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

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:

$\begin{array}{l}N(t)=P_c(t)+P_s(t)+S(t)+I_c(t)+Q_c(t)+I_{as}(t)+I_{ss}(t)\\ +T_s(t)+T_{cs}(t)+T(t).\end{array}$

TABLE 1

Table 1. Variables' definitions.

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:

$\begin{array}{ll}{\lambda }_c={\beta }_2(I_c+{\varphi }_1{I}_{cs}).& (1)\end{array}$

➢ 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:

$\begin{array}{ll}{\lambda }_s={\beta }_1(I_{as}+{\varphi }_2{I}_{ss}+{\varphi }_3{I}_{cs}).& (2)\end{array}$

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

Table 2. Parameter definitions.

FIGURE 1

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:

$\begin{array}{l}\frac{d{P}_c}{dt}={\tau }_1\Lambda -(\beta +{\lambda }_s+\mu )P_c,\\ \frac{d{P}_s}{dt}={\tau }_3\Lambda -(\pi +{\lambda }_c+\mu )P_s,\\ \frac{dS}{dt}={\tau }_2\Lambda +\beta P_c+\pi P_s+\delta T_s+\theta T-({\lambda }_s+{\lambda }_c+\mu )S,\\ \frac{d{I}_c}{dt}={\lambda }_{c}S+{\lambda }_c{P}_s-({\theta }_1{\lambda }_s+\rho +\mu +{\omega }_1)I_c,& (3)\\ \frac{d{Q}_c}{dt}=\rho I_c-\mu Q_c,\\ \frac{d{I}_{as}}{dt}={\lambda }_{s}S+{\lambda }_s{P}_c-({\theta }_2{\lambda }_c+ϵ+\mu )I_{as},\\ \frac{d{I}_{ss}}{dt}=ϵI_{as}-({\theta }_3{\lambda }_c+\gamma +\mu {+\omega }_2)I_{ss},\\ \frac{d{I}_{cs}}{dt}={\theta }_2{\lambda }_c{I}_{as}+{\theta }_3{\lambda }_c{I}_{ss}+{\theta }_1{\lambda }_s{I}_c-(\epsilon +\mu {+\omega }_3)I_{cs},\\ \frac{d{T}_s}{dt}=\gamma I_{ss}-(\delta +\mu )T_s,\\ \frac{dT}{dt}=\epsilon I_{cs}-(\theta +\mu )T.\end{array}$

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:

$\begin{array}{ll}\Omega =\left\{\begin{array}{c}(P_c,P_s,S,I_c,Q_c,I_{as},I_{ss},T_s,T_{cs},T)\in {ℝ}_{+}^{10},N\le \frac{\Lambda }{\mu }\end{array}\right\}& (4)\end{array}$

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

$\begin{array}{l}\frac{d{P}_c}{dt}+(\beta +{\lambda }_s+\mu )P_c={\tau }_1\Lambda .\end{array}$

After some calculations of integration, we got

$\begin{array}{l}{P}_c(\tau )=a_1{P}_c(0)+a_1{\displaystyle {\int }_0^{\tau }}{e}^{\int (\beta +{\lambda }_s+\mu )dt}{\tau }_1\Lambda dt>0,\text{ where}\\ a_1=e^{-\int (\beta +{\lambda }_s+\mu )dt}>0,P_c(0)>0,P_c(\tau )>0,\text{so that}\\ P_c(\tau )\ne 0.\end{array}$

From model (3) second equation, we have

$\begin{array}{l}\frac{d{P}_s}{dt}={\tau }_3\Lambda -(\pi +{\lambda }_c+\mu )P_s.\end{array}$

After some calculations of integration, we have

$\begin{array}{l}{P}_s(\tau )=b_1{P}_s(0)+b_1{\displaystyle {\int }_0^{\tau }}{e}^{\int (\pi +{\lambda }_c+\mu )dt}{\tau }_3\Lambda dt>0,\text{ where}\\ b_1=e^{-\int (\pi +{\lambda }_c+\mu )dt}>0,P_s(0)>0,P_s(\tau )>0,\text{so that}\\ P_s(\tau )\ne 0.\end{array}$

From model (3) third equation, we have

$\begin{array}{l}\frac{dS}{dt}={\tau }_2\Lambda +\beta P_c+\pi P_s+\delta \gamma T_s-S({\lambda }_s+{\lambda }_c+\mu ).\end{array}$

After some calculations, we have

$S(\tau )=c_{1}S(0)+c_1{\int }_0^{\tau }{e}^{\int ({\lambda }_s+{\lambda }_c+\mu )dt}({\tau }_2\Lambda +\beta P_c+\pi P_s+\delta \gamma T_s)dt>0$, where

$c_1=e^{-\int ({\lambda }_s+{\lambda }_c+\mu )dt}>0,S(0)>0$, 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:

$\begin{array}{l}N(t)=P_c(t)+P_s(t)+S(t)+I_c(t)+Q_c(t)+I_{as}(t)\\ +I_{ss}(t)+T_s(t)+T_{cs}(t)+T(t).\end{array}$

Differentiating both sides gives the following result

$\begin{array}{l}\frac{dN}{dt}=\frac{d{P}_c}{dt}+\frac{d{P}_s}{dt}+\frac{dS}{dt}+\frac{d{I}_c}{dt}+\frac{d{Q}_c}{dt}+\frac{d{I}_{as}}{dt}+\frac{d{I}_{ss}}{dt}+\frac{d{T}_s}{dt}\\ +\frac{d{T}_{cs}}{dt}+\frac{dT}{dt}.\\ =\Lambda -\mu N-{\omega }_1{I}_c-{\omega }_2{I}_{ss}-{\omega }_3{I}_{cs},\text{where }{\tau }_1+{\tau }_3+{\tau }_2=1.\\ \Rightarrow \frac{dN}{dt}\le \Lambda -\mu N.\end{array}$

After some steps, we have $0\le N(t)\le \frac{\Delta }{\mu }$, 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:

$\begin{array}{l}\frac{d{P}_c}{dt}={\tau }_1\Lambda -(\beta +\mu )P_c,\\ \frac{dS}{dt}={\tau }_2\Lambda +\beta P_c-({\lambda }_c+\mu )S,\\ \frac{d{I}_c}{dt}={\lambda }_{c}S-(\rho +\mu +{\omega }_1)I_c,& (5)\\ \frac{d{Q}_c}{dt}=\rho I_c-\mu Q_c.\end{array}$

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 $E_c^0=\left(P_c^0,S^{0}0,0,0\right)=(\frac{{\tau }_1\Lambda }{\beta +\mu },\frac{{\tau }_2\Lambda \left(\beta +\mu \right)+\beta {\tau }_1\Lambda }{\mu \left(\beta +\mu \right)},0,0)$.

