# Hopf-Bifurcation Analysis of Pneumococcal Pneumonia with Time Delays

Authors: Fulgensia Kamugisha Mbabazi; Joseph Y. T. Mugisha; Mark Kimathi
Journal: Abstract and Applied Analysis
 (2019)

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## Abstract

In this paper, a mathematical model of pneumococcal pneumonia with time delays is proposed. The stability theory of delay differential equations is used to analyze the model. The results show that the disease-free equilibrium is asymptotically stable if the control reproduction ratioR0 is less than unity and unstable otherwise. The stability of equilibria with delays shows that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions. The existence of Hopf-bifurcation is investigated and transversality conditions are proved. The model results suggest that, as the respective delays exceed some critical value past the endemic equilibrium, the system loses stability through the process of local birth or death of oscillations. Further, a decrease or an increase in the delays leads to asymptotic stability or instability of the endemic equilibrium, respectively. The analytical results are supported by numerical simulations.


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## Body

## 1. Introduction

Worldwide, pneumococcal pneumonia disease continues to be a major cause of morbidity and mortality in persons of all ages and the leading cause of bacterial childhood disease, despite a century of study and the development of antibiotics and vaccination [1, 2]. Pneumococci are different, with 90 recognized serotypes; several of these serotypes are capable of causing invasive disease [3]. Pneumococcal pneumonia infections may follow a viral infection, like a cold or flu (influenza) [4], and cause the following types of illnesses depending on the affected part of the body: invasive pneumococcal diseases (IPD) such as meningitis, bacteremia, and bacteremic pneumonia; lower respiratory tract infections (e.g., pneumonia), and upper respiratory tract infections (e.g., otitis media and sinusitis) [5]. The wide spread of the disease may be promoted by potentially asymptomatic persons (incubation individuals) [6, 7] and an individual remains in the exposed class for a certain latent period prior to becoming infective [8, 9].Diseases exhibit a lot of economic burden including productivity loss, health care related expenses, losses due to disease related mortality, and loss of employment [10]. Globally, an estimated 14.5 million episodes of serious pneumococcal disease occur each year among children under 5 years of age, resulting in approximately 500,000 deaths [11], most of which occur in low and middle-income countries [12, 13]. Pneumonia is the most common form of severe pneumococcal disease, accounting for 15 % of all deaths of children under 5 years and killing an estimated 922,000 in 2015, and is the leading cause of death in this age group [14].Vaccination is a highly efficient means of preventing diseases and death [15]. A vaccine consists of a killed or weakened form or derivative of the infectious germ. Once administered to a healthy person, the vaccine activates an immune response and makes the body to assume that it is being attacked by a specific organism [16]. Decrease of invasive pneumococcal disease (IPD) has been managed by pneumococcal conjugate vaccines (PCVs), and they are among the many ongoing stories of vaccine successes around the world. One dose of vaccine does not protect all receivers because vaccine-induced immunity is lost after some period of time [17, 18].Time delays are significant in the transmission process of epidemics and arise due to delayed feedback especially the period for waning vaccine-induced immunity, latent period of infection, the infectious period, and the immunity period [19–21]. Among the mathematical tools currently used, delay differential models with time delay have attracted attention in the field of science especially modeling infectious diseases. Delays change the dynamical systems’ stability by giving rise to Hopf-bifurcations [19, 22]. Works done by researchers, for example, [8, 23–26], demonstrate the role played by time delays in different capacities in controlling the spread of infectious diseases. Sharma et al. [27] discussed avian influenza transmission dynamics with two discrete time delays as incubation periods of avian influenza in the human and avian populations and found out that increment in time delays occurrence results into decrease in infected human population.In this paper, we explore the effect of two delays on pneumococcal pneumonia disease. We incorporate a time delay in the latent class because there is delayed time from the time an individual is infected and when one becomes infectious. A second time delay of seeking medical care is included in the infectious class. Not seeking medical attention leaves individuals’ behaviors unchanged not to respond to existing control measures and more individuals become infected.This paper is organized as follows. In Section2, we present the description and formulation of the time delay model of pneumococcal pneumonia dynamics. In Section 3, we present the stability of the steady states. Existence of Hopf-bifurcation is presented in Section 4. In Section 5, numerical simulations and results of the model are presented to support the analytical findings; a discussion is given in Section 6.


## 2. Model Description and Formulation

We formulate a model for the dynamics of the bacterial pneumonia (pneumococcal) in a human population with the total population size at timet, denoted by N(t). The population is subdivided into six mutually exclusive epidemiological classes: susceptible, vaccinated, exposed, carrier, and infected denoted by S(t), V(t), E(t), C(t), and I(t), respectively. The mathematical formulation adopts a mass-action incidence because it is important in deciding the dynamics of epidemic models [36], where the contact rate depends on the size of the total human population [37]. We assume a continuous vaccination strategy that is received by the recruited susceptible individuals at a rate ν and that vaccination does not affect the infectious [38]. We assume vaccination is not 100% efficient, which means there is a chance of being infectious or carrier in small proportions and the force of infection for the vaccinated class is ϑβI(t), where 0≤ϑ<1 is the proportion of the serotype not covered by vaccine [39]. The increase in the number of susceptible individuals comes from a constant recruitment b through birth or migration and recovery of individuals. Several vaccines wane with time, and so vaccinated individuals return to the susceptible compartment, at a waning rate ζ. The susceptible individuals become infected through a force of infection βI(t) and move to the latent class E(t).The latent class,E(t), accounts for a time delay τ1>0 of the exposed individuals, i.e., the period between the time of an infection onset and the time of developing pneumococcal clinical symptoms (assume that an individual is infectious upon exposure to influenza A disease that promotes severe pneumococcal pneumonia). The probability (survivorship function) of an individual surviving the natural mortality through the latent period [t-τ1,t] is e-μτ1 and exposed individuals transfer to the infectious class at a rate γ. Individuals in the carrier class C(t) become symptomatic and join the infected class at a rate ρ.The infectious classI(t) accounts for a time delay τ2>0: the time taken by infected individuals to seek medical care. We assume that infected individuals who survive the natural mortality through the infectious period [t-τ2,t] have a survivorship function e-μτ2. Moreover, infected individuals that delay to seek medical care die of pneumococcal pneumonia at a rate δ. Infectious individuals upon recovery transfer to the susceptible class at a rate ϕ. All classes exhibit a per capita natural mortality rate μ.The description of model variables and parameters is summarized in Tables1 and 2.Table 1
Description of variables.
Variable Description [unit] S(t) Number of susceptible individuals at timet individual V ( t ) Number of vaccinated individuals at timet individual E ( t ) Number of asymptomatic individuals at timet individual C ( t ) Number of individuals with one serotype not covered by the vaccine individual I ( t ) Number of infectious individuals at timet individualTable 2
Description of parameters.
Parameter Description value/unit Source b Recruitment rate 22day-1 estd ν Effective vaccination rate 2.53×10-5 [28] γ Transfer rate fromE to I class 3.3333 × 1 0 - 1   d a y - 1 assumed μ Natural mortality rate from causes unrelated to the infection 2.0547 × 1 0 - 3 [29] δ Disease–induced mortality rate 3.3×10-1day-1 [30] ρ Progression rate fromC to I class 1.096 × 1 0 - 2   d a y - 1 [31] ϕ Per capita rate of recovery 3.5714 × 1 0 - 2 d a y - 1 [32] ζ Waning rate of vaccine 5.4794 × 1 0 - 4 d a y - 1 [32] ϑ Proportion of the sero–type not covered by vaccine 0.54 [33] β Transmission coefficient 1.0102 × 1 0 - 4   d a y - 1 assumed τ 1 Delay for the incubating individuals 1–3 days [34] τ 2 Delay in seeking medical care 2 days [35]The compartmental diagram of the model is shown in Figure1.Figure 1
A schematic diagram showing the dynamics of pneumococcal pneumonia. The dotted lines represent contacts made by individuals in the respective classes and the solid lines show transfer from one class to another.Based on the description of model variables, parameters, and assumptions in Tables1 and 2., the dynamics of the model are governed by the following differential equations:(1)S˙t=b+ζVt+ϕIt-ν+μ+βItSt,V˙t=νSt-μ+ζVt-β1ItVt,E˙t=βItSt-γe-μτ1Et-τ1-μEt,C˙t=β1ItVt-ρ+μCt,I˙t=ρCt+γe-μτ1Et-τ1-δe-μτ2It-τ2-μ+ϕIt,where β1=ϑβ.


