EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 2, 2022, 537-556 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bifurcation Analysis of a Mathematical Model for the Covid-19 Infection among Pregnant and Non-Pregnant Women Jonathan Tsetimi1,∗, Marcus Ifeanyi Ossaiugbo1, Augustine Atonuje1 1 Department of Mathematics, Delta State University, Abraka, Delta State, Nigeria Abstract. A mathematical study and analysis of the covid-19 disease is essential in the control and eradication of the dreadful covid-19 infection. This research work is focused on the covid-19 infection among the women population. The women population was divided into seven compart- ments. These compartments were built into a deterministic model presented as a system of ordinary differential equations. Basic mathematical analyses such as positivity of solution, the disease-free equilibrium and the basic reproduction number, were performed on the model. The model assumes only positive solutions and the total population size is bounded. The next generation matrix ap- proach was employed in generating the basic reproduction number R0. The sensitivity analysis revealed that among the most sensitive parameters of R0 are the effective contact rate for covid-19 transmission, the modification parameter accounting for increased susceptibility to covid-19 infec- tion by pregnant women, and the transmission coefficient of the infectious pregnant women. The bifurcation analysis revealed a forward bifurcation which guarantees the stability of the disease-free equilibrium when R0 is lowered below unity. Numerical simulations, using Mathematica Version 12.0 package, are given to show the effects of the most sensitive parameters of the basic reproduc- tion number on the number of infectious cases of covid-19. 2020 Mathematics Subject Classifications: 34A34, 37D99, 37G99, 03C99 Key Words and Phrases: Covid-19, Bifurcation, Sensitivity, Simulation, Mathematical model 1. Introduction Maternal health is affected both directly and indirectly by the Covid-19 pandemic, and pregnant individuals were found to be at a heightened risk of more severe symptoms than people who are not pregnant [9]. Studies have suggested that pregnant women are more susceptible to COVID-19 compared to non-pregnant women [10]. [11] also revealed that ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i2.4312 Email addresses: jtsetimi@delsu.edu.ng; tsetimi@yahoo.com (J. Tsetimi), iossaiugbo@delsu.edu.ng; marcusossaiugbo@gmail.com (M. I. Ossaiugbo), atonuje@delsu.edu.ng (A. Atonuje) https://www.ejpam.com 537 © 2022 EJPAM All rights reserved. J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 538 COVID-19 may predispose pregnant women to higher risks of severe disease and poorer neonatal outcome. They also highlighted the importance of appropriate counselling by clinicians, and focused clarification on the effect of COVID-19 among pregnant women to be made for mental wellbeing and psychological support of pregnant women. [6] added that in unvaccinated pregnant women, Covid-19 can cause deadly harm to babies as the disease can attack and destroy the placenta. [10] suggested that education of pregnant women on COVID-19 preventive practices should be intensified at health facilities while improving upon the water, sanitation and hygiene need particularly in rural communities. The severity of the COVID-19 disease is significantly associated with increased cesarean delivery, admission to the intensive care unit, and cases of maternal mortality [18]. Mathematical modeling of infectious diseases has proven to be an effective tool in the management, control and eradication of infectious diseases. A mathematical model of an infectious disease gives clear dynamics of the disease. The relationship between the vari- ables and parameters involved in the dynamics of an infectious disease are clearly stated in its mathematical model, and are often supported by a schematic diagram. Different mathematical models exist for various infectious infectious diseases and these models often study various aspects of the diseases. Some of the models which exist in literature