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:

$\begin{array}{l}\frac{d{I}_c}{dt}={\lambda }_{c}S-(\rho +\mu +{\omega }_1)I_c,\\ ={\beta }_2{I}_{c}S-(\rho +\mu +{\omega }_1)I_c,\\ =({\beta }_{2}S-(\rho +\mu +{\omega }_1))I_c,\\ =(\frac{{\beta }_2\Lambda ({\tau }_2(\beta +\mu )+\beta {\tau }_1)}{\mu (\beta +\mu )}-(\rho +\mu +{\omega }_1))I_c,\\ =(\rho +\mu +{\omega }_1)(\frac{{\beta }_2\Lambda ({\tau }_2(\beta +\mu )+\beta {\tau }_1)}{\mu (\beta +\mu )(\rho +\mu +{\omega }_1))}-1)I_c,\\ =(\rho +\mu +{\omega }_1)({\Re }_0^c-1)I_c,\text{where}\\ {\Re }_0^c=\frac{{\beta }_2{\tau }_2\Lambda (\beta +\mu )+{\beta }_2\beta {\tau }_1\Lambda }{\mu (\beta +\mu )(\rho +\mu +{\omega }_1)}.\end{array}$

3.1.3. COVID-19 incidence equilibrium point

The COVID-19 incidence equilibrium point of the system (5) is $E_c^{*}=(P_c^{*},S^{*},I_c^{*},Q_c^{*})$, where

$\begin{array}{l}{P}_c^{*}=\frac{{\tau }_1\Lambda }{\beta +\mu },S^{*}=\frac{{\tau }_3\Lambda (\beta +\mu )+{\tau }_1\beta \Lambda ,}{({\lambda }_c^{*}+\mu )(\beta +\mu )},\\ I_c^{*}=\frac{({\tau }_3\Lambda (\beta +\mu )+{\tau }_1\beta \Lambda ){\lambda }_c^{*}}{({\lambda }_c^{*}+\mu )(\beta +\mu )(\rho +\mu +{\omega }_1)},\\ Q_c^{*}=\frac{({\tau }_3\Lambda (\beta +\mu )+{\tau }_1\beta \Lambda )\rho {\lambda }_c^{*}}{\mu ({\lambda }_c^{*}+\mu )(\beta +\mu )(\rho +\mu +{\omega }_1)}.\end{array}$

Theorem 3: The COVID-19 mono-infection model (5) has a unique COVID-19 incidence (endemic) equilibrium point whenever${\Re }_0^c>1$.

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

$\begin{array}{l}{\lambda }_c^{*}={\beta }_1{I}_c^{*}=\frac{{\beta }_1({\tau }_3\Lambda (\beta +\mu )+{\tau }_1\beta \Lambda ){\lambda }_c^{*}}{({\lambda }_c^{*}+\mu )(\beta +\mu )(\rho +\mu +{\omega }_1)}.\end{array}$

Then the non-zero value of ${\lambda }_c^{*}$ after a simple simplification is as follows:

${\lambda }_c^{*}=\mu ({{R}}_0^c-1)>0$, if and only if ${{R}}_0^c>1$.

Hence, the COVID-19 mono-infection model (5) has a unique incidence equilibrium point if and only if ${{R}}_0^c>1$.

Theorem 4: COVID-19 infection-free equilibrium point of the model (5) is locally asymptotically stable if${\Re }_0^c<1$; otherwise, it is unstable.

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

$\begin{array}{l}J(E_c^0)=\left(\begin{array}{cccc}-(\beta +\mu )& 0& 0& 0\\ \beta & -\mu & \frac{-{\beta }_2({\tau }_2\Lambda (\beta +\mu )+\beta {\tau }_1\Lambda )}{\mu (\beta +\mu )}& 0\\ 0& 0& \frac{{\beta }_2({\tau }_2\Lambda (\beta +\mu )+\beta {\tau }_1\Lambda )-\mu (\beta +\mu )(\rho +\mu +{\omega }_1)}{\mu (\beta +\mu )}& 0\\ 0& 0& \rho & -\mu \end{array}\right).\end{array}$

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

$\begin{array}{l}{\lambda }_1=-(\beta +\mu )<0or{\lambda }_2=-\mu <0,\text{or}\\ {\lambda }_3=(\rho +\mu +{\omega }_1)({{R}}_0^c-1).\end{array}$

Thus, each eigenvalue of the Jacobian matrix is negative if ${{R}}_0^c<1$ implies the COVID-19 infection-free equilibrium point is locally asymptotically stable whenever ${{R}}_0^c<1.$

Theorem 5: The COVID-19 infection-free equilibrium point denoted by $E_c^{*}$ of the COVID-19 mono-infection model is globally stable if ${{R}}_0^c<1$; otherwise, it is unstable.

Proof: Take the representative Lyapunov function l(I_c ) = aI_c, where $a=\frac{1}{\left(\rho +\mu +{\omega }_1\right))}$,

$\begin{array}{l}l\left(I_c\right)=a{I}_c=\frac{1}{(\rho +\mu +{\omega }_1)}{I}_c,\\ \frac{dl}{dt}=\frac{1}{(\rho +\mu +{\omega }_1)}(({\beta }_{2}S-(\rho +\mu +{\omega }_1))I_c),\\ \le \frac{1}{(\rho +\mu +{\omega }_1)}\\ (\frac{{\beta }_2\Lambda ({\tau }_2(\beta +\mu )+\beta {\tau }_1)-\mu (\beta +\mu )(\rho +\mu +{\omega }_1)}{\mu (\beta +\mu )})I_c,\\ \le (\frac{{\beta }_2\Lambda ({\tau }_2(\beta +\mu )+\beta {\tau }_1)-\mu (\beta +\mu )(\rho +\mu +{\omega }_1)}{\mu (\beta +\mu )(\rho +\mu +{\omega }_1)})I_c,\\ \le \mu (\beta +\mu )(\rho +\mu +{\omega }_1)(\frac{\frac{{\beta }_2\Lambda ({\tau }_2(\beta +\mu )+\beta {\tau }_1)}{\mu (\beta +\mu )(\rho +\mu +{\omega }_1)}-1}{\mu (\beta +\mu )(\rho +\mu +{\omega }_1)})I_c,\\ \le \mu (\beta +\mu )(\rho +\mu +{\omega }_1)(\frac{{{R}}_0^c-1}{\mu (\beta +\mu )(\rho +\mu +{\omega }_1)})I_c,\\ \le ({{R}}_0^c-1)I_c.\end{array}$