### 2.1. Positivity of Solutions

System (1) is a representation of the dynamics of the human populations; thus it is required that all solutions are nonnegative. We use the approach of Bodna [40] and Yang et al. [41]; we let C be a Banach space of continuous real valued functions ψ:[-τ,0]→R+5 equipped with the supremum norm, ψC=supt∈-τ,0⁡{ψ1,ψ2,ψ3,ψ4,ψ5}. The initial conditions of system (1) are represented by(2)St=ψ1t,Vt=ψ2t,Et=ψ3t,Ct=ψ4t,It=ψ5t,-τ≤t≤0,where τ=max⁡{τ1,τ2} and ψ=(ψ1,ψ2,ψ3,ψ4,ψ5)T∈C, such that ψi(t)=ψi(0)≥0(i=1,2,3,4,5). The following Lemma establishes the positivity of the solutions of system (1).Lemma 1.
Any solution of trajectories (1) with ψi(t)>0; t∈[-τ,0] remains positive whenever it exists.Proof.
SupposeS(t) was to lose positivity on some local existence interval [0,T) for some constant T>0; there would be a time at t1=sup⁡{t>0:S(t)>0} such that S(t1)=0.
From the first equation of system (1), it follows that(3)b+ζVt+ϕIt-ν+μSt-βItSt>0.This implies that S(t)<0 for t∈(t1-ε,t1), where ε is an arbitrary small positive constant. This leads to a contradiction; it thus follows that S(t) is always positive. Hence from the fundamental theory of differential equations, it is shown that there exists a unique solution for S(t) of system (1) with initial data in R+5 as follows:(4)ddtSte∫0tν+μ+βItdξ=e∫0tν+μ+βItdξb+ζVt+ϕIt,St=∫0t(b+ζVσ+ϕIσe-ν+μt-∫σtβIξdξdσ+ψ10e-ν+μt-∫0tβIξdξ.Therefore,(5)St1=ψ10e-ν+μt1-∫0t1βIξdξ+∫0t1(b+ζVσ+ϕIσe-ν+μt1-∫σt1βIξdξdσ>0.Since S(t1)>0, then S(t)>0, t≥0. This completes the proof.Similarly, it can be shown that(6)Vt2=ψ20e-μ+ζt2-∫0t2βIξdξ+∫0t2e∫σt2μ+ζ+β1IξdξνSσdσ>0.(7)Et3=e-μt3ψ30+e-μt3∫0t3eμξβIξSξ-γe-μτ1Eξ-τ1dξ>0.(8)Ct4=e-ρ+μt4ψ40+∫0t4β1IξVξeρ+μξdξ>0,and(9)It5=ψ50e-μ+ϕt5+e-μ+ϕt5∫0t5ρCσ+γe-μτ1Eσ-τ1-δe-μτ2Iσ-τ2eμ+ϕσdσTherefore, from the above integral forms of (5) to (9) all solution trajectories are positive for all time t>0 on [0,+∞].


### 2.2. Boundedness

For boundedness of system (1) with initial condition (2), we consider the following lemma.Lemma 2.
The closed set(10)Ωd=St,Vt,Et,Ct,It,Rt∈R+5:0≤St,Vt,Et,Ct,It;St+Vt+Et+Ct+It≤bμis positively invariant and absorbing with respect to the set of DDEs (1).Proof.
Summing all equations in system (1) yields(11)dNdt=b-μNt-δe-μτ2It.Therefore, dN/dt≤b-μN(t) which implies that dN/dt≤0 if N(t)≥b/μ. Using the standard comparison test in [42], we get N(t)≤N(0)e-μt+b/μ(1-e-μt). Particularly, N(t)≤b/μ if N(0)≤b/μ for all time t>0; hence Ωd is positively invariant. Further, if N(t)≥b/μ, then either the solution enters at finite time nor N(t) is close to π/μ and the infected variables E,C, and I tend to zero. Therefore, Ωd is attracting implying that all solutions in R+5 finally enter Ωd. Consequently, in Ωd, system (1) is mathematically and epidemiologically well-posed.


### 2.3. The Control Reproduction Ratio

The basic reproduction ratio identifies the number of secondary infections from the infected source and plays an important role in understanding the development of epidemics with a vaccination program in place. The control reproduction ratioR0 is computed using an approach in [43] and is given by(12)R0=R0u+R0v,where(13)R0u=βγe-μτ1S0μ+γe-μτ1ϕ+μ+δe-μτ2,R0v=βρϑV0ρ+μϕ+μ+δe-μτ2,provided the validity of (μ+ζ)(ν+μ)>ζν holds.The quantityR0u measures the expected number of secondary cases generated by an index case for the susceptible individuals and R0v represents new cases arising from the vaccination program.Remark 3.
The control reproduction ratio with no delays (τ1=0,τ2=0) is given by(14)R0u=βγS0μ+γϕ+μ+δ,R0v=βρϑV0ρ+μϕ+μ+δ.


## 2.1. Positivity of Solutions

System (1) is a representation of the dynamics of the human populations; thus it is required that all solutions are nonnegative. We use the approach of Bodna [40] and Yang et al. [41]; we let C be a Banach space of continuous real valued functions ψ:[-τ,0]→R+5 equipped with the supremum norm, ψC=supt∈-τ,0⁡{ψ1,ψ2,ψ3,ψ4,ψ5}. The initial conditions of system (1) are represented by(2)St=ψ1t,Vt=ψ2t,Et=ψ3t,Ct=ψ4t,It=ψ5t,-τ≤t≤0,where τ=max⁡{τ1,τ2} and ψ=(ψ1,ψ2,ψ3,ψ4,ψ5)T∈C, such that ψi(t)=ψi(0)≥0(i=1,2,3,4,5). The following Lemma establishes the positivity of the solutions of system (1).Lemma 1.
Any solution of trajectories (1) with ψi(t)>0; t∈[-τ,0] remains positive whenever it exists.Proof.
SupposeS(t) was to lose positivity on some local existence interval [0,T) for some constant T>0; there would be a time at t1=sup⁡{t>0:S(t)>0} such that S(t1)=0.
From the first equation of system (1), it follows that(3)b+ζVt+ϕIt-ν+μSt-βItSt>0.This implies that S(t)<0 for t∈(t1-ε,t1), where ε is an arbitrary small positive constant. This leads to a contradiction; it thus follows that S(t) is always positive. Hence from the fundamental theory of differential equations, it is shown that there exists a unique solution for S(t) of system (1) with initial data in R+5 as follows:(4)ddtSte∫0tν+μ+βItdξ=e∫0tν+μ+βItdξb+ζVt+ϕIt,St=∫0t(b+ζVσ+ϕIσe-ν+μt-∫σtβIξdξdσ+ψ10e-ν+μt-∫0tβIξdξ.Therefore,(5)St1=ψ10e-ν+μt1-∫0t1βIξdξ+∫0t1(b+ζVσ+ϕIσe-ν+μt1-∫σt1βIξdξdσ>0.Since S(t1)>0, then S(t)>0, t≥0. This completes the proof.Similarly, it can be shown that(6)Vt2=ψ20e-μ+ζt2-∫0t2βIξdξ+∫0t2e∫σt2μ+ζ+β1IξdξνSσdσ>0.(7)Et3=e-μt3ψ30+e-μt3∫0t3eμξβIξSξ-γe-μτ1Eξ-τ1dξ>0.(8)Ct4=e-ρ+μt4ψ40+∫0t4β1IξVξeρ+μξdξ>0,and(9)It5=ψ50e-μ+ϕt5+e-μ+ϕt5∫0t5ρCσ+γe-μτ1Eσ-τ1-δe-μτ2Iσ-τ2eμ+ϕσdσTherefore, from the above integral forms of (5) to (9) all solution trajectories are positive for all time t>0 on [0,+∞].