for the covid-19 infectious disease include [1, 4, 15]. [4] estimated the effective reproduction num- ber of COVID-19 from the next generation in the presence of intervention and opined that collaborated efforts from individual, media and integrated government interventions were the only way to reduce the effective reproduction number to a figure below 1. They added that individuals must adhere to the prevention protocols and the media must intensify its daily report on COVID-19 cases to improve behavioral change, which have beneficial impact on the incidence cases. [1] proposed a system of five non-linear ordinary differential equations to study the COVID-19 infection and mathematical analysis of the model were carried out, where boundedness, computation of equilibria, calculation of the basic reproduction ratio and stability analysis of the equilibria were carried out. [15] showed the suitability of proposed COVID-19 model for the outbreak that occured in Wuhan, China. The model divides the entire population into eight compartments. By the parameter values used in their research, they obtained the basic reproduction number as R0 = 0.945. [23] implemented an attractive reliable analytical technique for constructing numerical solutions for the fractional Lienard’s model enclosed with suitable nonhomogeneous initial conditions, which are often designed to demonstrate the behavior of weakly nonlinear waves arising in the oscillating circuits. [24] proposed the Atangana–Baleanu fractional methodology for fathoming the Van der Pol damping model by using the reproducing kernel algorithm. In this research work, we shall be considering the bifurcation and sensitivity analy- ses of a deterministic model for the covid-19 infection among pregnant and non-pregnant women. The effects of the effective contact rate, delivery rate, transmission coefficient of the class of infectious pregnant women and modification parameter accounting for in- creased susceptibility to covid-19 infection by pregnant women over the population of infectious individuals shall also be considered. J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 539 2. Model Description and Formulation In the model, we let Sp(t) denote the total number of pregnant women who are sus- ceptible to the covid-19 infection at time t, while Sn(t) represents the population of non- pregnant women who are susceptible to the covid-19 infection at time t. The total number of pregnant women who are infectious of the covid-19 disease at time t is represented by Ip(t), while the population of non-pregnant women at time t is represented by In(t). H(t) denotes the population of women who are both infectious of the disease and are hospi- talised at time t. We let the total number of pregnant women who have recovered from the disease at time t be represented by Rp(t) and the total population of non-pregnant women at time t be represented by Rn(t). The model divides the total women population into compartments namely Sn, In, Rn,H, Sp, Ip, Rp. The total population is given as N(t) = Sn(t) + In(t) +Rn(t) +H(t) + Sp(t) + Ip(t) +Rp(t). (1) The model is based on the following assumptions: 1. Both infectious expectant and infectious non-expectant mothers are kept in the same health care facility. 2. Disease-induced death rate is same for both pregnant and non-pregnant women. 3. The conception rate is the same for the susceptible, infectious and recovered com- partments. 4. The delivery rate is the same for the susceptible, infectious and recovered compart- ments. The susceptible class for non-pregnant women Non-pregnant women are recruited into the susceptible class Sn at the rate Λ. This class is also increased by women from the class Sp who have been delivered of their babies at the rate θ. Non-pregnant women who have recovered from the covid-19 disease can become susceptible to the disease at the rate Ψ. Women leave this compartment when they become infected at the rate χ and they join the infectious class In for the non-pregnant women. The rate χ is frequency of occurrence of new instances of the covid-19 infection within the given population under