Thus, $\frac{dl}{dt}<0$, if ${{R}}_0^c<1$, and the equality $\frac{dl}{dt}=0$ holds if ${{R}}_0^c=1$, and hence the COVID-19 infection-free equilibrium point is globally asymptotically stable if ${{R}}_0^c<1.$

Theorem 6: The COVID-19 incidence denoted by $E_C^{*}$ of the COVID-19 mono-infection model (5) is locally asymptotically stable whenever${{R}}_0^c>1$; otherwise, it is unstable.

Proof: The Jacobean of the system (5) at $E_C^{*}$

$\begin{array}{l}J\left(E_C^{\ast }\right)=\left(\begin{array}{cccc}-\left(\beta +\mu \right)& 0& 0& 0\\ \beta & -\left({\beta }_2{I}_c{}^{*}+\mu \right)& -{\beta }_2{S}^{\ast }& 0\\ 0& {\beta }_2{I}_c^{*}& {\beta }_2{S}^{*}-\left(\rho +\mu +{\omega }_1\right)& 0\\ 0& 0& \rho & -\mu \end{array}\right).\end{array}$

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

$\begin{array}{l}\left(-\left(\beta +\mu \right)-\lambda \right)(-\mu -\lambda )[\left(-\left({\beta }_2{I}_c^{\ast }+\mu \right)-\lambda \right)({\beta }_2{S}^{*}-(\rho +\mu \\ +{\omega }_1)-\lambda )+{\beta }_2{S}^{\ast }{\beta }_2{I}_c^{\ast }]=0.\end{array}$

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

$\begin{array}{l}{a}_0{\lambda }^2+a_1\lambda +a_2=0where\\ a_0\text{= 1,}\\ a_1=\left(\frac{\begin{array}{l}{\beta }_2\left({\tau }_3\Lambda \left(\beta +\mu \right)+{\tau }_1\beta \Lambda \right)\mu ({{R}}_0^c-1)\hfill \\ +\mu ({\lambda }_c^{\ast }+\mu )(\beta +\mu )(\rho +\mu +{\omega }_1)\hfill \end{array}}{({\lambda }_c^{\ast }+\mu )(\beta +\mu )(\rho +\mu +{\omega }_1)}\right)\\ +\frac{({\beta }_2{\tau }_2\Lambda (\beta +\mu )+{\tau }_1\beta \Lambda )[({\lambda }_c^{\ast }+\mu )\mu {{R}}_0^c-1]}{({\lambda }_c^{\ast }+\mu )(\beta +\mu )},\\ a_2=\frac{{\beta }_2({\tau }_3\Lambda (\beta +\mu )+{\tau }_1\beta \Lambda )\mu ({{R}}_0^c-1)}{({\lambda }_c^{\ast }+\mu )(\beta +\mu )(\rho +\mu +{\omega }_1)}\\ +\frac{({\beta }_2{\tau }_2\Lambda (\beta +\mu )+{\tau }_1\beta \Lambda )[({\lambda }_c^{\ast }+\mu )\mu {{R}}_0^c-1]}{({\lambda }_c^{\ast }+\mu )(\beta +\mu )}\\ +\frac{{\beta }_2({\tau }_2\Lambda (\beta +\mu )+{\tau }_1\beta \Lambda )\mu ({{R}}_0^c-1)}{({\lambda }_c^{\ast }+\mu )(\beta +\mu )(\rho +\mu +{\omega }_1)}.\end{array}$

Hence, all the coefficients of the characteristics equations are positive when ${{R}}_0^c>1;$ thus, the COVID-19 incidence equilibrium point has local asymptotic stability when ${{R}}_0^c>1$.

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:

$\begin{array}{l}\frac{d{P}_s}{dt}={\tau }_3\Lambda -(\pi +\mu )P_s,\\ \frac{dS}{dt}={\tau }_2\Lambda +\pi P_s+\delta \gamma T_s-S({\lambda }_s+\mu ),\\ \frac{d{I}_{as}}{dt}={\lambda }_{s}S-I_{as}(ϵ+\mu ),& (6)\\ \frac{d{I}_{ss}}{dt}=ϵI_{as}-I_{ss}(\gamma +\mu {+\omega }_2),\\ \frac{d{T}_s}{dt}=\gamma I_{ss}-T_s(\delta \gamma +\mu ).\end{array}$

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 $E_s^0=(P_s^0,S^0,I_{as}^0,I_{ss}^0,T_s^0)=\left(\frac{{\tau }_3\Lambda }{\pi +\mu },\frac{{\tau }_2\Lambda (\pi +\mu )+{\tau }_3\pi \Lambda }{\mu (\pi +\mu )},0,0,0\right).$

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:

$\begin{array}{l}F=\left(\begin{array}{cc}{\beta }_1{S}^0& {\beta }_1{\varphi }_2{S}^0\\ 0& 0\end{array}\right),\\ \Rightarrow F=\left(\begin{array}{cc}\frac{{\beta }_1\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu (\pi +\mu )}& \frac{{\beta }_1{\varphi }_2\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu (\pi +\mu )}\\ 0& 0\end{array}\right),\end{array}$

and

$\begin{array}{l}V=\left(\begin{array}{cc}ϵ+\mu & 0\\ -ϵ& \gamma +\mu {+\omega }_2\end{array}\right).\end{array}$