## 2.2. Boundedness

For boundedness of system (1) with initial condition (2), we consider the following lemma.Lemma 2.
The closed set(10)Ωd=St,Vt,Et,Ct,It,Rt∈R+5:0≤St,Vt,Et,Ct,It;St+Vt+Et+Ct+It≤bμis positively invariant and absorbing with respect to the set of DDEs (1).Proof.
Summing all equations in system (1) yields(11)dNdt=b-μNt-δe-μτ2It.Therefore, dN/dt≤b-μN(t) which implies that dN/dt≤0 if N(t)≥b/μ. Using the standard comparison test in [42], we get N(t)≤N(0)e-μt+b/μ(1-e-μt). Particularly, N(t)≤b/μ if N(0)≤b/μ for all time t>0; hence Ωd is positively invariant. Further, if N(t)≥b/μ, then either the solution enters at finite time nor N(t) is close to π/μ and the infected variables E,C, and I tend to zero. Therefore, Ωd is attracting implying that all solutions in R+5 finally enter Ωd. Consequently, in Ωd, system (1) is mathematically and epidemiologically well-posed.


## 2.3. The Control Reproduction Ratio

The basic reproduction ratio identifies the number of secondary infections from the infected source and plays an important role in understanding the development of epidemics with a vaccination program in place. The control reproduction ratioR0 is computed using an approach in [43] and is given by(12)R0=R0u+R0v,where(13)R0u=βγe-μτ1S0μ+γe-μτ1ϕ+μ+δe-μτ2,R0v=βρϑV0ρ+μϕ+μ+δe-μτ2,provided the validity of (μ+ζ)(ν+μ)>ζν holds.The quantityR0u measures the expected number of secondary cases generated by an index case for the susceptible individuals and R0v represents new cases arising from the vaccination program.Remark 3.
The control reproduction ratio with no delays (τ1=0,τ2=0) is given by(14)R0u=βγS0μ+γϕ+μ+δ,R0v=βρϑV0ρ+μϕ+μ+δ.


## 3. Stability of Equilibria

Let(S∗,V∗,E∗,C∗,I∗) be the corresponding partial populations at the eventual equilibrium point. Given that the values of the partial populations at the equilibrium are stable, the delay-dependency vanishes so that limt→∞⁡I(t-τ2)=limt→∞⁡I(t)=I∗ and limt→∞⁡E(t-τ1)=limt→∞⁡E(t)=E∗, such that, at equilibrium, we have(15)b+ζV∗+ϕI∗-ν+μ+βI∗S∗=0,νS∗-μ+ζV∗-β1I∗V∗=0,βI∗S∗-γe-μτ1+μE∗=0,β1I∗V∗-ρ+μC∗=0,ρC∗+γe-μτ1E∗-μ+δe-μτ2+ϕI∗=0,S˙∗+V˙∗+E˙∗+C˙∗+I˙∗=b-μN∗-δe-μτ2I∗=0.Hence, from system (15), we obtain the disease-free equilibrium P0=(S0,V0,0,0,0), where(16)S0=bμ+ζμ+ζν+μ-ζν,V0=bνμ+ζν+μ-ζν,provided (μ+ζ)(ν+μ)>ζν.It should be noted that, forν>0, the disease-free equilibrium is biologically feasible for any epidemiological parameters, whereas in the absence of vaccination strategy, i.e., for ν=0, E0 is only feasible for epidemiological parameters in the susceptible class. From system (15) the endemic equilibrium P∗=(S∗,V∗,E∗,C∗,I∗) is given as(17)S∗=b+ζV∗+ϕI∗ν+μ+βI∗,V∗=νb+ϕI∗ν+μ+βI∗μ+ζ+β1I∗-νζ,E∗=βζν+a1bI∗+ϕI∗2a1γe-μτ1+μν+μ+βI∗,C∗=νβ1I∗b+ϕI∗a1ρ+μ,I∗=I∗,where a1=(ν+μ+βI∗)(μ+ζ+β1I∗)-νζ


### 3.1. Local Stability of the Disease-Free Equilibrium Point

Suppose thatP0=(S0,V0,0,0,0) is a disease-free equilibrium point of system (1), then the linearization matrix JP0 is given by (18)JP0=-μ+νζ00ϕ-βS0ν-mu+ζ00-βϑV000-μ0βS0000-ρ+μβϑV0000ρ-μ+ϕ=0.Clearlyy1=-μ is one of the negative roots (eigenvalues) that guarantee local stability of the disease-free equilibrium P0. The remaining eigenvalues are obtained from the characteristic polynomial given by(19)gy=y4+e3y3+e2y2+e1y+e0=0,where e3=4μ+ν+ζ+ρ+ϕ, e2=(μ+ζ)(2μ+ρ+ν)+(μ+ϕ)(2μ+ν+ζ)-βϑρV0, e1=(μ+ζ)(μ+ρ)(2μ+ν+ϕ)+(μ+ν)(μ+ϕ)(2μ+ζ+ρ)-ζν(2μ+ρ+ϕ)-βρϑ(2μ+ν+ζ), and e0=(μ+ν)(μ+ζ)((ρ+μ)(μ+ϕ)-βρϑV0)+ζν(βρϑV0-(ρ+μ)(μ+ϕ).Thus computing the roots of polynomial (19) gives(20)y2=-μ,y3=-μ+ζ+ν,y4=-122μ+ρ+ϕ+ρ-ϕ2+4βρϑV0,y5=-122μ+ρ+ϕ-ρ-ϕ2+4βρϑV0.Since the rest of the roots are negative, root y5 is negative provided (μ+ϕ)(ρ+μ)>βρϑV0 holds implying that R0v=βρϑbν/(μ+ϕ)(ρ+μ)((μ+ζ)(ν+μ)-ζν)<1.Thus we have the result belowProposition 4.
The disease-free equilibriumP0 is locally asymptotically stable if the control reproduction ratio R0<1, whenever conditions (μ+ζ)(μ+ν)>ζν and R0v<1 are satisfied and unstable otherwise.To illustrate the stability of disease-free equilibrium, we use parameter values in Table2 with corresponding population estimates of S(0)=10604,V(0)=103,E(0)=C(0)=I(0)=0, and the resulting simulation is given in Figure 2.Figure 2
Simulation of model (1), the disease-free equilibrium, with populations parameters: ϕ=3.57144×10-1, β=1.0102×10-5, ν=2.53×10-2, and γ=3.3333×10-2 (with R0=0.7873, R0u=0.1382, and R0v=0.6490).The biological implication of Proposition4 means that in the long run the vaccinated and susceptible populations will be stable and pneumococcal pneumonia will be under control.


### 3.2. The Transcendental Equation

We obtain the expression for the transcendental equation by linearizing system (1) around P∗=(S∗,V∗,E∗,C∗,I∗), to obtain(21)S˙tV˙tE˙tC˙tI˙t=a1a200a3a4a500a6a70a80a90a100a11a12000a13a14StVtEtCtIt+000000000000a15000000000a160a1700Eτ10Iτ2,a1=-((μ+ν)+βI∗), a2=ζ,a3=ϕ-βS∗, a4=ν, a5=-((μ+ζ)+βϑI∗), a6=-βϑV∗, a7=βI∗, a8=-μ, a9=βS∗, a10=βϑI∗, a11=-(ρ+μ), a12=βϑV∗, a13=ρ, a14=-(μ+ϕ), a15=-γe-μτ1, a16=γe-μτ1, a17=-δe-μτ2, Eτ1=E(t-τ1), and Iτ2=I(t-τ2).The variational matrix of (21) is given by(22)gλ,e-λτ1,e-λτ2=a1-λa200a3a4a5-λ00a6a70a8-γe-μ+λτ1-λ0a9000a11-λa1200γe-μ+λτ1a13a14-δe-μ+λτ2-λ=0.Then, we obtain the transcendental equation of the linearized system at P∗:(23)
g
λ
,
e
-
λ
τ
1
,
e
-
λ
τ
2
=
λ
5
+
k
4
λ
4
+
k
3
λ
3
+
k
2
λ
2
+
k
1
λ
+
k
0
+
λ
4
+
l
3
λ
3
+
l
2
λ
2
+
l
1
λ
+
l
0
γ
e
-
μ
+
λ
τ
1
+
λ
4
+
m
3
λ
3
+
m
2
λ
2
+
m
1
λ
+
m
0
δ
e
-
μ
+
λ
τ
2
+
λ
3
+
n
2
λ
2
+
n
1
λ
+
n
0
γ
δ
e
-
μ
+
λ
τ
1
+
τ
2
=
0
,with coefficients of the transcendental equation (23) given in Appendix A.