consideration during a specific period of time. This parameter is very important in this model since it is directly proportional to the average number of infections. χ is the force of infection of the covid-19 disease and it is defined as χ = k(1−$) In+%Ip+ιH N where k is the effective contact rate for covid-19 transmission, $ is the rate of adherence of women to government’s regulations on vaccination, social distancing, hand-washing and the use of nose mask and sanitizer. A high value of k means an increased force of infection and this can boost the number of infectious cases of J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 540 the covid-19 disease among the women population. % and ι are transmission coefficients for the pregnant infectious women and the hospitalised women respectively. The value of ι is expected to be relatively small compared to the value of % since the women in the hospitalised compartment are constrained to follow strict preventive measures and possible isolation, hence their transmission coefficient must be relatively low. Women who become pregnant also leave this compartment and are joined to the susceptible class (Sp) for pregnant women. The population of this class Sn is also reduced by natural death at the rate µ. Thus, the equation for this class is given as dSn dt = Λ+ θSp +ΨRn − (δ + µ+ χ)Sn. (2) The infectious class for non-pregnant women The infectious class In of non-pregnant women is increased by non-pregnant women who have caught the covid-19 disease by the force of infection χ and are now infectious. Infectious pregnant women who have been delivered of their babies also join this class at the rate θ. Women in this class conceive at the rate δ and move into the infectious class for pregnant women. Non-pregnant infectious women are hospitalised/quarantined at the rate ε. The recovery rate of women in this infectious class Ip is η. The higher the value of η, the faster the elimination of the covid-19 infection from the pregnant women population. The size of this class is also reduced by disease-induced deaths at the rate ς. Natural death also occurs in this infectious class at the rate µ. The equation for the compartment is given as dIn dt = χSn + θIp − (δ + ε+ η + µ+ ς)In. (3) The recovered class for non-pregnant women Individuals from the infectious class In and the hospitalised class H join the recovered class Rn at the rates η and pϕ respectively, where p is the proportion of hospitalised individuals who have recovered from the disease but are not pregnant. Pregnant women in the class Rp who have recovered from the disease and have been delivered of their babies are also added to this class Rn. As a result of loss of immunity, individuals in this compartment return to the class Sn at the rate Ψ. Individuals also leave this class for the class Rp due to conception at the rate δ. Natural death occurs here at the rate µ. dRn dt = θRp + pϕH + ηIn − (δ + µ+Ψ)Rn. (4) The hospitalised class The population of this class is increased by individuals from the compartments In and Ip. Individuals in the infectious classes In and Ip are hospitalised at the rates ε and α respectively. The recovery rate of these women who have been hospitalised is ϕ, while J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 541 disease-induced death also occurs in this class at the rate ς. Individuals in this class can also die due to natural death at the rate µ. dH dt = εIn + αIp − (µ+ ς + ϕ)H. (5) The susceptible class for pregnant women The recruitment rate of pregnant women into the susceptible class Sp is denoted by Π. This compartment is also increased by pregnant women who have recovered from the disease but lost their immunity. This women return to the susceptible class at the rate σ. Individuals in the class Sn move into the class Sp due to conception at the rate δ. Individuals in this compartment can become infected at the rate βχ and join the infectious class Ip where β is the modification parameter accounting for increased susceptibility to covid-19 infection by pregnant women. This modification parameter of the force of infection is very important