Then, we applied Mathematica coding; we have

$\begin{array}{l}{V}^{-1}=\left[\begin{array}{cc}\frac{1}{ϵ+\mu }& 0\\ \frac{ϵ}{(\gamma +\mu +{\omega }_2)(ϵ+\mu )}& \frac{1}{(\gamma +\mu +{\omega }_2)}\end{array}\right],\text{ and}\\ F{V}^{-1}=[\frac{{\beta }_1\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu \left(\pi +\mu \right)(ϵ+\mu )}\\ +\begin{array}{cc}\frac{ϵ{\beta }_1{\varphi }_2\Lambda \left({\tau }_2\left(\pi +\mu \right)+{\tau }_3\pi \right)}{\mu \left(\pi +\mu \right)(\gamma +\mu +{\omega }_2)\left(ϵ+\mu \right)}& \frac{{\beta }_1{\varphi }_2\Lambda \left({\tau }_2\left(\pi +\mu \right)+{\tau }_3\pi \right)}{\mu \left(\pi +\mu \right)(\gamma +\mu +{\omega }_2)}\\ 0& 0\end{array}].\end{array}$

Thus, the reproduction number of the syphilis sub-model (6) is given by ${{R}}_0^s=\frac{{\beta }_1\Lambda \left[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ\right]({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu (\pi +\mu )(\gamma +\mu {+\omega }_2)(ϵ+\mu )}$.

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 $E_s^{*}=(P_s^{*},S^{*},I_{as}^{*},I_{ss}^{*},T_s^{*})$, where

$\begin{array}{l}{P}_s^{*}=\frac{{\tau }_3\Lambda }{(\pi +\mu )},\\ S^{*}=\frac{(\gamma +\mu {+\omega }_2)(ϵ+\mu )(\delta \gamma +\mu )({\tau }_2\Lambda (\pi +\mu )+{\tau }_3\Lambda \pi )}{(\pi +\mu )((\gamma +\mu {+\omega }_2)(ϵ+\mu )({\lambda }_s+\mu )(\delta \gamma +\mu )-\gamma \delta \gamma ϵ{\lambda }_s)},\\ I_{as}^{*}=\frac{{\lambda }_s(\gamma +\mu {+\omega }_2)(\delta \gamma +\mu )({\tau }_2\Lambda (\pi +\mu )+{\tau }_3\Lambda \pi )}{(\pi +\mu )((\gamma +\mu {+\omega }_2)(ϵ+\mu )({\lambda }_s+\mu )(\delta \gamma +\mu )-\gamma \delta \gamma ϵ{\lambda }_s)},\\ I_{ss}^{*}=\frac{ϵ{\lambda }_s(\delta \gamma +\mu )({\tau }_2\Lambda (\pi +\mu )+{\tau }_3\Lambda \pi )}{(\pi +\mu )((\gamma +\mu {+\omega }_2)(ϵ+\mu )({\lambda }_s+\mu )(\delta \gamma +\mu )-\gamma \delta \gamma ϵ{\lambda }_s)},\\ T_s^{*}=\frac{ϵ{\lambda }_s\gamma ({\tau }_2\Lambda (\pi +\mu )+{\tau }_3\Lambda \pi )}{(\pi +\mu )((\gamma +\mu {+\omega }_2)(ϵ+\mu )({\lambda }_s+\mu )(\delta \gamma +\mu )-\gamma \delta \gamma ϵ{\lambda }_s)}.\end{array}$

Theorem 7: The syphilis incidence equilibrium point of syphilis in the model (6) is unique whenever ${{R}}_s^0>1.$

Proof: From the syphilis infection rate, we have

$\begin{array}{l}{\lambda }_s^{\ast }={\beta }_1(I_{as}^{\ast }+{\varphi }_2{I}_{ss}^{\ast }).\\ {\lambda }_s^{\ast }={\beta }_1(\frac{{\lambda }_s(\gamma +\mu +{\omega }_2)(\delta \gamma +\mu )({\tau }_2\Lambda \left(\pi +\mu \right)+{\tau }_3\Lambda \pi )}{\begin{array}{c}\left(\pi +\mu \right)((\gamma +\mu +{\omega }_2)\left(ϵ+\mu \right)\left({\lambda }_s^{\ast }+\mu \right)\left(\delta \gamma +\mu \right)\\ -ϵ\gamma \delta \gamma {\lambda }_s^{\ast })\end{array}}\\ +\frac{ϵ{\lambda }_s^{\ast }{\varphi }_2(\delta \gamma +\mu )({\tau }_2\Lambda \left(\pi +\mu \right)+{\tau }_3\Lambda \pi )}{\begin{array}{c}\left(\pi +\mu \right)((\gamma +\mu +{\omega }_2)\left(ϵ+\mu \right)\left({\lambda }_s^{\ast }+\mu \right)\left(\delta \gamma +\mu \right)\\ -\gamma \delta \gamma ϵ{\lambda }_s^{\ast })\end{array}}).\\ \Rightarrow {\lambda }_s^{\ast }=\left(\delta \gamma +\mu \right)\\ \frac{\begin{array}{c}{\beta }_1\Lambda [(\gamma +\mu +{\omega }_2)\left({\tau }_2\left(\pi +\mu \right)+{\tau }_3\pi \right)+ϵ{\varphi }_2({\tau }_2\left(\pi +\mu \right)\\ +{\tau }_3\pi )]-\mu \left(\pi +\mu \right)[(\gamma +\mu +{\omega }_2)\left(ϵ+\mu \right)]\end{array}}{\left(\pi +\mu \right)\left((\gamma +\mu +{\omega }_2)\left(ϵ+\mu \right)\left(\delta \gamma +\mu \right)-\gamma \delta \gamma ϵ\right)}.\\ \Rightarrow {\lambda }_s^{\ast }=\frac{\left(\delta \gamma +\mu \right)\mu (\gamma +\mu +{\omega }_2)\left(ϵ+\mu \right)({{R}}_0^s-1)}{\left((\gamma +\mu +{\omega }_2)\left(ϵ+\mu \right)\left(\delta \gamma +\mu \right)-\gamma \delta \gamma ϵ\right)}.\\ \Rightarrow {\lambda }_s^{\ast }={\text{k}}_1({{R}}_0^s-1),\text{where}\\ {\text{k}}_1=\frac{\left(\delta \gamma +\mu \right)\mu \left(\pi +\mu \right)(\gamma +\mu +{\omega }_2)\left(ϵ+\mu \right)}{\left(\pi +\mu \right)\left((\gamma +\mu +{\omega }_2)\left(ϵ+\mu \right)\left(\delta \gamma +\mu \right)-\gamma \delta \gamma ϵ\right)}.\end{array}$

Hence, the syphilis sub-model (6) has a unique incidence equilibrium if ${{R}}_0^s>1$.