### 3.3. Delay-Free System

Here, to show the local stability ofP∗, we consider a situation where there are no delays during the latent period (τ1=0) and in seeking medical care (τ2=0). By letting τ1=τ2=0, (23) reduces to(24)gλ=λ5+b4λ4+b3λ3+b2λ2+b1λ+b0=0,with coefficients of polynomial equation in Appendix A.Proposition 5.
The endemic equilibriumP∗ is locally asymptotically stable in the absence of delays τ1=τ2=0, iff the following Routh–Hurwitz conditions are satisfied:(25)b0>0,b4b3-b2>0,b2b4b3-b2-b4b4b1-b0>0,b1b2b4b3-b2-b4b4b1-b0-b0b3b4b3-b2-b4b1-b0>0,with b4, b3, b2, b1, and b0 defined in Appendix A.2.Numerically, using parameter values in Table2 the characteristic equation (24) is given as (26)λ5+0.7364λ4-148.4λ3-4.9408λ2-0.3965λ-0.0001806=0.The resulting eigenvalues are given by λ1=11.8, λ2=-0.0005, λ3=-12.5357, and λ4,5=-0.01641±0.4885i.Since there exists a positive root for model (1), there is a stability change from unstable to stable of the endemic equilibrium point P∗=(S∗,V∗,E∗,C∗,I∗)=(2099,6,54,2,100) that gives rise to a Hopf-bifurcation.


## 3.1. Local Stability of the Disease-Free Equilibrium Point

Suppose thatP0=(S0,V0,0,0,0) is a disease-free equilibrium point of system (1), then the linearization matrix JP0 is given by (18)JP0=-μ+νζ00ϕ-βS0ν-mu+ζ00-βϑV000-μ0βS0000-ρ+μβϑV0000ρ-μ+ϕ=0.Clearlyy1=-μ is one of the negative roots (eigenvalues) that guarantee local stability of the disease-free equilibrium P0. The remaining eigenvalues are obtained from the characteristic polynomial given by(19)gy=y4+e3y3+e2y2+e1y+e0=0,where e3=4μ+ν+ζ+ρ+ϕ, e2=(μ+ζ)(2μ+ρ+ν)+(μ+ϕ)(2μ+ν+ζ)-βϑρV0, e1=(μ+ζ)(μ+ρ)(2μ+ν+ϕ)+(μ+ν)(μ+ϕ)(2μ+ζ+ρ)-ζν(2μ+ρ+ϕ)-βρϑ(2μ+ν+ζ), and e0=(μ+ν)(μ+ζ)((ρ+μ)(μ+ϕ)-βρϑV0)+ζν(βρϑV0-(ρ+μ)(μ+ϕ).Thus computing the roots of polynomial (19) gives(20)y2=-μ,y3=-μ+ζ+ν,y4=-122μ+ρ+ϕ+ρ-ϕ2+4βρϑV0,y5=-122μ+ρ+ϕ-ρ-ϕ2+4βρϑV0.Since the rest of the roots are negative, root y5 is negative provided (μ+ϕ)(ρ+μ)>βρϑV0 holds implying that R0v=βρϑbν/(μ+ϕ)(ρ+μ)((μ+ζ)(ν+μ)-ζν)<1.Thus we have the result belowProposition 4.
The disease-free equilibriumP0 is locally asymptotically stable if the control reproduction ratio R0<1, whenever conditions (μ+ζ)(μ+ν)>ζν and R0v<1 are satisfied and unstable otherwise.To illustrate the stability of disease-free equilibrium, we use parameter values in Table2 with corresponding population estimates of S(0)=10604,V(0)=103,E(0)=C(0)=I(0)=0, and the resulting simulation is given in Figure 2.Figure 2
Simulation of model (1), the disease-free equilibrium, with populations parameters: ϕ=3.57144×10-1, β=1.0102×10-5, ν=2.53×10-2, and γ=3.3333×10-2 (with R0=0.7873, R0u=0.1382, and R0v=0.6490).The biological implication of Proposition4 means that in the long run the vaccinated and susceptible populations will be stable and pneumococcal pneumonia will be under control.


## 3.2. The Transcendental Equation

We obtain the expression for the transcendental equation by linearizing system (1) around P∗=(S∗,V∗,E∗,C∗,I∗), to obtain(21)S˙tV˙tE˙tC˙tI˙t=a1a200a3a4a500a6a70a80a90a100a11a12000a13a14StVtEtCtIt+000000000000a15000000000a160a1700Eτ10Iτ2,a1=-((μ+ν)+βI∗), a2=ζ,a3=ϕ-βS∗, a4=ν, a5=-((μ+ζ)+βϑI∗), a6=-βϑV∗, a7=βI∗, a8=-μ, a9=βS∗, a10=βϑI∗, a11=-(ρ+μ), a12=βϑV∗, a13=ρ, a14=-(μ+ϕ), a15=-γe-μτ1, a16=γe-μτ1, a17=-δe-μτ2, Eτ1=E(t-τ1), and Iτ2=I(t-τ2).The variational matrix of (21) is given by(22)gλ,e-λτ1,e-λτ2=a1-λa200a3a4a5-λ00a6a70a8-γe-μ+λτ1-λ0a9000a11-λa1200γe-μ+λτ1a13a14-δe-μ+λτ2-λ=0.Then, we obtain the transcendental equation of the linearized system at P∗:(23)
g
λ
,
e
-
λ
τ
1
,
e
-
λ
τ
2
=
λ
5
+
k
4
λ
4
+
k
3
λ
3
+
k
2
λ
2
+
k
1
λ
+
k
0
+
λ
4
+
l
3
λ
3
+
l
2
λ
2
+
l
1
λ
+
l
0
γ
e
-
μ
+
λ
τ
1
+
λ
4
+
m
3
λ
3
+
m
2
λ
2
+
m
1
λ
+
m
0
δ
e
-
μ
+
λ
τ
2
+
λ
3
+
n
2
λ
2
+
n
1
λ
+
n
0
γ
δ
e
-
μ
+
λ
τ
1
+
τ
2
=
0
,with coefficients of the transcendental equation (23) given in Appendix A.


## 3.3. Delay-Free System

Here, to show the local stability ofP∗, we consider a situation where there are no delays during the latent period (τ1=0) and in seeking medical care (τ2=0). By letting τ1=τ2=0, (23) reduces to(24)gλ=λ5+b4λ4+b3λ3+b2λ2+b1λ+b0=0,with coefficients of polynomial equation in Appendix A.Proposition 5.
The endemic equilibriumP∗ is locally asymptotically stable in the absence of delays τ1=τ2=0, iff the following Routh–Hurwitz conditions are satisfied:(25)b0>0,b4b3-b2>0,b2b4b3-b2-b4b4b1-b0>0,b1b2b4b3-b2-b4b4b1-b0-b0b3b4b3-b2-b4b1-b0>0,with b4, b3, b2, b1, and b0 defined in Appendix A.2.Numerically, using parameter values in Table2 the characteristic equation (24) is given as (26)λ5+0.7364λ4-148.4λ3-4.9408λ2-0.3965λ-0.0001806=0.The resulting eigenvalues are given by λ1=11.8, λ2=-0.0005, λ3=-12.5357, and λ4,5=-0.01641±0.4885i.Since there exists a positive root for model (1), there is a stability change from unstable to stable of the endemic equilibrium point P∗=(S∗,V∗,E∗,C∗,I∗)=(2099,6,54,2,100) that gives rise to a Hopf-bifurcation.


## 4. Existence of Hopf-Bifurcation

Under this subsection, we discuss the stability of the endemic equilibrium point of model (1). We use the approach of Song and Wei [44] to prove the conditions for continuation of unstable or stable switches at the endemic equilibrium point, by choosing time delay τ=max={τ1,τ2} as the bifurcation parameter.


