λτ3μ(ρ+μ+ω1)=ℜ0c+n or λ4=β1(τ2Λ(β+μ)(π+μ)+τ1βΛ(π+μ)+τ3πΛ(β+μ)+τ1Λμ(π+μ)μ(β+μ)(π+μ)(ε+μ)) =ℜ0s-m where , n=β2Λτ3μ(ρ+μ+ω1), m=β1Λ[ϵϕ2(τ2(π+μ)+τ3π)-τ1((π+μ)γ+μ+ω2)]μ(π+μ)(γ+μ+ω2)(ϵ+μ ).

Therefore, the COVID-19–syphilis complete model (3) reproduction number denoted by ℜ0cs is given by ℜ0cs=max{ℜ0c+n ,ℜ0s-m }.

3.3.3. Model (3) disease-free equilibrium local stability

Theorem 10: The full-model (3) disease-free equilibrium point has local asymptotic stability if ℜ0cs<1; otherwise, it is unstable.

Proof: The Jacobian of the COVID-19 and syphilis co-infection model (3) at E⁰ is as follows:

J(E0)= (a0000bc0d00e0f0000g0βπhi0jklm0000n0000o0000ρh0000000000pq0r000000ϵs000000000γt0000000000u000000000εh),

where a =-(β+μ), b=-β1Pc0, c= -β1ϕ2Pc0, d=-β1ϕ3Pc0, e=-(π+μ), f=-β2Ps0, g=-β2ϕ1Ps0, h=-μ, i=-β2 S0, j=-β1 S0, k=-β1ϕ2 S0, l=δγ, m=-(β1ϕ3+β2ϕ1) S0, n=β2(S+Ps)-(ρ+μ+ω1), o=β2ϕ1(S+Ps), p=β1(S+Pc)-(ϵ+μ), q=β1ϕ2(S+Pc), r=β1ϕ3( S0+Pc0), s=-(γ+μ+ω2), t= -(δγ+μ), u=-(ε+μ+ω3 ).

⇒|a-λ0000bc0d00e-λ0f0000g0βπh-λi0jklm0000n-λ0000o0000ρh-λ0000000000p-λq0r000000ϵs-λ000000000γt-λ0000000000u-λ000000000εh-λ|= 0.

By using square block matrix properties, we rewrite the above determinant as follows:

|AB0C|=0, whereA=[a-λ00000e-λ0f0βπh-λi0000n-λ0000ρh-λ],B=[bc0d0000g0jklm00000000000],C=[p-λq0r0ϵs-λ0000γt-λ00000u-λ0000εh-λ],O=[0000000000000000000000000].|a-λ0000bc0d00e-λ0f0000g0βπh-λi0jklm0000n-λ0000o0000ρh-λ0000000000p-λq0r000000ϵs-λ000000000γt-λ0000000000u-λ000000000εh-λ|=|A||C|= 0.

From this, we do have

|A|=(a-λ)(e-λ)(h-λ)(n-λ)(h-λ),|C|=(t-λ)(u-λ)(h-λ)((p-λ)(s-λ)-qϵ),|A||C|=[(a-λ)(e-λ)(h-λ)(n-λ)(h-λ)] [(t-λ)(u-λ)(h-λ)((p-λ)(s-λ)-qϵ)]= 0.

Then, the eigenvalue of the full model is as follows:

λ1=a or λ2=e or λ3=h or λ4=n or λ5=h or λ6=t or λ7=u or λ8=h or a0λ2+a1λ+a2=0, where,a0= 1,a1=(ρ+μ+ω1)(ϵ+μ)(1- ℜ0cs)> 0,a2=(ϵ+μ)ϕ2ϵ[1-ℜ0cs]>0,ifℜ0cs< 1.

Therefore, the co-infection model disease-free equilibrium point has local asymptotic stability if ℜ0cs< 1.