when considering the dynamics of the covid-19 infection among pregnant women. It is expected that β raises the value of the force of infection for the pregnant women. Women in this compartment can also move into the class Sn due to delivery at the rate θ. Natural death also occurs in this class at the rate µ. dSp dt = Π+ δSn + σRp − (βχ+ θ + µ)Sp. (6) The infectious class for pregnant women The population of individuals in this class Ip is increased by individuals from the classes Sp and In at the rates βχ and δ respectively. Individuals in this class are hospitalised at the rate α. Infectious individuals who have been delivered of their babies leave this class and join the class In at the rate θ. The recovery rate of infectious pregnant women is γ. Pregnant women in this compartment die due to the disease and also due to natural death at the rates ς and µ respectively. The severity of the covid-19 infection among pregnant women directly influences the value of ς. dIp dt = δIn + βχSp − (α+ γ + θ + µ+ ς)Ip. (7) Recovered class for pregnant women This class is increased by individuals from the classes H, Rn and Ip at the rates (1−p)ϕ, δ and γ respectively, where the parameters have been defined earlier and are clearly stated in table 1. Individuals in this class who have been delivered of their babies join the class Rn at the rate θ. Recovered pregnant women can return to the susceptible class Sp at the rate σ, due to loss of immunity. Natural death also occurs in this compartment at the rate µ. dRp dt = (1− p)ϕH + δRn + γIp − (θ + µ+ σ)Rp. (8) J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 542 Thus the mathematical model is given as: dSn dt = Λ++θSp +ΨRn − (δ + µ+ χ)Sn dIn dt = χSn + θIp − (δ + ε+ η + µ+ ς)In dRn dt = ηIn + θRp + pϕH − (δ + µ+Ψ)Rn dH dt = εIn + αIp − (µ+ ς + ϕ)H dSp dt = Π+ δSn + σRp − (βχ+ θ + µ)Sp dIp dt = βχSp + δIn − (α+ γ + θ + µ+ ς)Ip dRp dt = (1− p)ϕH + δRn + γIp − (θ + µ+ σ)Rp χ = k(1−$) In+%Ip+ιH N N(t) = Sn(t) + In(t) +Rn(t) +H(t) + Sp(t) + Ip(t) +Rp(t) (9) The schematic diagram is shown in figure 1, while the description of state variables and parameters are given in table 1. Figure 1: Schematic Diagram J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 543 Table 1: Variables and Parameters Description Variables/ Parameters Descriptions Sn Susceptible class for non-pregnant women In Infectious class for non-pregnant women Rn Recovered class for non-pregnant women H Class of hospitalised individuals SP Class of susceptible pregnant women IP Class of infectious pregnant women RP Class of recovered pregnant women Λ Recruitment rate into the susceptible class for non-pregnant women Π Recruitment rate into the susceptible class for pregnant women χ Force of infection κ Effective contact rate for covid-19 transmission $ Rate of adherence of individuals to government’s regulations like the compulsory wearing of good nose masks, washing of hands, social distancing, etc. β Modification parameter accounting for increased susceptibility to covid-19 infection by pregnant women µ Natural death rate ς Disease-induced death rate ε Rate at which infectious non-pregnant women move into the hospitalised class α Rate at which infectious pregnant women move into the class of hospitalised individuals η Recovery rate of non-pregnant infectious individuals γ Recovery rate of infectious pregnant women ϕ Recovery rate of hospitalised individuals p Proportion of hospitalised individuals who recover from the disease but are not pregnant Ψ Rate at which recovered non-pregnant women become susceptible σ Rate at which pregnant women who have recovered from the covid-19 infection move into the susceptible class δ Conception rate θ Delivery rate ι Transmission coefficient of the hospitalised class % Transmission coefficient of the class of infectious pregnant women ζ Transmission coefficient of the class of infectious non-pregnant mothers 3. Basic Analysis of the Model In order to establish the relevance of the disease model (9), we shall perform some basic analyses such as existence of the invariant region and positivity of solutions, disease-free J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 544 equilibrium, disease-endemic equilibrium and basic reproduction number. 