### 4.1. Delay Only in Latent Period(τ1>0,τ2=0)

In such a situation the characteristic equation (23) reduces to(27)λ5+k4λ4+k3λ3+k2λ2+k1λ+k0+γλ4+h3λ3+h2λ2+h1λ+h0e-μ+λτ1=0,where(28)q=e-μτ1,h0=qγl0+n0δ,h1=qγl1+n1δ,h2=qγl2+n2δ,h3=qγl3+δ.Suppose the endemic equilibrium of system (1) is stable in the absence of delay (τ2) to seek medical care, implying that Re⁡(λ)=ξ(0)<0. The bifurcation value of τ10>0 occurs when λ(τ10)=ξ(τ10)+iw(τ10) is purely imaginary, for ξ(τ10)=0. Hence, defining the eigenvalue λ=wi, with infection rate oscillation frequency (w>0) and making a substitution in (27) and expressing the exponential in terms of trigonometric ratios, we get(29)Im≔a1cos⁡wτ1+a2sin⁡wτ1=R1,Re:a2cos⁡wτ1-a1sin⁡wτ1=R2,where(30)a1=wh1-h3w2,a2=wγw3-h2w+h0,R1=wk3w2-w4+k1,R2=k2w2-k4w4+k0.By eliminatingτ1 from (27), squaring and adding these two equations, and putting w2=z, we obtain the Hopf frequency below:(31)Hz=z5+B4z4+B3z3+B2z2+B1z+B0=0,where(32)B4=k4-2k3+γ,B3=k32+2k1+2h2γ-k2k4+h32,B2=k2+22h1h3+k4k0-k1k3+2h0γ+h22,B1=k12+22h0h2-h12+k2k0,B0=2h02+k02.The two propositions about stability and critical delay in Wesley et al. [45] are written as lemmasLemma 6.
If theBi(i=0,1,2,3,4) guarantee the Routh–Hurwitz criteria, then all eigenvalues of (31) have negative real parts for all delay τ1≥0. Thus the endemic equilibrium P∗ if it exists is locally asymptotically stable whenever τ1≥0, provided the endemic steady state is stable in the absence of the latent period delay; specifically τ1 will not affect the stability of the dynamical system, for (31) without positive real roots.Lemma 7.
IfBi(i=0,1,2,3,4) do not satisfy Routh–Hurwitz criteria, thus B0<0 or B0>0 implies that (47) has at least one positive root and suppose that it has a pair of imaginary roots say ±iw10 for such a value of w10.Consequently to obtain the main results in this paper, we assume (31) has at least one positive root w10. By squaring and summing together the imaginary and real parts in (29), we get(33)τ1=1warccos⁡w2h1-h3w2k3w2-w4+k1+wγw3-h2w+h0k2w2-k4w4+k0wh1-h3w22+wγw3-h2w+h02+2nπw.By denoting(34)τ1nm=1w1narccos⁡w1n2h1-h3w1n2k3w1n2-w1n4+k1+w1nγw1n3-h2w1n+h0k2w1n2-k4w1n4+k0w1nh1-h3w1n22+w1nγw1n3-h2w1n+h02+2nπw1n,m=1,2,…,m~,n∈Nthis allows us to define(35)τ10=τn00=min1≤n≤5⁡τ1n0,w10=wn0and state the result as follows.Lemma 8.
Ifτ10 and w10 are defined as (35) and H′(z=w2)>0. The endemic equilibrium point P∗ is linearly asymptotically stable for τ1<τ10 and unstable for τ1>τ10 and undergoes a Hopf-bifurcation at τ1=τ10.To ensure the occurrence of the Hopf-bifurcation, it is desirable to verify the transversality condition. Without loss of generality, the delayτ1 is chosen as the bifurcation parameter. The essential condition for existence of the Hopf-bifurcation is that the threshold eigenvalues traverse the imaginary axis with nonzero velocity.Proposition 9.
IfΦ2(w10)>0, where Φ2(w10) satisfies (47), system (1) undergoes a Hopf-bifurcation at the endemic equilibrium as τ1 increases through τ10.Proof (transversality condition for Hopf-bifurcation).
Differentiating (27) with respect to τ1 we obtain(36)dτ1dλ=5λ4+4k4λ3+3k3λ2+k1eλτ1+4γλ3+3h3λ2+2h2λ+h1γλ5+h3λ4+h2λ3+h1λ2+h0λ-τ1λ,(37)signdRe⁡λdτ1τ1=τ10=signRedλdτ1-1λ=iw10=signRe⁡N1+signRe⁡N2=signc3c1cos⁡wτ1+c2sin⁡wτ1+c4c1sin⁡wτ1+c2cos⁡wτ1+h1-3h3w2+c4w2h2-4γw3c32+c42.with(38)N1=c3c1cos⁡wτ10+c2sin⁡wτ10+c4c1sin⁡wτ10+c2cos⁡wτ10c32+c42+i(c3c2cos⁡wτ1+c10sin⁡wτ10-c4c2sin⁡wτ10+c1cos⁡wτ10c32+c42,N2=h1-3h3w2+c4w2h2-4γw3+ic3w2h2-4γw2+h1-3h3w2c32+c42.
Remark 10. Any linear combination of a sine and cosine of equal periods is equal to a single sine with the same period, however, with an infection rate oscillation phase shift Ψ [46].
Therefore, we get(39)signdRe⁡λdτ1τ1=τ10=signRedλdτ1-1λ=iw10=D0sin⁡wτ1+Ψ2+h1-3h3w2+c4w2h2-4γw3c32+c42,where(40)c1=5w4-3k3w2+k1,c2=4k4w3,c3=wγw4-h2w2+h0,c4=w2h3w2-h1,D0=c3c1+c2c12+c3c2+c4c12,D01=c4w102h2-4γw103,Ψ2=arctan⁡c3c1+c4c2c3c2+c4c1Let Φ2(w10)=D0sin⁡(w10τ10+Ψ)+(h1-3h3w102)+c4w10(2h2-4γw103)>0, if conditions (w10τ10+Ψ2)∈(π,π/2), h1>3h3w102, and h2>2γw102 hold. Clearly(41)signdRe⁡λdτ1τ1=τ10=signRedλdτ1-1λ=iw10=D0sin⁡w10τ10+Ψ2+h1-3h3w102+D01c32+c42has the same sign as Φ2(w10). This completes the proof.Therefore, Proposition9 implies that given m>0, the eigenvalue λm(τ1) of the characteristic equation (27) close to τ1m crosses the imaginary axis from the left to the right as τ1 continuously changes from a value less than τ1m to one greater than τ1m.