3.3.4. The full-model endemic equilibrium and stabilities

The COVID-19 and syphilis co-infection model endemic equilibrium point is denoted by

Ecs*=(Pc*,Ps*, S*,Ic*,Qc*,Ias*,Iss*,Ics*,, Ts*,T*). The analysis of the COVID-19-only mono-infection system (5) and the syphilis-only sub-model (6) shows that there is no endemic equilibrium point whenever ℜ0c<1 and ℜ0s<1, respectively, which means there is no endemic equilibrium point if ℜ0cs<1 for the co-infection model (3). In other words, the COVID-19 and syphilis co-infection disease-free equilibrium point have global stability if ℜ0cs< 1.

The explicit calculation of the co-infection model endemic equilibrium in terms of model parameters is tedious analytically; however, the model (3) endemic equilibriums correspond to

E1*=(Pc*,0, S*,Ic*,Qc*,0,0,0,, 0,0), if ℜ0c>1 is the syphilis-free (COVID-19 persistence) equilibrium point.

The analysis of the equilibrium E1* is similar to the endemic equilibrium Ec* in the model (5).

E2*=(0,Ps*, S*,0,0,Ias*,Iss*,0,, Ts*,0), if ℜ0s>1 is the COVID-19-free (syphilis persistence) equilibrium point. The analysis of the equilibrium E2* is similar to the endemic equilibrium Es* in Equation (6).

Ecs*=(Pc*,Ps*, S*,Ic*,Qc*,Ias*,Iss*,Ics*,, Ts*,T*) is the COVID-19 and syphilis co-existence persistence equilibrium point. It exists when each component of Ecs* is positive whenever ℜ0cs>1 for this case, we have shown its stability in the numerical simulation part given in Section 4.

4. Sensitivity analysis and numerical simulations

In this section, we carried out the sensitivity analysis to examine the most sensitive parameters in the disease spreading and numerical simulations to verify the qualitative results of the mathematical model (3). Particularly, some numerical verification is considered to illustrate the qualitative analysis and results of the preceding sections. Here, we have taken some parameter values from literature and assumed some of the parameter values that are not from real data since there is a lack of mathematical modeling analysis literature which have studied the COVID-19 and syphilis co-infection transmission dynamics in the community. The fundamental problem of numerical analysis of a mathematical model is how to estimate parameters. Randomly choosing the values of parameters in the model in plausible intervals followed by sensitivity to the parameters is possible partially to overcome the limitations of parameters [41].

Here, the numerical simulation is used for checking the behaviors of the full-model (3) solutions and the effects of parameters in the transmission as well as the controlling of COVID-19 infection, syphilis infection, and co-infection of COVID-19 and syphilis. For numerical simulation purposes, we have applied MATLAB ode45 code with parameter values given in Table 3.

Table 3

Parameter values for numerical simulations.

Parameter	Value	References
μ	0.0000559 year⁻¹	[17]
Λ	500 day⁻¹	[45]
β	0.30 day⁻¹	Assumed
π	0.21 day⁻¹	Assumed
ρ	0.5 day⁻¹	[45]
ϵ	0.40 year⁻¹	[12]
ε	0.3 year⁻¹	Assumed
γ	0.021 year⁻¹	[38]
δ	0.2482 year⁻¹	[34]
β₁	8 year⁻¹	[38]
β₂	0.6 day⁻¹	[45]
θ₁	1.1 dimensionless	Assumed
θ₂	1.1 dimensionless	Assumed
θ₃	1.1 dimensionless	Assumed
τ₁	0.27 dimensionless	Assumed
τ₂	0.41 dimensionless	Assumed
τ₃	0.32 dimensionless	Assumed
ω₁	0.023 day⁻¹	[45]
ω₂	0.06849 year⁻¹	[17]
ω₃	0.07 year⁻¹	Assumed

4.1. Analysis of sensitivity

Definition: The syphilis and COVID-19 co-infection model (3) normalized forward sensitivity index for its variable reproduction number is denoted by R0cs its derivative depends on a parameter ζ is defined by SEI(p)=∂R0cs∂ζ*ζR0cs [20, 21, 42].