3.1. Invariant Region & Positivity of solutions Consider the total population N at time t is given by N(t) = Sn(t) + In(t) +Rn(t) +H(t) + Sp(t) + Ip(t) +Rp(t). We want to establish that the total population is bounded for infinite time t. Theorem 1. The set Γ = { (Sn, In, Rn,H, Sp, Ip, Rp) ∈ R7 + : 0 ≤ Sn + InH ++Rn +Rp + Sp + Ip = N ≤ Λ µ + Π µ } , is positively-invariant for the model 9. Proof. Since the total population N at time t is given by Γ = { (Sn, In, Rn,H, Sp, Ip, Rp) ∈ R7 + : 0 ≤ Sn + In +H +Rn +Rp + Sp + Ip = N ≤ Λ µ + Π µ } , (10) it follows that dNt dt = −ς (H + In + Ip) + Λ + Π− µNt ≤ Λ +Π− µNt. (11) ∴ dNt dt + µNt ≤ Λ +Π. (12) ∴ Nt ≤ ke−µt + Λ+Π µ . (13) Taking limit as t → ∞, N ≤ Λ µ + Π µ = Λ+Π µ . (14) The inequality 14 which is referred to as the threshold population level shows that if the total women population is greater than the threshold population level, then the total women population reduces asymptotically to the carrying capacity, and if this population is less than the threshold population level, then the solution of the model remains in Γ for all t > 0. Thus, the region Γ is positively invariant � Theorem 2. Suppose Γ = { (Sn, In, Rn,H, Sp, Ip, Rp) ∈ R7 + : Sn(0) > 0, In(0) > 0, Rn(0) > 0,H(0) > 0, Sp(0) > 0, Ip(0) > 0, Rp(0) > 0 } , then the solution set {Sn, In, Rn,H, Sp, Ip, Rp} , is positive for all t ≥ 0. J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 545 Proof. The first equation of the model is dSn dt = Λ+ΨRn − Sn(δ + µ+ χ) + θSp ≥ Sn(−(δ + µ+ χ)). (15) That is dSn dt ≥ Sn(−(δ + µ+ χ)), (16) ln(ln)Sn ≥ − ∫ dt(δ + µ+ χ). (17) ∴ Sn ≥ e ∫ (χ+δ+µ)dt. (18) The constant of integration from the integral is a function of the initial data Sn(0). It follows that Sn > 0. Similarly, we can show that In(t) > 0, Rn(t) > 0,H(t) > 0, Sp(t) > 0, Ip(t) > 0, Rp(t) > 0 for all time t ≥ 0 � 3.2. Disease-free equilibrium By setting the right hand side of the model to zero we obtain the following disease-free equilibrium: E0 = ( θ(Λ + Π) + Λµ µ(δ + θ + µ) , 0, 0, 0, δ(Λ + Π) + µΠ µ(δ + θ + µ) , 0, 0 ) . (19) 3.3. Basic reproduction number The infected classes of the model (1) are In, H and Ip. We separate the right hand side of the equations of these classes as follows χSn 0 βχSp −  In(δ + ε+ η + µ+ ς)− θIp H(µ+ ς + ϕ)− εIn − αIp Ip(α+ γ + θ + µ+ ς)− δIn  . (20) Let F =  χSn 0 βχSp  ,V =  In(δ + ε+ η + µ+ ς)− θIp H(µ+ ς + ϕ)− εIn − αIp Ip(α+ γ + θ + µ+ ς)− δIn  . (21) The rows of matrix F shows the new infection terms appearing in each infected class of the model, while the rows of matrix V shows the transfer terms (old infection terms) also appearing in each infected class of the model. This notation is in line with Driessche and Watmough [19]. This is the next generation matrix approach for finding the basic reproduction number R0. The Jacobian of the matrices F and V at the disease-free equilibrium (E0) are respectively given as F =  − (k($−1))(θ(Λ+Π)+Λµ) (Λ+Π)(δ+θ+µ) − (kι($−1))(θ(Λ+Π)+Λµ) (Λ+Π)(δ+θ+µ) −%(k($−1))(θ(Λ+Π)+Λµ) (Λ+Π)(δ+θ+µ) 0 0 0 − (kβ($−1))(δ(Λ+Π)+µΠ) (Λ+Π)(δ+θ+µ) − (kβι($−1))(δ(Λ+Π)+µΠ) (Λ+Π)(δ+θ+µ) −%(kβ($−1))(δ(Λ+Π)+µΠ) (Λ+Π)(δ+θ+µ)  , (22) J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 546 V =  δ + ε+ η + µ+ ς 0 −θ −ε µ+ ς + ϕ −α −δ 0 α+ γ + θ + µ+ ς  , (23) V −1 =  α+γ+θ+µ+ς (α+γ+θ+µ+ς)(δ+ε+η+µ+ς)−δθ 0 θ (α+γ+θ+µ+ς)(δ+ε+η+µ+ς)−δθ ε(α+γ+θ+µ+ς)+αδ (µ+ς+ϕ)((α+γ+θ+µ+ς)(δ+ε+η+µ+ς)−δθ) 1 µ+ς+ϕ α(δ+ε+η+µ+ς)+εθ (µ+ς+ϕ)((α+γ+θ+µ+ς)(δ+ε+η+µ+ς)−δθ) δ (α+γ+θ+µ+ς)(δ+ε+η+µ+ς)−δθ 0 δ+ε+η+µ+ς (α+γ+θ+µ+ς)(δ+ε+η+µ+ς)−δθ  , (24) FV −1 =  a b c 0 0 0 d e f  , (25)  a = (k($−1))(θ(Λ+Π)+Λµ) ( − ι(ε(α+γ+θ+µ+ς)+αδ) µ+ς+ϕ −α−γ−δ%−θ−µ−ς ) (Λ+Π)(δ+θ+µ)((α+γ+θ+µ+ς)(δ+ε+η+µ+ς)−δθ) , b = − (kι($−1))(θ(Λ+Π)+Λµ) (Λ+Π)(δ+θ+µ)(µ+ς+ϕ) , c = (k($−1))(θ(Λ+Π)+Λµ) ( − ι(α(δ+ε+η+µ+ς)+εθ) µ+ς+ϕ −%(δ+ε+η+µ+ς)−θ ) (Λ+Π)(δ+θ+µ)((α+γ+θ+µ+ς)(δ+ε+η+µ+ς)−δθ) , d = (kβ($−1))(δ(Λ+Π)+µΠ) ( − ι(ε(α+γ+θ+µ+ς)+αδ) µ+ς+ϕ −α−γ−δ%−θ−µ−ς ) (Λ+Π)(δ+θ+µ)((α+γ+θ+µ+ς)(δ+ε+η+µ+ς)−δθ) , e = − (kβι($−1))(δ(Λ+Π)+µΠ) (Λ+Π)(δ+θ+µ)(µ+ς+ϕ) , f = (kβ($−1))(δ(Λ+Π)+µΠ) ( − ι(α(δ+ε+η+µ+ς)+εθ) µ+ς+ϕ −%(δ+ε+η+µ+ς)−θ ) (Λ+Π)(δ+θ+µ)((α+γ+θ+µ+ς)(δ+ε+η+µ+ς)−δθ) . (26) The spectral radius of the matrix FV −1 is obtained as: ρ(FV −1) =− ((k($ − 1))(α((βι(Π(δ + µ) + δΛ))(δ + ε+ η + µ+ ς) + (Λ(θ + µ) + θΠ) (ι(δ + ε) + µ+ ς + ϕ)) + βδ2Λµ%+ βδ2Λ%ς + βδ2µΠ%+ βδ2Π%ς + ϕ((β(Π(δ + µ) + δΛ)) (%(δ + ε+ η + µ+ ς) + θ) + (δ%+ θ + µ+ ς)(Λ(θ + µ) + θΠ)) + βµ3Π%+ 2βµ2Π%ς + 2βδµ2Π%+ βδεθιΛ + βδεθιΠ+ βδεΛµ%+ βδεΛ%ς + βδεµΠ%+ βδεΠ%ς + βδηΛµ% + βδηΛ%ς + βδηµΠ%+ βδηΠ%ς + βδθΛµ+ βδθΛς + βδθµΠ+ βδθΠς + βδΛµ2% + 2βδΛµ%ς + βδΛ%ς2 + 3βδµΠ%ς + βδΠ%ς2 + βεµ2Π%+ βεθιµΠ+ βεµΠ%ς + βηµ2Π% + βηµΠ%ς + βθµ2Π+ βθµΠς + βµΠ%ς2 + (ει+ µ+ ς + ϕ)(γ(θ(Λ + Π) + Λµ)) + δθΛµ%+ δθΛ%ς + δθµΠ%+ δθΠ%ς + δΛµ2%+ δΛµ%ς + εθ2ιΛ + εθ2ιΠ+ 2εθιΛµ + εθιΛς + εθιµΠ+ εθιΠς + ειΛµ2 + ειΛµς + θ2Λµ+ θ2Λς + θ2µΠ+ θ2Πς + θµ2Π + 2θΛµ2 + θΛς2 + 3θΛµς + 2θµΠς + θΠς2 + Λµ3 + 2Λµ2ς + Λµς2))/ ((Λ + Π)(δ + θ + µ)(µ+ ς + ϕ)(α(δ + ε+ η + µ+ ς) + γ(δ + ε+ η + µ+ ς) + δ(µ+ ς) + (θ + µ+ ς)(ε+ η + µ+ ς))). Substituting the parameter values in table 2 into the spectral radius ρ(FV −1), we obtain the basic reproduction R0 as J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 547 R0 = ρ(FV −1) = 0.58599. (27) The basic reproduction number represents the average number of secondary cases of the covid-19 infection generated by an infected woman in a completely susceptible women population. If the initial sizes of the compartments of the model are within the basin of attraction of the disease-free equilibrium, then the covid-19 disease can be eliminated from the population when R0 < 1. In this case we say that the disease-free equilibrium is locally asymptotically stable if the basic reproduction number is less than one. It is unstable if R0 > 1. Table 2: Parameter Values Parameters Value Source κ 1.082 [5] β 1.05 Assumed. δ 0.245 [20] µ 0.0182 [3] p 0.6 Assumed Λ 0.0755 Estimated from [5] and [20] ς 0.022 [3] Ψ 0.00755 Assumed ε 0.154 citeEnahoroOluwaseunCalistusAbba ι 0.1 Assumed η 1/7 per day [16] θ 36.440 births per 1000 [12] $ 0.45 Assumed % 0.04 Assumed ϕ 1/14 per day [16, 22] Π 0.0245 Estimated from [5] and [20] σ 0.00245 Assumed γ 1/7 per day [16, 22] α 0.154 [5] ζ 0.07 Assumed 4. Sensitivity Analysis We want to show the sensitivity of the parameters of the basic reproduction number on the value of R0. We employ the approach used by Kizito and Tumwiine [7] to compute the sensitivity of the parameters of R0. The sensitivity of a parameter, say µ, of R0 is defined as ζR0 µ = ∂R0 ∂µ × µ R0 . (28) J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 548 The sensitivity analysis analysis of the parameters of the basic reproduction number R0 reveals the level of sensitivity of each parameter of R0 to very small changes in their values. This analysis is important because for proper disease control and possible elimination of the covid-19 infection, the sensitivity analysis helps to channel employed control measures to some targeted parameters. If a parameter of R0 has high positive sensitivity, then any small increment in the value of the parameter can greatly increase the value of R0 which might cause stability for the disease endemic equilibrium of the the covid-19 infection. Table 3: Sensitivity of Parameters of R0. Parameters Value Sesitivity κ 1.082 +1.0000 β 1.05 +0.6177 ι 0.1 +0.4330 θ 36.440 births per 1000 +0.3075 % 0.04 +0.1071 α 0.154 +0.0825 Λ 0.0755 +0.0173 Π 0.0245 -0.0173 µ 0.0182 -0.0683 ε 0.154 -0.0958 η 1/7 per day -0.1477 ς 0.022 -0.1491 γ 1/7 per day -0.2662 ϕ 1/14 per day -0.2771 δ 0.245 -0.3859 $ 0.45 -0.8182 The sensitivity indices are shown in Table 3. From the sensitivity analysis, it is seen that the most sensitive parameters of the basic reproduction number, R0, are the effec- tive contact rate κ for covid-19 transmission, the modification parameter β accounting for increased susceptibility to covid-19 infection by pregnant women, the transmission co- efficient ι of the hospitalised class, the rate of delivery θ of newborns, the transmission coefficient % of the infectious pregnant women, and amongst the less sensitive parameters are the conception rate δ, the recovery rate ϕ of hospitalised individuals, the recovery rate γ of infectious pregnant women. 