Theorem 8: Syphilis infection-free equilibrium point of the model (3) has local asymptotic stability if${{R}}_0^s<1$; otherwise, it is unstable.

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

$\begin{array}{l}J(E_0^s)=\left(\begin{array}{ccccc}-(\pi +\mu )& 0& 0& 0& 0\\ \pi & -\mu & \frac{-{\beta }_1\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu (\pi +\mu )}& \frac{-{\beta }_1{\varphi }_2\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu (\pi +\mu )}& \delta \gamma \\ 0& 0& \frac{{\beta }_1\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )-\mu (\pi +\mu )(ϵ+\mu )}{\mu (\pi +\mu )}& \frac{{\beta }_1{\varphi }_2\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu (\pi +\mu )}& 0\\ 0& 0& ϵ& -(\gamma +\mu {+\omega }_2)& 0\\ 0& 0& 0& \gamma & -(\delta \gamma +\mu )\end{array}\right).\end{array}$

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

$\begin{array}{l}\left(-\left(\pi +\mu \right)\lambda \right)\left(-\mu -\lambda \right)(-\left(\delta \gamma +\mu \right)\lambda )\\ [\left(\frac{{\beta }_1\Lambda \left({\tau }_2\left(\pi +\mu \right)+{\tau }_3\pi \right)-\mu \left(\pi +\mu \right)\left(ϵ+\mu \right)}{\mu \left(\pi +\mu \right)}\lambda \right)\\ (-(\gamma +\mu +{\omega }_2)\lambda )-(\frac{{\beta }_1{\varphi }_2\Lambda ϵ\left({\tau }_2\left(\pi +\mu \right)+{\tau }_3\pi \right)}{\mu \left(\pi +\mu \right)})]=0.\end{array}$

Then the eigenvalues are λ₁ = −(π + μ) < 0 or λ₂ = −μ < 0 or λ₃ = −(δγ + μ) < 0 or $a_0{\lambda }^2+a_1\lambda +a_2=0$.

where,

$\begin{array}{l}{a}_0=1,\\ a_1=(\gamma +\mu {+\omega }_2)+(ϵ+\mu )+\frac{{\beta }_1\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu (\pi +\mu )},\\ a_2=(ϵ+\mu )(\gamma +\mu {+\omega }_2)(1-{{R}}_0^s).\end{array}$

Applying Routh–Hurwitz criteria stated in [33], each eigenvalue of the matrix is negative whenever${{R}}_0^s<1$; thus, the syphilis infection-free equilibrium point has local asymptotic stability if ${{R}}_0^s<1$.

Theorem 9: Syphilis infection-free equilibrium point $E_s^0$ of the model (6) has global stability if ${{R}}_0^s<1$; otherwise, it is unstable.

Proof: Let the Lyapunov representative function be given as l(I_as, I_ss ) = aI_as + bI_ss, where $=\frac{\left[(\gamma +\mu )+{\varphi }_2ϵ\right]}{(ϵ+\mu )(\gamma +\mu {+\omega }_2)}$, $b=\frac{{\varphi }_2}{(\gamma +\mu {+\omega }_2)}$.

$\begin{array}{l}\Rightarrow l(I_{as},I_{ss})=\frac{[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]}{(ϵ+\mu )(\gamma +\mu {+\omega }_2)}{I}_{as}+\frac{{\varphi }_2}{(\gamma +\mu {+\omega }_2)}{I}_{ss}.\\ \Rightarrow \frac{dl}{dt}=\frac{[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]}{(ϵ+\mu )(\gamma +\mu {+\omega }_2)}({\lambda }_{s}S-I_{as}(ϵ+\mu ))\\ +\frac{{\varphi }_2}{(\gamma +\mu )}(ϵI_{as}-I_{ss}(\gamma +\mu {+\omega }_2)),\\ \le \frac{[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]}{(ϵ+\mu )(\gamma +\mu {+\omega }_2)}({\beta }_1(I_{as}+{\varphi }_2{I}_{ss})S^{*}-I_{as}(ϵ+\mu ))\\ +\frac{{\varphi }_2}{(\gamma +\mu )}(ϵI_{as}-I_{ss}(\gamma +\mu {+\omega }_2)),\\ \le \frac{[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]}{(ϵ+\mu )(\gamma +\mu {+\omega }_2)}({\beta }_1{I}_{as}{S}^{*}+{\beta }_1{\varphi }_2{I}_{ss}{S}^{*}-I_{as}(ϵ+\mu ))\\ +\frac{{\varphi }_2}{(\gamma +\mu )}(ϵI_{as}-I_{ss}(\gamma +\mu {+\omega }_2)),\\ \le ([{\beta }_1{S}^{*}-(ϵ+\mu )]\frac{[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]}{(ϵ+\mu )(\gamma +\mu {+\omega }_2)}+\frac{{\varphi }_2ϵ}{(\gamma +\mu {+\omega }_2)})I_{as}\\ +(\frac{{\beta }_1{\varphi }_2[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]S^{*}}{(ϵ+\mu )(\gamma +\mu {+\omega }_2)}-{\varphi }_2)I_{ss},\\ \le ([\frac{[(\gamma +\mu )+{\varphi }_2ϵ]{\beta }_1{S}^{*}}{(ϵ+\mu )(\gamma +\mu {+\omega }_2)}-\frac{[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]}{(\gamma +\mu {+\omega }_2)}])I_{as}\\ +(\frac{{\beta }_1{\varphi }_2[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]S^{*}}{(ϵ+\mu )(\gamma +\mu {+\omega }_2)}-{\varphi }_2)I_{ss},\\ \le \frac{[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]{\beta }_1\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu (\pi +\mu )(ϵ+\mu )(\gamma +\mu {+\omega }_2)}{I}_{as}\\ -\frac{[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]}{(\gamma +\mu {+\omega }_2)}{I}_{as}+\\ (\frac{{\beta }_1{\varphi }_2[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu (\pi +\mu )(ϵ+\mu )(\gamma +\mu {+\omega }_2)}-{\varphi }_2)I_{ss}.\\ \le ([\frac{[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]{\beta }_1\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu (\pi +\mu )(ϵ+\mu )(\gamma +\mu {+\omega }_2)}-m])I_{as}\\ +(\frac{{\beta }_1{\varphi }_2[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]\Lambda ({\tau }_2(\pi +\mu )+{\tau }_3\pi )}{\mu (\pi +\mu )(ϵ+\mu )(\gamma +\mu {+\omega }_2)}-{\varphi }_2),where\\ m=\frac{[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]}{(\gamma +\mu {+\omega }_2)}>1.\\ \le m([\frac{{{R}}_0^s}{m}-1])I_{as}+{\varphi }_2(\frac{{{R}}_0^s}{{\varphi }_2}-1)I_{ss},\\ \frac{{{R}}_0^s}{m}<1,\frac{{{R}}_s^0}{{\varphi }_2}<1,\\ \frac{{{R}}_0^s}{m}<1,\frac{{{R}}_0^s}{{\varphi }_2}<1\text{ implies }{{R}}_0^s<1\text{ since m}\\ =\frac{[(\gamma +\mu {+\omega }_2)+{\varphi }_2ϵ]}{(\gamma +\mu {+\omega }_2)}>1,\text{ and }{\varphi }_2>1.\end{array}$