The syphilis and COVID-19 co-infection model sensitivity index values justify the significance of different parameters in the single infections and co-infection spreading in the community. The parameter which has the highest magnitude of the sensitivity index value compared to other parameter index values is the most sensitive. Here, we have calculated the sensitivity index values in terms of the basic reproduction number R0cs=max{R0s,R0c}. Using parameter values stated in Table 3, the sensitivity index values of the model (3) are calculated in Tables 4, 5.

Table 4

Sensitivity indexes of R0cs=R0c.

Sensitivity index	Values
SEI(β₂)	1
SEI(Λ)	1
SEI(τ₂)	0.50
SEI(β)	0.09
SEI(τ₁)	–0.56
SEI(μ)	–0.13
SEI(ρ)	–0.65
SEI(ω₁)	–0.07

Table 5

Sensitivity indexes of R0cs=R0s.

Sensitivity index	Values
SEI(β₁)	1
SEI(Λ)	1
SEI(γ)	–0.65
SEI(ω₂)	–0.32
SEI(π)	0.12
SEI(τ₂)	0.56
SEI(τ₃)	–0.64
SEI(ϕ₂)	0.41
SEI(ϵ)	0.46

Using parameter values in Table 3, we have computed ℜ0cs=max{ℜ0c,ℜ0s}=max{2.7, 3.2}=3.2 and biologically, it means that syphilis infection, COVID-19 infection, and their co-infection are spreading in the population. The sensitivity index values stated in Table 4 explain that the recruitment rate Λ and the COVID-19 spreading rate β₂ have a high direct impact on the COVID-19 basic reproduction ℜ0c. That means the recruitment rate and the COVID-19 transmission rates are the most sensitive parameters where stakeholders can control the transmission rate by applying prevention and control measures. Similarly, the COVID-19 protection portion τ₁ and the quarantine with treatment rate ρ also have an indirect impact on the COVID-19 reproduction number ℜ0c.

Sensitivity indices stated in Table 5 explain that the recruitment rate Λ and syphilis spreading rate β₁ have a high direct impact on the syphilis basic reproduction ℜ0s. That means the recruitment rate and syphilis transmission rates are the most sensitive parameters where stakeholders can control the transmission rate by applying prevention and control measures. Similarly, the syphilis protection portion τ₃ and syphilis treatment rate γ have a high indirect effect on the syphilis reproduction number ℜ0s.

4.2. Results of numerical simulations 4.2.1. Behaviors of solutions of model (3) whenever <inline-formula><mml:math id="M129"><mml:msubsup><mml:mrow><mml:mi>ℜ</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="bold"><mml:mn>0</mml:mn></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>c</mml:mtext></mml:mstyle><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow></mml:msubsup><mml:mstyle mathvariant="bold"><mml:mo><</mml:mo><mml:mn>1</mml:mn></mml:mstyle></mml:math></inline-formula>

In the numerical simulation given in Figure 2, we observed that all the COVID-19 and syphilis co-infection model (3) solutions converge toward the disease-free equilibrium point whenever ℜ0c=0.71 and ℜ0s=0.34 with β₁ = 0.3 and β₂ = 0.08, respectively. At the co-infection disease-free equilibrium point, each solution curve of the model converges to zero while the susceptible group increases and then becomes constant, implying that the disease-free equilibrium point of the COVID-19 and syphilis co-infection model has global asymptotic stability if ℜ0cs<1. Biologically it means the COVID-19 and syphilis co-infection diseases have been eradicated from the community through time whenever ℜ0cs=max{ℜ0c, ℜ0s}=0.71< 1.

Figure 2

The complete model solutions behavior if ℜ0cs<1 at β₁ = 0.3 and β₂ = 0.08.