5. Bifurcation Analysis We now examine the type of bifurcation exhibited by the model (9). By this knowledge, we are equipped to know if fixing R0 < 1 is enough to ‘keep-off’ the disease-endemic equilibrium. The proof is established based on the Centre Manifold Theorem as presented by Castillo-Chavez and Song [2]. J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 549 Theorem 3. [2]: Consider the following general system of ODEs with a parameter φ. dy dt = f(y, φ), f : Rn × R and f ∈ C2 (Rn × R) , (29) where 0 is an equilibrium point of the system (that is,f(0, φ) ≡ 0 for all φ and assume A1: A = Dyf(0, 0) = ( ∂fi ∂yj (0, 0) ) is the linearization matrix of the system (29) around the equilibrium point 0 with φ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts; A2: Matrix A has a right eigenvector w and a left vector v (each corresponding to the zero eigenvalue). Let fk be the kth component of f and a = n∑ i,j,k=1 vkwiwj ∂2fk ∂xi∂xj (0, 0), (30) b = n∑ i,k=1 vkwi ∂2fk ∂xi∂φ (0, 0). (31) The local dynamics of the system (29) around 0 is totally determined by the signs of a and b: 1 a > 0, b > 0. When φ < 0 with |φ| ≤ 1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0 < φ � 1, 0 is unstable and there exists a negative and locally asymptotically stable equilibrium. 2 a < 0, b < 0.When φ < 0 with |φ| � 1, 0 is unstable; when 0 < φ � 1, 0 is locally asymptotically stable, and there exists a negative unstable equilibrium. 3 a > 0, b < 0. When φ < 0 with |φ| � 1, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0 < φ � 1, 0 is stable, and a positive unstable equilibrium appears. 4 a < 0, b > 0.When φ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable. Particularly, if a > 0 and b > 0, then a backward bifurcation occurs at φ = 0. Proof. We set Sn = x1, In = x2, Rn = x3,H = x4, Sp = x5, Ip = x6, Rp = x7. (32) J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 550 Thus, the model (9) becomes · x1 = Λ− x1(δ + µ+ χ) + θx5 + x3Ψ · x2 = −x2(δ + ε+ η + µ+ ς) + θx6 + χx1 · x3 = pϕx4 − x3(δ + µ+Ψ) + ηx2 + θx7 · x4 = αx6 + εx2 − x4(µ+ ς + ϕ) · x5 = Π− x5(βχ+ θ + µ) + δx1 + σx7 · x6 = −x6(α+ γ + θ + µ+ ς) + βχx5 + δx2 · x7 = (1− p)ϕx4 + γx6 + δx3 − x7(θ + µ+ σ) (33) Now, the Jacobian of the system (33) evaluated at the disease-free equilibrium (E0), is given by: JE0 =  −δ − µ (k($−1))(θ(Λ+Π)+Λµ) (Λ+Π)(δ+θ+µ) Ψ (kι($−1))(θ(Λ+Π)+Λµ) (Λ+Π)(δ+θ+µ) θ %(k($−1))(θ(Λ+Π)+Λµ) (Λ+Π)(δ+θ+µ) 0 0 −δ − ε− η − (k($−1))(θ(Λ+Π)+Λµ) (Λ+Π)(δ+θ+µ) − µ− ς 0 − (kι($−1))(θ(Λ+Π)+Λµ) (Λ+Π)(δ+θ+µ) 0 θ − %(k($−1))(θ(Λ+Π)+Λµ) (Λ+Π)(δ+θ+µ) 0 0 η −δ − µ−Ψ pϕ 0 0 θ 0 ε 0 −µ− ς − ϕ 0 α 0 δ (kβ($−1))(δ(Λ+Π)+µΠ) (Λ+Π)(δ+θ+µ) 0 (kβι($−1))(δ(Λ+Π)+µΠ) (Λ+Π)(δ+θ+µ) −θ − µ %(kβ($−1))(δ(Λ+Π)+µΠ) (Λ+Π)(δ+θ+µ) σ 0 δ − (kβ($−1))(δ(Λ+Π)+µΠ) (Λ+Π)(δ+θ+µ) 0 − (kβι($−1))(δ(Λ+Π)+µΠ) (Λ+Π)(δ+θ+µ) 0 −α− γ − θ − %(kβ($−1))(δ(Λ+Π)+µΠ) (Λ+Π)(δ+θ+µ) − µ− ς 0 0 0 δ ϕ− pϕ 0 γ −θ − µ− σ  Let $ = $∗ be the bifurcation parameter. We consider the parameter values in table 2 and see that $ = 1− 0.9386R0. When R0 = 1, we obtain $ = $∗ = 0.06141. From the characteristic equation of JE0 given by |JE0 − λI| = 0, we obtain the eigenvalues: λ1 = −0.5978 λ2 = −0.3065 λ3 = −0.2996 λ4 = −0.2638 λ5 = −0.0213 λ6 = −0.0182 λ7 = −1.1001× 1017 ≈ 0. (34) The right eigenvector of J(E0)|$=$∗ is given by (g1, g2, g3, g4, g5, g6, g7) T where g1 = −14.6948g2, g2 = g2 > 0, g3 = 10.9219g2, g4 = 9.41465g2, g5 = −98.2523g2, g6 = 5.82431g2, g7 = 66.1568g2. (35) J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 551 The left eigenvector, (h1, h2, h3, h4, h5, h6), is also obtained with h1 = h2 = h5 = 0, h3 = h3 > 0, h4 = 0.403387521843334h3, h6 = −0.1809562986293365h3, h7 = 0.6382904186372395h3. (36) From the equations, a = n∑ i,j,k=1 vkwiwj ∂2fk ∂xi∂xj (S0, 0, 0, 0), (37) b = n∑ i,k=1 vkwi ∂2fk ∂xi∂φ (S0, 0, 0, 0). (38) we now compute a and b. Considering only the non-zero components (h1, h2, h3, h4, h5, h6) of the left eigenvectors, we obtained: a = 18.823262325556414g22h3 > 0, b = −16.73833317110245g2h3 < 0. Since a > 0 and b < 0, when R0 < 1, E0 is stable, and there exists a locally asymp- totically stable negative disease