Hence, the syphilis-free equilibrium point is globally stable if ${{R}}_0^s<1$.

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 $E^0=\left(P_c^0,P_s^0,S^0,I_c^0,Q_c^0,I_{as}^0,I_{ss}^0,T_s^0,T_{cs}^0,T^0\right)=$ $(\frac{{\tau }_1\Lambda }{\left(\beta +\mu \right)},\frac{{\tau }_3\Lambda }{(\pi +\mu )},\frac{\Lambda ({\tau }_2\left(\beta +\mu \right)\left(\pi +\mu \right)+{\tau }_1\beta \left(\pi +\mu \right)+{\tau }_3\pi \left(\beta +\mu \right))}{\mu \left(\beta +\mu \right)(\pi +\mu )},0,0,0,0,0,0,0)$.

3.3.2. The model (3) reproduction number

The COVID-19 and syphilis co-infection model (3) reproduction number denoted by ${\Re }_0^{cs}$ 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

$\begin{array}{l}{f}_i=\left[\begin{array}{c}{\beta }_2(I_c+{\varphi }_1{I}_{cs})(S+P_s)\\ {\beta }_1(I_{as}+{\varphi }_2{I}_{ss}+{\varphi }_3{I}_{cs})(S+P_c)\\ 0\\ {\beta }_2({\theta }_2{I}_{as}+{\theta }_3{I}_{ss})(I_c+{\varphi }_1{I}_{cs})+{\theta }_1{\beta }_1(I_{as}+{\varphi }_2{I}_{ss}+{\varphi }_3{I}_{cs})I_c\end{array}\right].\\ \Rightarrow f=\left[\begin{array}{cccc}{\beta }_2(S^0+P_s^0)& 0& 0& {\varphi }_1(S^0+P_s^0)\\ 0& {\beta }_1(S^0+P_c^0)& {\beta }_1{\varphi }_2(S^0+P_c^0)& {\beta }_1{\varphi }_3(S^0+P_c^0)\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],\end{array}$

and

$\begin{array}{l}{v}_i={v_i}^{-}(x)-{v_i}^{+}(x)\\ =\left[\begin{array}{c}{I}_c({\theta }_1{\beta }_1(I_{as}+{\varphi }_2{I}_{ss}+{\varphi }_3{I}_{cs})+\rho +\mu {+\omega }_1)\\ I_{as}({\theta }_2{\beta }_2(I_c+{\varphi }_1{I}_{cs})+ϵ+\mu )\\ I_{ss}({\theta }_3{\beta }_2(I_c+{\varphi }_1{I}_{cs})+\gamma +\mu {+\omega }_2)-ϵI_{as}\\ I_{cs}(\epsilon +\mu {+\omega }_3)\end{array}\right].\\ \Rightarrow v=\left[\begin{array}{cccc}(\rho +\mu {+\omega }_1)& 0& 0& 0\\ 0& (ϵ+\mu )& 0& 0\\ 0& -ϵ& (\gamma +\mu {+\omega }_2)& 0\\ 0& 0& 0& (\epsilon +\mu {+\omega }_3)\end{array}\right].\end{array}$

Then applying Mathematica, we have got

$\begin{array}{l}{v}^{-1}=\left[\begin{array}{cccc}\frac{1}{(\rho +\mu {+\omega }_1)}& 0& 0& 0\\ 0& \frac{1}{(\epsilon +\mu )}& 0& 0\\ 0& 0& \frac{1}{(\gamma +\mu {+\omega }_2)}& 0\\ 0& 0& 0& \frac{1}{(\epsilon +\mu {+\omega }_3)}\end{array}\right],\end{array}$

and

$\begin{array}{l}f{v}^{-1}=\left[\begin{array}{cccc}\frac{{\beta }_2(S^0+{P^0}_s)}{(\rho +\mu {+\omega }_1)}& 0& 0& \frac{{\varphi }_1(S^0+{P^0}_s)}{(\epsilon +\mu {+\omega }_3)}\\ 0& \frac{{\beta }_1(S^0+{P^0}_c)}{(\epsilon +\mu )}& \frac{{\beta }_1{\varphi }_2(S^0+{P^0}_c)}{(\gamma +\mu {+\omega }_2)}& \frac{{\beta }_1{\varphi }_3(S^0+{P^0}_c)}{(\epsilon +\mu {+\omega }_3)}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right].\\ \Rightarrow \left|\begin{array}{cccc}\frac{{\beta }_2(S^0+{P^0}_s)}{(\rho +\mu {+\omega }_1)}-\lambda & 0& 0& \frac{{\varphi }_1(S^0+{P^0}_s)}{(\epsilon +\mu {+\omega }_3)}\\ 0& \frac{{\beta }_1(S^0+{P^0}_c)}{(\epsilon +\mu )}-\lambda & \frac{{\beta }_1{\varphi }_2(S^0+{P^0}_c)}{(\gamma +\mu {+\omega }_2)}& \frac{{\beta }_1{\varphi }_3(S^0+{P^0}_c)}{(\epsilon +\mu {+\omega }_3)}\\ 0& 0& 0-\lambda & 0\\ 0& 0& 0& 0-\lambda \end{array}\right|\\ =0.\end{array}$