4.2.2. Behaviors of the model solutions whenever <inline-formula><mml:math id="M135"><mml:msubsup><mml:mrow><mml:mi>ℜ</mml:mi></mml:mrow><mml:mrow><mml:mstyle mathvariant="bold"><mml:mn>0</mml:mn></mml:mstyle></mml:mrow><mml:mrow><mml:mstyle mathvariant="bold"><mml:mtext>c</mml:mtext></mml:mstyle><mml:mstyle mathvariant="bold"><mml:mtext>s</mml:mtext></mml:mstyle></mml:mrow></mml:msubsup><mml:mstyle mathvariant="bold"><mml:mo>></mml:mo><mml:mn>1</mml:mn></mml:mstyle></mml:math></inline-formula>

Figure 3 shows that all the COVID-19 and syphilis co-infection model (3) solutions converge toward the endemic equilibrium point whenever ℜ0c=3.2 and ℜ0s=2.1 with β₁ = 8 and β₂ = 11, respectively. After 10 years, the full-model solutions converge to the endemic equilibrium, while the susceptible population decreases and then remains constant means the COVID-19 and syphilis co-infection model endemic equilibrium point has local asymptotic stability if ℜ0cs=max{ℜ0c,ℜ0s}=3.2>1. Biologically, it means that COVID-19 and syphilis co-infection disease spreads throughout the community under consideration.

Figure 3

Behaviors of the full model solutions whenever ℜ0cs>1 at β₁ = 8 and β₂ = 11.

4.2.3. Effects of protection measures on reproduction numbers

The numerical simulation represented by Figure 4 shows that when we maximize the COVID-19 rate of protection τ₁, the reproduction number ℜ0c decreases, implying that the COVID-19 spreading rate decreases. Its biological meaning is that whenever the COVID-19 rate of protection τ₁ > 0.7 the reproduction number ℜ0c<1, that is, the COVID-19 infection will be eradicated throughout the community.

Figure 4

Effect of COVID-19 protection rate on ℜ0c.

Here, the numerical simulation represented by Figure 5 shows that whenever we maximize the syphilis protection rate τ₃, the syphilis reproduction number ℜ0s decreases, implying that the syphilis spreading rate decreases. Whenever τ₃ > 0.686 then ℜ0s<1, biologically, it means the syphilis infection eradicate from the community.

Figure 5

Effect of syphilis protection rate τ₃ on ℜ0s.

4.2.4. Impact of treatment on co-infected population

The numerical simulation given in Figure 6 shows that whenever the combined treatment rate ε of the COVID-19 virus and syphilis microorganism Treponema pallidum bacterium co-infected individuals I_cs increases, the number of co-infected individuals decreases; that is, whenever the value of ε increases from 0.3 to 0.8, then the co-infected group I_cs going down.

Figure 6

Impact of treatment rate ε on I_cs.

The numerical simulation given in Figure 7 shows that if the treatment rate ρ of COVID-19 increases, then the number of infections in the population decreases; that is, whenever ρ value increases from 0.2 to 0.8 then the infected group I_c decreases.

Figure 7

Impact of treatment rate ρ on I_c.

5. Discussions and conclusion

In this study, we have formulated and analyzed a new deterministic mathematical model for gaining insight into the effects of protections and treatments on the transmission dynamics of COVID-19 and syphilis co-infection. Both the positivity and boundedness of the complete model solutions have been discussed to show that the model is both mathematically and biologically meaningful. COVID-19 infection-free equilibrium point, COVID-19 incidence equilibrium point, and local and global stabilities of COVID-19 infection-free and COVID-19 incidence equilibrium points have been examined. Syphilis infection-free equilibrium point, syphilis incidence equilibrium point, and local and global stabilities of syphilis-free and syphilis incidence equilibrium points have been carried out. Using data stated in Table 3, we have carried out and discussed both sensitivity and numerical analyses of the full COVID-19 and syphilis co-infection model. From the analytical and numerical results, we observed that the model disease-free equilibrium points have global asymptotic stability when the basic reproduction numbers are less than unity. Biologically, this means that diseases die out in the community, with the full-model solutions converging to their endemic equilibrium point whenever their basic reproduction number is greater than unity, the reproduction numbers of both the COVID-19 infection and syphilis infection sub-models decreasing when the corresponding protection and treatment rates are maximized, and the numbers of co-infected individuals decreasing when the co-infection treatment rate is increased.

Based on the findings of this study, we recommend public health stakeholders concentrate on increasing both the COVID-19 and syphilis protection rates, as well as the syphilis treatment rate, the COVID-19 isolation with treatment rate, and the co-infection treatment rate, in order to reduce and possibly eradicate syphilis and COVID-19 co-infection transmission in the community. Finally, since no other COVID-19 and syphilis mathematical modeling approach literature has been formulated and analyzed, this study is not exhaustive. Interested researchers can extend this study in different manners, such as including syphilis mother-to-child transmission, COVID-19 vaccination as a new compartment, two-strain COVID-19 co-infection with syphilis, age structure for both infections, the four infection stages of syphilis (primary, secondary, latent, and tertiary), optimal control approach, stochastic method, fractional order method, and applying appropriate real population data.

Data availability statement

All relevant data is contained within the article: The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.

Ethics statement

Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements. The patients/participants provided their written informed consent to participate in this study.

Author contributions

Both authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

The authors thank Mr. Sitotaw Eshete for his great Wi-Fi contribution that we have used in the study.

Conflict of interest

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.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

References 1. Panovska-Griffiths

. Can mathematical modelling solve the current COVID-19 crisis? BMC Public Health. (2020) 20:1–3. 10.1186/s12889-020-08671-z

32331516

2. Adiga

Dubhashi

Lewis

Marathe

Venkatramanan

Vullikanti

. Mathematical models for COVID-19 pandemic: a comparative analysis. J Indian Inst Sci. (2020) 100:793–807. 10.1007/s41745-020-00200-6

33144763

3. Musuuza

Watson

Parmasad

Putman-Buehler

Christensen

Safdar

. Prevalence and outcomes of co-infection and superinfection with SARS-CoV-2 and other pathogens: a systematic review and meta-analysis. PLoS ONE. (2021) 16:e0251170. 10.1371/journal.pone.0251170

33956882

4. Kumar

Erturk

Murillo-Arcila

. A new fractional mathematical modelling of COVID-19 with the availability of vaccine. Results Phys. (2021) 24:104213. 10.1016/j.rinp.2021.104213

33898210

5. Omame

Sene

Nometa

Nwakanma

Nwafor

Iheonu

. Analysis of COVID-19 and comorbidity co-infection model with optimal control. Optimal Control Appl Methods. (2021) 42:1568–90. 10.1002/oca.2748

34226774

6. Tuite

Fisman

Greer

. Mathematical modelling of COVID-19 transmission and mitigation strategies in the population of Ontario, Canada. CMAJ. (2020) 192:E497–505. 10.1503/cmaj.200476

32269018

7. Boukanjime

Caraballo

El Fatini

El Khalifi

. Dynamics of a stochastic coronavirus (COVID-19) epidemic model with Markovian switching. Chaos Solitons Fractals. (2020) 141:110361. 10.1016/j.chaos.2020.110361

33100608

8. Din

Khan

Baleanu

. Stationary distribution and extinction of stochastic coronavirus (COVID-19) epidemic model. Chaos Solitons Fractals. (2020) 139:110036. 10.1016/j.chaos.2020.110036

32834596

9. Abdela

Berhanu

Ferede

van Griensven

. Essential healthcare services in the face of COVID-19 prevention: experiences from a referral hospital in Ethiopia. Am J Trop Med Hyg. (2020) 103:1198. 10.4269/ajtmh.20-0464

32762799

10. Gumel

Lubuma

Sharomi

Terefe

. Mathematics of a sex-structured model for syphilis transmission dynamics. Math Methods Appl Sci. (2018) 41:8488–513. 10.1002/mma.4734

34131428

11. Momoh

Bala

Washachi

Déthié

. Mathematical analysis and optimal control interventions for sex structured syphilis model with three stages of infection and loss of immunity. Adv Diff Equ. (2021) 2021:1–26. 10.1186/s13662-021-03432-7