endemic equilibrium; when R0 > 1, E0 is unstable, and a positive stable disease endemic equilibrium appears. This is shown in figure 2. Figure 2: Bifurcation diagram From the bifurcation plot, it is seen that reducing the value of the basic reproduction number below unity, i.e. R0 < 1 is enough to minimize the spread of the disease and also eradicate the disease from the population. The parameters pertaining to pregnant women play a major role in controlling the value of the basic reproduction number. From the sensitivity analysis, it can be seen that reducing the values of the effective contact rate κ for covid-19 transmission, the modification parameter β accounting for increased suscep- tibility to covid-19 infection by pregnant women, Transmission coefficient % of the class J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 552 of infectious pregnant women we can successfully achieve R0 < 1. Below are simulations showing the effect of the parameters κ, β, %, θ, on the number of infections. Figure 3: Effect of κ on the infectious class Figure 4: Effect of β on the infectious class Figure 5: Effect of % on the infectious class J. Tsetimi, M. I. Ossaiugbo, A. Atonuje / Eur. J. Pure Appl. Math, 15 (2) (2022), 537-556 553 Figure 6: Effect of θ on the infectious class 6. Discussion and Conclusion A deterministic mathematical model is considered in this research paper. The model divides the women population into seven mutually exclusive compartments according to their disease status, namely the pregnant susceptible women Sp, the non-pregnant suscep- tible women Sn, the pregnant infectious women Ip, the non-pregnant infectious women In, the recovered pregnant women Rp, the recovered non-pregnant women Rn, and the hospitalized women H. The size of each class is a function of time t. The system of equations is shown in system (9) and the dynamics is clearly displayed in the schematic diagram contained as figure 1. By the basic mathematical analyses, it is revealed that the solution of the system of equations is positive for all time t and the invariant region Γ was given. The proof which was clearly given also confirmed the existence of the invariant region. The disease-free equilibrium was obtained. The expression for the average number of secondary cases generated by a single infectious individual in an entirely susceptible population, which is given as R0, was obtained via the next-generation matrix approach, and the parameter values contained in table 2 resulted in R0 = 0.58599. From the sen- sitivity analysis, it is seen that the most sensitive parameters of the basic reproduction number, R0, are κ, β, ι, θ, %, which are the effective contact rate for covid-19 transmission, the modification parameter accounting for increased susceptibility to covid-19 infection by pregnant women, the transmission coefficient of the hospitalised class, the rate of delivery of newborns, and the transmission coefficient of the infectious pregnant women respec- tively. The less sensitive parameters are the conception rate δ, the recovery rate ϕ of hospitalised individuals, the recovery rate γ of infectious pregnant women. A small incre- ment in the values of the parameters β and %, causes a great increment in the value of R0 and hence in the number of infectious cases. Thus, the values of certain parameters of the class of pregnant women affect the number of infectious cases. The bifurcation analysis which was done via the centre manifold theorem revealed a forward bifurcation. That is, R0 < 1 guarantees the stability of the disease-free equilibrium. This can be achieved by minimizing the most sensitive parameters of the basic reproduction number. The numeri- cal simulations shown in figures 4 – 6 show the effects of the most sensitive parameters of REFERENCES 554 R0 on the infectious class. We observed that an increase in the values of the parameters causes an increase in the size of the infectious class. From figure 3, we see that setting the effective contact rate κ > 1.082, we obtain a spontaneous increase in the number of infec- tious cases. 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