### 4.2. Delay Only in Seeking Medical Care by the Infectious(τ1=0,τ2>0)

To understand the influence of time delay in seeking medical care, we setτ1=0 in (23) yielding(42)gλ,e-λτ2=λ5+p4λ4+p3λ3+p2λ2+p1λ+p0+q4λ4+q3λ3+q2λ2+q1λ+q0pγδe-λτ2=0,where(43)p=e-μτ2,p4=k4+γ,p3=k3+l3γ,p2=k2+γ,p1=k1+l1γ,p0=k0+l0γq4=δp,q3=m3+γ+δp,q2=m2δ+n2γδ,q1=m1+n1γδp,q0=m0+n0γδpProposition 11.
The endemic equilibrium pointP∗ is locally asymptotically stable (LAS) for τ2<τ20 where τ20 is the minimum positive value of (44)τ-20=1w20arccos⁡p2w202-p4w204-p0q4w204-q2w202+q0+q3w203-q1w20p3w203-w205-p1w20pγδq4w204-q2w202+q02-q1w20-q3w2032.Proof.
Letλ=iw,w>0 be a root of (42) to obtain(45)Pλ,τ2=w5i+p4w4-p3w3i-p2w2+p1wi+p0+q4w4-q3w3i-q2w2+q1wi+q0pγδe-iwτ2.Using Euler expansion and separating real and imaginary parts, we obtain(46)pγδq4w4-q2w2+q0cos⁡wτ2+q1w-q3w3sin⁡wτ2=p2w2-p4w4-p0,pγδq1w-q3w3cos⁡wτ2+q4w4-q2w2+q0sin⁡wτ2=p3w3-w5-p1w.
Eliminatingτ2 from (46), by squaring and adding these two equations and putting w2=z, we obtain the Hopf frequency below:(47)z5+A4z4+A3z3+A2z2+A1z+A0=0,with coefficients in (47) in Appendix A.
Let us denoteg(z)=z5+A4z4+A3z3+A2z2+A1z+A0. Since limz→+∞⁡g(z)=+∞ and A0<0, then (47) has at least one positive root. Assuming (47) has n~ positive roots, given by n~(1≤n~≤5), denote by z1<z2<…zn~, respectively. Then, (47) has n~ positive roots if(48)w1=z1,w2=z2,⋮wn~=zn~.From (46), the corresponding τ2n>0, for which the characteristic equation (23) has a pair of purely imaginary roots, is derived to have(49)cos⁡wτ2=p2w2-p4w4-p0q4w4-q2w2+q0+q3w3-q1wp3w3-w5-p1wq4w4-q2w2+q02+q3w3-q1wq1w-q3w3pγδ.Thus, denoting(50)τ2nk=1wnarccos⁡p2wn2-p4wn4-p0q4wn4-q2wn2+q0+q3wn3-q1wnp3wn3-wn5-p1wnq4wn4-q2wn2+q02+q3wn3-q1wnq1wn-q3wn3pγδ+2πk-1wn,where n=1,2,…n~,k=1,2,…, then ±iwn are a pair of purely imaginary roots of (23). This allows us to define the Hopf-bifurcation threshold time delay value as(51)τ20=1w20arccosp2w202-p4w204-p0q4w204-q2w202+q0+q3w203-q1w20p3w203-w205-p1w20pγδq4w204-q2w202+q02-q1w20-q3w2032.This completes the proof.Proposition 12.
If conditions(52)5w204q1+2q2w20+3p3w202+p13q3w202+4q3w203>5w2043q3w202+4q4w203+q1+2q2w203p3w202+p1,q3w202q1>1,w204q4+q0q2w202>1hold, such that Φ1(w20)>0, then system (1) undergoes a Hopf-bifurcation at the endemic equilibrium point as τ2 increases through τ20, where expressions of Φ1(w20) satisfy (58).Proof (transversality condition for Hopf-bifurcation).
In order to establish whether the endemic equilibrium pointP∗ actually undergoes a Hopf-bifurcation at τ2=τ20, we let λ(τ2)=β(τ2)+iw(τ2) be a root of (23) near τ2=τ20(k) and β(τ2)(k)=0, as w(τ2)(k)=w20. Making a substitution into the L.H.S. of (23) and taking a derivative with respect to λ, we have(53)dτ2dλ=5λ4+4p4λ3+3p3λ2+2p2λ+p1eμλτ2q4λ5+q3λ4+q2λ3+q1λ2+q0λpγδ+4q4λ3+3q3λ2+2q2λ+q1pγδq4λ5+q3λ4+q2λ3+q1λ2+q0λ-τ2λ.
Computing the Sign ofd[Re⁡(λ)]/dτ2, by differentiating the characteristic equation (23) with respect to τ2 and evaluating (53) at τ2=τ20 with λ=iw20 and expressing sin(w20τ20) and cos⁡(w20τ20), we obtain sign[d(Re⁡λ)/dτ2]τ2=τ20=sign[Re⁡(dλ/dτ2)-1]λ=iw20,(54)=signRe⁡f1cos⁡d0+f2sin⁡d0g1+ig2+Re⁡if3cos⁡d0+f4sin⁡d0g1+ig2+Re⁡f5g1+ig2-Re⁡τ2iw20,(55)=signRe⁡g1f1cos⁡d0+f2sin⁡d0-ig2f1cos⁡d0+f2sin⁡d0g12+g22+signRe⁡g2f2cos⁡d0+f4sin⁡d0+ig1f3cos⁡d0+f4sin⁡d0g12+g22+signRe⁡f5g1g12+g22,with coefficients in Appendix A.
By Remark 10, (55) gives(56)signg1f12+f22sin⁡d0+Ψ0+g2f22+f42sin⁡d0+Ψ1+f5g1g12+g22,with(57)Ψ0=arctan⁡f1f2,Ψ1=arctan⁡f2f4,f2≠0,f4≠0Let(58)Φ1w20=g1f12+f22sin⁡d0+Ψ0+g2f22+f42sin⁡d0+Ψ1+f5g1.If Φ1(w20)>0, with (d0+Ψ{i=0,1})∈(π,π/2], then sign[d(Re⁡λ)/dτ2]τ2=τ20>0, and hence the transversality condition holds and the system undergoes Hopf-bifurcation.


### 4.3. Delay in Latent Period and Seeking Medical Care(τ1=τ2=τ>0)

Making a substitution ofτ1=τ2=τ in (23), we get(59)gλ,e-λτ=λ5+k4λ4+k3λ3+k2λ2+k1λ+k0+s4λ4+s3λ3+s2λ2+s1λ+s0e-λτ+s3′λ3+s2′λ2+s1′λ+s0′e-2λτ=0with(60)s4=γ+δe-μτ,s3=γl3+m3δe-μτ,s2=l2γ+m2δe-μτ,s1=l1γ+m1δe-μτ,s0=l0γ+m0δe-μτ,s3′=γδe-2μτ,s2′=n2δγe-2μτ,s1′=n1γδe-2μτ,s0′=n0γe-2μτIn order to examine whether or not the endemic equilibrium loses stability and undergoes Hopf-bifurcation as an outcome with inclusion of the time delays, a pair of purely imaginary root of the transcendental equation (59) is found. Suppose the pair of the imaginary root is given as λ=iv with infection rate oscillation frequency (v>0), using Euler’s expansion and making a substitution into (59), separating real and imaginary parts, we obtain(61)g0cos⁡vτ+g1sin⁡vτ+g2sin⁡2vτ=G1,(62)-g1cos⁡vτ+g0sin⁡vτ+g3sin⁡2vτ=G2,where(63)g0=s1v-s3v3,g1=s2v2-s4v4-s0,g2=s2v2,g3=s3′v3+s1′v,G1=v5+k3+s3+s3′v3-k1+s1′v,G2=k2+s2′v2-k4v4+k0+s0′.Squaring and adding (61) and (62), we get following equation:(64)g02+g12-G12-G22=-12g22+g321-cos⁡4vτ.Supposing cos⁡4vτ<1, (64) leads to(65)G12+G22-g02+g12=0,which reduces to(66)v10+2k3+s3+s3′+k42-s42v8+k3+s3+s3′2+2k1+s1′-2k4k2+s2′+2s2s4-s32v6+2k1+s1′k3+s3+s3′+2k4k0+s0′+2s1s3-s22+2s0s4v4+k1+s1′2+2s0s2-2k2+s2′k0+s0′-s12v2+k0+s0′2=0.Let z=v2 such that we obtain (66) in terms of z:(67)Lz=z5+u4z4+u3z3+u2z2+u1z+u0,with(68)u4=2k3+s3+s3′+k42-s42,u3=k3+s3+s3′2+2k1+s11-2k4k2+s21+2s2s4-s32,u2=2k1+s1′k3+s3+s3′+2k4k0+s0′+2s1s3-s22+2s0s4,u0=k0+s0′2u1=k1+s1′2+2s0s2-2k2+s2′k0+s0′-s12.Since (67) has a high degree polynomial we compute the eigenvalues numerically by using parameter values in Table 2. The resulting polynomial is(69)z5-295.18z4-130.18z3+92.52z2-0.15038z+2.6×10-12=0.Therefore, the following eigenvalues are obtained:(70)z1=295.62,z2=0,z3=0.001629,z4=0.38024,z5=-0.8212.We observe that there is only one negative real root which does not guarantee stability of model (1) in the presence of time delays τ=τ1=τ2>0; thus by Lemma 7 there exists a pure imaginary root wc such that a critical time delay τc is achieved for which there is death or birth of period oscillations (Hopf-bifurcation).Equation (64) yields (71)τc=14v0arccos⁡g22+g32+2g02+g12-G12+G22g22+g32+jπ2v0;j=0,1,3,…with λ=iv (a purely imaginary root of (59)), if condition g02+g12=G12+G22 and τ∈[0,τc) holds. Without loss of generality, let v0 represent the value v0 corresponding to τc. We thus state the result below.Proposition 13.
If conditiong02+g12=G12+G22 holds, then the chronic steady state P∗ is locally asymptotically stable for τ∈[0,τc) and unstable when τ>τc and undergoes a Hopf-bifurcation.