34131428

12. Peeling

Mabey

Kamb

Chen

Radolf

. “Benzaken.” AS Syphilis. Nat Rev Dis Prim. (2017) 3:17073. 10.1038/nrdp.2017.73

29022569

13. Peeling

Hook

III. The pathogenesis of syphilis: the Great Mimicker, revisited. J Pathol. (2006) 208:224–32. 10.1002/path.1903

16362988

14. Saad-Roy

Shuai

Van den Driessche

. A mathematical model of syphilis transmission in an MSM population. Math Biosci. (2016) 277:59–70. 10.1016/j.mbs.2016.03.017

27071977

15. Tessema

Yismaw

Kassu

Amsalu

Mulu

Emmrich

. Seroprevalence of HIV, HBV, HCV and syphilis infections among blood donors at Gondar University Teaching Hospital, Northwest Ethiopia: declining trends over a period of five years. BMC Infect Dis. (2010) 10:1–7. 10.1186/1471-2334-10-111

20459703

16. World Health Organization. WHO Guidelines for the Treatment of Treponema pallidum (Syphilis). Geneva: WHO (2016). 17. Iboi

Okuonghae

. Population dynamics of a mathematical model for syphilis. Appl Math Model. (2016) 40:3573–90. 10.1016/j.apm.2015.09.090 18. Milner

Zhao

. A new mathematical model of syphilis. Math Model Nat Phenom. (2010) 5:96–108. 10.1051/mmnp/20105605 19. Tuite

Fisman

Mishra

. Screen more or screen more often? Using mathematical models to inform syphilis control strategies. BMC Public Health. (2013) 13:1–9. 10.1186/1471-2458-13-606

23800206

20. Teklu

Rao

. HIV/AIDS-Pneumonia codynamics model analysis with vaccination and treatment. Comput Math Methods Med. (2022) 2022:3105734. 10.1155/2022/3105734

35069778

21. Teklu

Mekonnen

. HIV/AIDS-pneumonia coinfection model with treatment at each infection stage: mathematical analysis and numerical simulation. J Appl Math. (2021) 2021:5444605. 10.1155/2021/5444605

35069778

22. Danane

Allali

Hammouch

Nisar

. Mathematical analysis and simulation of a stochastic COVID-19 Lévy jump model with isolation strategy. Results Phys. (2021) 23:103994. 10.1016/j.rinp.2021.103994

33686366

23. Bonyah

Chukwu

Juga

. Modeling fractional order dynamics of syphilis via Mittag-Leffler law. medRxiv. (2021). 10.3934/math.2021485 24. Nabi

Abboubakar

Kumar

. Forecasting of COVID-19 pandemic: from integer derivatives to fractional derivatives. Chaos Solitons Fractals. (2020) 141:110283. 10.1016/j.chaos.2020.110283

32982078

25. Nabi

Kumar

Erturk

. Projections and fractional dynamics of COVID-19 with optimal control strategies. Chaos Solitons Fractals. (2021) 145:110689. 10.1016/j.chaos.2021.110689

33531738

26. Kumar

Erturk

Murillo-Arcila

Banerjee

Manickam

. A case study of 2019-nCOV cases in Argentina with the real data based on daily cases from March 03, 2020 to March 29, 2021 using classical and fractional derivatives. Adv Diff Equ. (2021) 2021:1–21. 10.1186/s13662-021-03499-2

34306044

27. Etemad

Avci

Kumar

Baleanu

Rezapour

. Some novel mathematical analysis on the fractal-fractional model of the AH1N1/09 virus and its generalized Caputo-type version. Chaos Solitons Fractals. (2022) 162:112511. 10.1016/j.chaos.2022.112511 28. Zarin

Khan

Kumar

. Fractional-order dynamics of Chagas-HIV epidemic model with different fractional operators. AIMS Math. (2022) 7:18897–924. 10.3934/math.20221041 29. Gao