Then, the corresponding eigenvalues are λ₁ = 0 or λ₂ = 0 or ${\lambda }_3=\frac{{\beta }_2(S^0+{P^0}_s)}{(\rho +\mu {+\omega }_1)}=\frac{{\beta }_2\Lambda {\tau }_2}{\mu (\rho +\mu {+\omega }_1)}+\frac{{\beta }_2\Lambda {\tau }_1\beta }{\mu (\beta +\mu )(\rho +\mu {+\omega }_1)}+\frac{{\beta }_2\Lambda {\tau }_3}{\mu (\rho +\mu {+\omega }_1)}={\Re }_0^c+n$ or ${\lambda }_4={\beta }_1(\frac{{\tau }_2\Lambda (\beta +\mu )(\pi +\mu )+{\tau }_1\beta \Lambda (\pi +\mu )+{\tau }_3\pi \Lambda (\beta +\mu )+{\tau }_1\Lambda \mu (\pi +\mu )}{\mu (\beta +\mu )(\pi +\mu )(\epsilon +\mu )})$ $={\Re }_0^s-m$ where $,n=\frac{{\beta }_2\Lambda {\tau }_3}{\mu (\rho +\mu {+\omega }_1)}$, $m=\frac{{\beta }_1\Lambda \left[ϵ{\varphi }_2({\tau }_2(\pi +\mu )+{\tau }_3\pi )-{\tau }_1((\pi +\mu )\gamma +\mu {+\omega }_2)\right]}{\mu (\pi +\mu )(\gamma +\mu {+\omega }_2)(ϵ+\mu )}$.

Therefore, the COVID-19–syphilis complete model (3) reproduction number denoted by ${\Re }_0^{cs}$ is given by ${\Re }_0^{cs}=max\left\{{\Re }_0^c+n,{\Re }_0^s-m\right\}.$

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 ${\Re }_0^{cs}<1;$ otherwise, it is unstable.

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

$\begin{array}{l}J(E^0)=\left(\begin{array}{cccccccccc}a& 0& 0& 0& 0& b& c& 0& d& 0\\ 0& e& 0& f& 0& 0& 0& 0& g& 0\\ \beta & \pi & h& i& 0& j& k& l& m& 0\\ 0& 0& 0& n& 0& 0& 0& 0& o& 0\\ 0& 0& 0& \rho & h& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& p& q& 0& r& 0\\ 0& 0& 0& 0& 0& ϵ& s& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \gamma & t& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& u& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \epsilon & h\end{array}\right),\end{array}$

where a $=-(\beta +\mu )$, $b=-{\beta }_1{P_c}^0$, $c=-{\beta }_1{\varphi }_2{P_c}^0$, $d=-{\beta }_1{\varphi }_3{P_c}^0$, $e=-(\pi +\mu )$, $f=-{\beta }_2{P_s}^0$, $g=-{\beta }_2{\varphi }_1{P_s}^0$, $h=-\mu$, $i=-{\beta }_2{S}^0$, $j=-{\beta }_1{S}^0$, $k=-{\beta }_1{\varphi }_2{S}^0$, $l=\delta \gamma$, $m=-({{\beta }_1\varphi }_3{+{\beta }_2\varphi }_1)S^0$, $n={\beta }_2(S+P_s)-(\rho +\mu {+\omega }_1)$, $o={{\beta }_2\varphi }_1(S+P_s)$, $p={\beta }_1(S+P_c)-(ϵ+\mu )$, $q={\beta }_1{\varphi }_2(S+P_c)$, $r={\beta }_1{\varphi }_3(S^0+{P_c}^0)$, $s=-(\gamma +\mu {+\omega }_2)$, $t=-(\delta \gamma +\mu )$, $u=-(\epsilon +\mu {+\omega }_3)$.

$\begin{array}{l}\Rightarrow \left|\begin{array}{cccccccccc}a-\lambda & 0& 0& 0& 0& b& c& 0& d& 0\\ 0& e-\lambda & 0& f& 0& 0& 0& 0& g& 0\\ \beta & \pi & h-\lambda & i& 0& j& k& l& m& 0\\ 0& 0& 0& n-\lambda & 0& 0& 0& 0& o& 0\\ 0& 0& 0& \rho & h-\lambda & 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& p-\lambda & q& 0& r& 0\\ 0& 0& 0& 0& 0& ϵ& s-\lambda & 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \gamma & t-\lambda & 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& u-\lambda & 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \epsilon & h-\lambda \end{array}\right|\\ =0.\end{array}$

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

$\begin{array}{l}\left|\begin{array}{cc}A& B\\ 0& C\end{array}\right|=0,\text{ where}\\ A=\left[\begin{array}{ccccc}a-\lambda & 0& 0& 0& 0\\ 0& e-\lambda & 0& f& 0\\ \beta & \pi & h-\lambda & i& 0\\ 0& 0& 0& n-\lambda & 0\\ 0& 0& 0& \rho & h-\lambda \end{array}\right],B=\left[\begin{array}{ccccc}b& c& 0& d& 0\\ 0& 0& 0& g& 0\\ j& k& l& m& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right],\\ C=\left[\begin{array}{ccccc}p-\lambda & q& 0& r& 0\\ ϵ& s-\lambda & 0& 0& 0\\ 0& \gamma & t-\lambda & 0& 0\\ 0& 0& 0& u-\lambda & 0\\ 0& 0& 0& \epsilon & h-\lambda \end{array}\right],O=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right].\\ \left|\begin{array}{cccccccccc}a-\lambda & 0& 0& 0& 0& b& c& 0& d& 0\\ 0& e-\lambda & 0& f& 0& 0& 0& 0& g& 0\\ \beta & \pi & h-\lambda & i& 0& j& k& l& m& 0\\ 0& 0& 0& n-\lambda & 0& 0& 0& 0& o& 0\\ 0& 0& 0& \rho & h-\lambda & 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& p-\lambda & q& 0& r& 0\\ 0& 0& 0& 0& 0& ϵ& s-\lambda & 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \gamma & t-\lambda & 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& u-\lambda & 0\\ 0& 0& 0& 0& 0& 0& 0& 0& \epsilon & h-\lambda \end{array}\right|\\ =\left|A\right|\left|C\right|=0.\end{array}$