## 4.1. Delay Only in Latent Period(τ1>0,τ2=0)

In such a situation the characteristic equation (23) reduces to(27)λ5+k4λ4+k3λ3+k2λ2+k1λ+k0+γλ4+h3λ3+h2λ2+h1λ+h0e-μ+λτ1=0,where(28)q=e-μτ1,h0=qγl0+n0δ,h1=qγl1+n1δ,h2=qγl2+n2δ,h3=qγl3+δ.Suppose the endemic equilibrium of system (1) is stable in the absence of delay (τ2) to seek medical care, implying that Re⁡(λ)=ξ(0)<0. The bifurcation value of τ10>0 occurs when λ(τ10)=ξ(τ10)+iw(τ10) is purely imaginary, for ξ(τ10)=0. Hence, defining the eigenvalue λ=wi, with infection rate oscillation frequency (w>0) and making a substitution in (27) and expressing the exponential in terms of trigonometric ratios, we get(29)Im≔a1cos⁡wτ1+a2sin⁡wτ1=R1,Re:a2cos⁡wτ1-a1sin⁡wτ1=R2,where(30)a1=wh1-h3w2,a2=wγw3-h2w+h0,R1=wk3w2-w4+k1,R2=k2w2-k4w4+k0.By eliminatingτ1 from (27), squaring and adding these two equations, and putting w2=z, we obtain the Hopf frequency below:(31)Hz=z5+B4z4+B3z3+B2z2+B1z+B0=0,where(32)B4=k4-2k3+γ,B3=k32+2k1+2h2γ-k2k4+h32,B2=k2+22h1h3+k4k0-k1k3+2h0γ+h22,B1=k12+22h0h2-h12+k2k0,B0=2h02+k02.The two propositions about stability and critical delay in Wesley et al. [45] are written as lemmasLemma 6.
If theBi(i=0,1,2,3,4) guarantee the Routh–Hurwitz criteria, then all eigenvalues of (31) have negative real parts for all delay τ1≥0. Thus the endemic equilibrium P∗ if it exists is locally asymptotically stable whenever τ1≥0, provided the endemic steady state is stable in the absence of the latent period delay; specifically τ1 will not affect the stability of the dynamical system, for (31) without positive real roots.Lemma 7.
IfBi(i=0,1,2,3,4) do not satisfy Routh–Hurwitz criteria, thus B0<0 or B0>0 implies that (47) has at least one positive root and suppose that it has a pair of imaginary roots say ±iw10 for such a value of w10.Consequently to obtain the main results in this paper, we assume (31) has at least one positive root w10. By squaring and summing together the imaginary and real parts in (29), we get(33)τ1=1warccos⁡w2h1-h3w2k3w2-w4+k1+wγw3-h2w+h0k2w2-k4w4+k0wh1-h3w22+wγw3-h2w+h02+2nπw.By denoting(34)τ1nm=1w1narccos⁡w1n2h1-h3w1n2k3w1n2-w1n4+k1+w1nγw1n3-h2w1n+h0k2w1n2-k4w1n4+k0w1nh1-h3w1n22+w1nγw1n3-h2w1n+h02+2nπw1n,m=1,2,…,m~,n∈Nthis allows us to define(35)τ10=τn00=min1≤n≤5⁡τ1n0,w10=wn0and state the result as follows.Lemma 8.
Ifτ10 and w10 are defined as (35) and H′(z=w2)>0. The endemic equilibrium point P∗ is linearly asymptotically stable for τ1<τ10 and unstable for τ1>τ10 and undergoes a Hopf-bifurcation at τ1=τ10.To ensure the occurrence of the Hopf-bifurcation, it is desirable to verify the transversality condition. Without loss of generality, the delayτ1 is chosen as the bifurcation parameter. The essential condition for existence of the Hopf-bifurcation is that the threshold eigenvalues traverse the imaginary axis with nonzero velocity.Proposition 9.
IfΦ2(w10)>0, where Φ2(w10) satisfies (47), system (1) undergoes a Hopf-bifurcation at the endemic equilibrium as τ1 increases through τ10.Proof (transversality condition for Hopf-bifurcation).
Differentiating (27) with respect to τ1 we obtain(36)dτ1dλ=5λ4+4k4λ3+3k3λ2+k1eλτ1+4γλ3+3h3λ2+2h2λ+h1γλ5+h3λ4+h2λ3+h1λ2+h0λ-τ1λ,(37)signdRe⁡λdτ1τ1=τ10=signRedλdτ1-1λ=iw10=signRe⁡N1+signRe⁡N2=signc3c1cos⁡wτ1+c2sin⁡wτ1+c4c1sin⁡wτ1+c2cos⁡wτ1+h1-3h3w2+c4w2h2-4γw3c32+c42.with(38)N1=c3c1cos⁡wτ10+c2sin⁡wτ10+c4c1sin⁡wτ10+c2cos⁡wτ10c32+c42+i(c3c2cos⁡wτ1+c10sin⁡wτ10-c4c2sin⁡wτ10+c1cos⁡wτ10c32+c42,N2=h1-3h3w2+c4w2h2-4γw3+ic3w2h2-4γw2+h1-3h3w2c32+c42.
Remark 10. Any linear combination of a sine and cosine of equal periods is equal to a single sine with the same period, however, with an infection rate oscillation phase shift Ψ [46].
Therefore, we get(39)signdRe⁡λdτ1τ1=τ10=signRedλdτ1-1λ=iw10=D0sin⁡wτ1+Ψ2+h1-3h3w2+c4w2h2-4γw3c32+c42,where(40)c1=5w4-3k3w2+k1,c2=4k4w3,c3=wγw4-h2w2+h0,c4=w2h3w2-h1,D0=c3c1+c2c12+c3c2+c4c12,D01=c4w102h2-4γw103,Ψ2=arctan⁡c3c1+c4c2c3c2+c4c1Let Φ2(w10)=D0sin⁡(w10τ10+Ψ)+(h1-3h3w102)+c4w10(2h2-4γw103)>0, if conditions (w10τ10+Ψ2)∈(π,π/2), h1>3h3w102, and h2>2γw102 hold. Clearly(41)signdRe⁡λdτ1τ1=τ10=signRedλdτ1-1λ=iw10=D0sin⁡w10τ10+Ψ2+h1-3h3w102+D01c32+c42has the same sign as Φ2(w10). This completes the proof.Therefore, Proposition9 implies that given m>0, the eigenvalue λm(τ1) of the characteristic equation (27) close to τ1m crosses the imaginary axis from the left to the right as τ1 continuously changes from a value less than τ1m to one greater than τ1m.