Veeresha

Baskonus

Prakasha

Kumar

. A new study of unreported cases of 2019-nCOV epidemic outbreaks. Chaos Solitons Fractals. (2020) 138:109929. 10.1016/j.chaos.2020.109929

33519103

30. Kumar

Erturk

Abboubakar

Nisar

. Prediction studies of the epidemic peak of coronavirus disease in Brazil via new generalised Caputo type fractional derivatives. Alexandr Eng J. (2021) 60:3189–204. 10.1016/j.aej.2021.01.032 31. Vellappandi

Kumar

Govindaraj

. Role of fractional derivatives in the mathematical modeling of the transmission of Chlamydia in the United States from 1989 to 2019. Nonlinear Dyn. (2022) 2022:1–15. 10.1007/s11071-022-08073-3

36373036

32. Abbas

Tyagi

Kumar

Ertürk

Momani

. Stability and bifurcation analysis of a fractional-order model of cell-to-cell spread of HIV-1 with a discrete time delay. Math Methods Appl Sci. (2022) 45:7081–95. 10.1002/mma.8226 33. Teklu

Terefe

. Mathematical modeling analysis on the dynamics of university students animosity towards mathematics with optimal control theory. Sci Rep. (2022) 12:1–19. 10.1038/s41598-022-15376-3

35803995

34. Babaei

Ahmadi

Jafari

Liya

. A mathematical model to examine the effect of quarantine on the spread of coronavirus. Chaos Solitons Fractals. (2021) 142:110418. 10.1016/j.chaos.2020.110418

33288973

35. Basnarkov

. SEAIR. Epidemic spreading model of COVID-19. Chaos Solitons Fractals. (2021) 142:110394.34539151 36. Mohamadou

Halidou

Kapen

. A review of mathematical modeling, artificial intelligence and datasets used in the study, prediction and management of COVID-19. Appl Intell. (2020) 50:3913–25. 10.1007/s10489-020-01770-9

34764546

37. Ndaïrou

Area

Nieto

Torres

. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos Solitons Fractals. (2020) 135:109846. 10.1016/j.chaos.2020.109846

32994672

38. Nwankwo

Okuonghae

. Mathematical analysis of the transmission dynamics of HIV syphilis co-infection in the presence of treatment for syphilis. Bull Math Biol. (2018) 80:437–92. 10.1007/s11538-017-0384-0

29282597

39. Teklu

Kotola

. The impact of protection measures and treatment on pneumonia infection: a mathematical model analysis supported by numerical simulation. bioRxiv. (2022). 10.1101/2022.02.21.481255 40. Vellappandi

Kumar

Govindaraj

. Role of vaccination, the release of competitor snails, chlorination of water, and treatment controls on the transmission of bovine schistosomiasis disease: a mathematical study. Phys Script. (2022) 97:074006. 10.1088/1402-4896/ac7421 41. Oshinubi

Buhamra

Al-Kandari

Waku

Rachdi

Demongeot

. Age dependent epidemic modeling of COVID-19 outbreak in Kuwait, France, and Cameroon. In: Send

, editor. Healthcare (Vol. 10, No. 3). MDPI (2022), p. 482.35326960 42. Teklu

. Mathematical analysis of the transmission dynamics of COVID-19 infection in the presence of intervention strategies. J Biol Dyn. (2022) 16:640–64. 10.1080/17513758.2022.2111469

35972455

43. Castillo-Chavez

Feng

Huang

. On the computation of ro and its role on. In: Castillo-Chavez

Blower

Driessche

Kirschner

Yakubu

A-A

, editors. Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction. vol 1, 229 (2002). 10.1007/978-1-4757-3667-0_13 44. Van den Driessche

Watmough

. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci. (2002) 180:29–48. 10.1016/S0025-5564(02)00108-6

12387915

45. Mekonen

Balcha

Obsu

Hassen

. Mathematical modeling and analysis of TB and COVID-19 coinfection. J Appl Math. (2022) 2022:1–20. 10.1155/2022/2449710