From this, we do have

$\begin{array}{l}\left|A\right|=(a-\lambda )(e-\lambda )(h-\lambda )(n-\lambda )(h-\lambda ),\\ \left|C\right|=(t-\lambda )(u-\lambda )(h-\lambda )((p-\lambda )(s-\lambda )-qϵ),\left|A\right|\\ \left|C\right|=\left[(a-\lambda )(e-\lambda )(h-\lambda )(n-\lambda )(h-\lambda )\right]\\ \left[(t-\lambda )(u-\lambda )(h-\lambda )((p-\lambda )(s-\lambda )-qϵ)\right]=0.\end{array}$

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

$\begin{array}{l}{\lambda }_1=a\text{ or }{\lambda }_2=e\text{ or }{\lambda }_3=h\text{ or }{\lambda }_4=n\text{ or }{\lambda }_5=h\text{ or }{\lambda }_6=t\text{ or}\\ {\lambda }_7=u\text{ or }{\lambda }_8=h\text{ or }{a}_0{\lambda }^2+a_1\lambda +a_2=0,\text{ where},\\ a_0=1,\\ a_1=(\rho +\mu {+\omega }_1)(ϵ+\mu )(1-{\Re }_0^{cs})>0,\\ a_2=(ϵ+\mu ){\varphi }_2ϵ\left[1-{\Re }_0^{cs}\right]>0,if{\Re }_0^{cs}<1.\end{array}$

Therefore, the co-infection model disease-free equilibrium point has local asymptotic stability if ${\Re }_0^{cs}<1.$

3.3.4. The full-model endemic equilibrium and stabilities

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

$E_{cs}^{*}=(P_c^{*},P_s^{*},S^{*},I_c^{*},Q_c^{*},I_{as}^{*},I_{ss}^{*},I_{cs}^{*},,T_s^{*},T^{*})$. 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 ${\Re }_0^c<1$ and ${\Re }_0^s<1$, respectively, which means there is no endemic equilibrium point if ${\Re }_0^{cs}<1$ for the co-infection model (3). In other words, the COVID-19 and syphilis co-infection disease-free equilibrium point have global stability if ${\Re }_0^{cs}<1.$

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

1. $E_1^{*}=(P_c^{*},0,S^{*},I_c^{*},Q_c^{*},0,0,0,,0,0),$ if ${\Re }_0^c>1$ is the syphilis-free (COVID-19 persistence) equilibrium point.

The analysis of the equilibrium $E_1^{*}$ is similar to the endemic equilibrium $E_c^{*}$ in the model (5).

2. $E_2^{*}=(0,P_s^{*},S^{*},0,0,I_{as}^{*},I_{ss}^{*},0,,T_s^{*},0)$, if ${\Re }_0^s>1$ is the COVID-19-free (syphilis persistence) equilibrium point. The analysis of the equilibrium $E_2^{*}$ is similar to the endemic equilibrium $E_s^{*}$ in Equation (6).

3. $E_{cs}^{*}=(P_c^{*},P_s^{*},S^{*},I_c^{*},Q_c^{*},I_{as}^{*},I_{ss}^{*},I_{cs}^{*},,T_s^{*},T^{*})$ is the COVID-19 and syphilis co-existence persistence equilibrium point. It exists when each component of $E_{cs}^{*}$ is positive whenever ${\Re }_0^{cs}>1$ 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

Table 3. Parameter values for numerical simulations.

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 ${{R}}_0^{cs}$ its derivative depends on a parameter ζ is defined by SEI$(p)={\frac{\partial {{R}}_0^{cs}}{\partial \zeta }}^{*}\frac{\zeta }{{{R}}_0^{cs}}$ [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 ${{R}}_0^{cs}=max\left\{{{R}}_0^s,{{R}}_0^c\right\}$. Using parameter values stated in Table 3, the sensitivity index values of the model (3) are calculated in Tables 4, 5.

TABLE 4

Table 4. Sensitivity indexes of ${{R}}_0^{cs}={{R}}_0^c$.

TABLE 5

Table 5. Sensitivity indexes of ${{R}}_0^{cs}={{R}}_0^s$.

Using parameter values in Table 3, we have computed ${\Re }_0^{cs}=max\left\{{\Re }_0^c,{\Re }_0^s\right\}=max\left\{2.7,3.2\right\}=3.2$ 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 ${\Re }_0^c$. 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 ${\Re }_0^c$.

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 ${\Re }_0^s$. 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${\Re }_0^s$.

4.2. Results of numerical simulations

4.2.1. Behaviors of solutions of model (3) whenever ${\Re }_0^{\text{c}\text{s}}<1$

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 ${\Re }_0^c=0.71$ and ${\Re }_0^s=0.34$ 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 ${\Re }_0^{cs}<1.$ Biologically it means the COVID-19 and syphilis co-infection diseases have been eradicated from the community through time whenever ${\Re }_0^{cs}=max\left\{{\Re }_0^c,{\Re }_0^s\right\}=0.71<1$.

FIGURE 2

Figure 2. The complete model solutions behavior if ${\Re }_0^{cs}<1$ at β₁ = 0.3 and β₂ = 0.08.

4.2.2. Behaviors of the model solutions whenever ${\Re }_0^{\text{c}\text{s}}>1$

Figure 3 shows that all the COVID-19 and syphilis co-infection model (3) solutions converge toward the endemic equilibrium point whenever ${\Re }_0^c=3.2$ and ${\Re }_0^s=2.1$ 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 ${\Re }_0^{cs}=max\left\{{\Re }_0^c,{\Re }_0^s\right\}=3.2>1.$ Biologically, it means that COVID-19 and syphilis co-infection disease spreads throughout the community under consideration.

FIGURE 3

Figure 3. Behaviors of the full model solutions whenever ${\Re }_0^{cs}>1$ 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 ${\Re }_0^c$ 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 ${\Re }_0^c<1$, that is, the COVID-19 infection will be eradicated throughout the community.

FIGURE 4

Figure 4. Effect of COVID-19 protection rate on ${\Re }_0^c$.

Here, the numerical simulation represented by Figure 5 shows that whenever we maximize the syphilis protection rate τ₃, the syphilis reproduction number ${\Re }_0^s$ decreases, implying that the syphilis spreading rate decreases. Whenever τ₃ > 0.686 then ${\Re }_0^s<1$, biologically, it means the syphilis infection eradicate from the community.

FIGURE 5

Figure 5. Effect of syphilis protection rate τ₃ on ${\Re }_0^s$.

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

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

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.

Acknowledgments

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.

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