## 4.2. Delay Only in Seeking Medical Care by the Infectious(τ1=0,τ2>0)

To understand the influence of time delay in seeking medical care, we setτ1=0 in (23) yielding(42)gλ,e-λτ2=λ5+p4λ4+p3λ3+p2λ2+p1λ+p0+q4λ4+q3λ3+q2λ2+q1λ+q0pγδe-λτ2=0,where(43)p=e-μτ2,p4=k4+γ,p3=k3+l3γ,p2=k2+γ,p1=k1+l1γ,p0=k0+l0γq4=δp,q3=m3+γ+δp,q2=m2δ+n2γδ,q1=m1+n1γδp,q0=m0+n0γδpProposition 11.
The endemic equilibrium pointP∗ is locally asymptotically stable (LAS) for τ2<τ20 where τ20 is the minimum positive value of (44)τ-20=1w20arccos⁡p2w202-p4w204-p0q4w204-q2w202+q0+q3w203-q1w20p3w203-w205-p1w20pγδq4w204-q2w202+q02-q1w20-q3w2032.Proof.
Letλ=iw,w>0 be a root of (42) to obtain(45)Pλ,τ2=w5i+p4w4-p3w3i-p2w2+p1wi+p0+q4w4-q3w3i-q2w2+q1wi+q0pγδe-iwτ2.Using Euler expansion and separating real and imaginary parts, we obtain(46)pγδq4w4-q2w2+q0cos⁡wτ2+q1w-q3w3sin⁡wτ2=p2w2-p4w4-p0,pγδq1w-q3w3cos⁡wτ2+q4w4-q2w2+q0sin⁡wτ2=p3w3-w5-p1w.
Eliminatingτ2 from (46), by squaring and adding these two equations and putting w2=z, we obtain the Hopf frequency below:(47)z5+A4z4+A3z3+A2z2+A1z+A0=0,with coefficients in (47) in Appendix A.
Let us denoteg(z)=z5+A4z4+A3z3+A2z2+A1z+A0. Since limz→+∞⁡g(z)=+∞ and A0<0, then (47) has at least one positive root. Assuming (47) has n~ positive roots, given by n~(1≤n~≤5), denote by z1<z2<…zn~, respectively. Then, (47) has n~ positive roots if(48)w1=z1,w2=z2,⋮wn~=zn~.From (46), the corresponding τ2n>0, for which the characteristic equation (23) has a pair of purely imaginary roots, is derived to have(49)cos⁡wτ2=p2w2-p4w4-p0q4w4-q2w2+q0+q3w3-q1wp3w3-w5-p1wq4w4-q2w2+q02+q3w3-q1wq1w-q3w3pγδ.Thus, denoting(50)τ2nk=1wnarccos⁡p2wn2-p4wn4-p0q4wn4-q2wn2+q0+q3wn3-q1wnp3wn3-wn5-p1wnq4wn4-q2wn2+q02+q3wn3-q1wnq1wn-q3wn3pγδ+2πk-1wn,where n=1,2,…n~,k=1,2,…, then ±iwn are a pair of purely imaginary roots of (23). This allows us to define the Hopf-bifurcation threshold time delay value as(51)τ20=1w20arccosp2w202-p4w204-p0q4w204-q2w202+q0+q3w203-q1w20p3w203-w205-p1w20pγδq4w204-q2w202+q02-q1w20-q3w2032.This completes the proof.Proposition 12.
If conditions(52)5w204q1+2q2w20+3p3w202+p13q3w202+4q3w203>5w2043q3w202+4q4w203+q1+2q2w203p3w202+p1,q3w202q1>1,w204q4+q0q2w202>1hold, such that Φ1(w20)>0, then system (1) undergoes a Hopf-bifurcation at the endemic equilibrium point as τ2 increases through τ20, where expressions of Φ1(w20) satisfy (58).Proof (transversality condition for Hopf-bifurcation).
In order to establish whether the endemic equilibrium pointP∗ actually undergoes a Hopf-bifurcation at τ2=τ20, we let λ(τ2)=β(τ2)+iw(τ2) be a root of (23) near τ2=τ20(k) and β(τ2)(k)=0, as w(τ2)(k)=w20. Making a substitution into the L.H.S. of (23) and taking a derivative with respect to λ, we have(53)dτ2dλ=5λ4+4p4λ3+3p3λ2+2p2λ+p1eμλτ2q4λ5+q3λ4+q2λ3+q1λ2+q0λpγδ+4q4λ3+3q3λ2+2q2λ+q1pγδq4λ5+q3λ4+q2λ3+q1λ2+q0λ-τ2λ.
Computing the Sign ofd[Re⁡(λ)]/dτ2, by differentiating the characteristic equation (23) with respect to τ2 and evaluating (53) at τ2=τ20 with λ=iw20 and expressing sin(w20τ20) and cos⁡(w20τ20), we obtain sign[d(Re⁡λ)/dτ2]τ2=τ20=sign[Re⁡(dλ/dτ2)-1]λ=iw20,(54)=signRe⁡f1cos⁡d0+f2sin⁡d0g1+ig2+Re⁡if3cos⁡d0+f4sin⁡d0g1+ig2+Re⁡f5g1+ig2-Re⁡τ2iw20,(55)=signRe⁡g1f1cos⁡d0+f2sin⁡d0-ig2f1cos⁡d0+f2sin⁡d0g12+g22+signRe⁡g2f2cos⁡d0+f4sin⁡d0+ig1f3cos⁡d0+f4sin⁡d0g12+g22+signRe⁡f5g1g12+g22,with coefficients in Appendix A.
By Remark 10, (55) gives(56)signg1f12+f22sin⁡d0+Ψ0+g2f22+f42sin⁡d0+Ψ1+f5g1g12+g22,with(57)Ψ0=arctan⁡f1f2,Ψ1=arctan⁡f2f4,f2≠0,f4≠0Let(58)Φ1w20=g1f12+f22sin⁡d0+Ψ0+g2f22+f42sin⁡d0+Ψ1+f5g1.If Φ1(w20)>0, with (d0+Ψ{i=0,1})∈(π,π/2], then sign[d(Re⁡λ)/dτ2]τ2=τ20>0, and hence the transversality condition holds and the system undergoes Hopf-bifurcation.


## 4.3. Delay in Latent Period and Seeking Medical Care(τ1=τ2=τ>0)

Making a substitution ofτ1=τ2=τ in (23), we get(59)gλ,e-λτ=λ5+k4λ4+k3λ3+k2λ2+k1λ+k0+s4λ4+s3λ3+s2λ2+s1λ+s0e-λτ+s3′λ3+s2′λ2+s1′λ+s0′e-2λτ=0with(60)s4=γ+δe-μτ,s3=γl3+m3δe-μτ,s2=l2γ+m2δe-μτ,s1=l1γ+m1δe-μτ,s0=l0γ+m0δe-μτ,s3′=γδe-2μτ,s2′=n2δγe-2μτ,s1′=n1γδe-2μτ,s0′=n0γe-2μτIn order to examine whether or not the endemic equilibrium loses stability and undergoes Hopf-bifurcation as an outcome with inclusion of the time delays, a pair of purely imaginary root of the transcendental equation (59) is found. Suppose the pair of the imaginary root is given as λ=iv with infection rate oscillation frequency (v>0), using Euler’s expansion and making a substitution into (59), separating real and imaginary parts, we obtain(61)g0cos⁡vτ+g1sin⁡vτ+g2sin⁡2vτ=G1,(62)-g1cos⁡vτ+g0sin⁡vτ+g3sin⁡2vτ=G2,where(63)g0=s1v-s3v3,g1=s2v2-s4v4-s0,g2=s2v2,g3=s3′v3+s1′v,G1=v5+k3+s3+s3′v3-k1+s1′v,G2=k2+s2′v2-k4v4+k0+s0′.Squaring and adding (61) and (62), we get following equation:(64)g02+g12-G12-G22=-12g22+g321-cos⁡4vτ.Supposing cos⁡4vτ<1, (64) leads to(65)G12+G22-g02+g12=0,which reduces to(66)v10+2k3+s3+s3′+k42-s42v8+k3+s3+s3′2+2k1+s1′-2k4k2+s2′+2s2s4-s32v6+2k1+s1′k3+s3+s3′+2k4k0+s0′+2s1s3-s22+2s0s4v4+k1+s1′2+2s0s2-2k2+s2′k0+s0′-s12v2+k0+s0′2=0.Let z=v2 such that we obtain (66) in terms of z:(67)Lz=z5+u4z4+u3z3+u2z2+u1z+u0,with(68)u4=2k3+s3+s3′+k42-s42,u3=k3+s3+s3′2+2k1+s11-2k4k2+s21+2s2s4-s32,u2=2k1+s1′k3+s3+s3′+2k4k0+s0′+2s1s3-s22+2s0s4,u0=k0+s0′2u1=k1+s1′2+2s0s2-2k2+s2′k0+s0′-s12.Since (67) has a high degree polynomial we compute the eigenvalues numerically by using parameter values in Table 2. The resulting polynomial is(69)z5-295.18z4-130.18z3+92.52z2-0.15038z+2.6×10-12=0.Therefore, the following eigenvalues are obtained:(70)z1=295.62,z2=0,z3=0.001629,z4=0.38024,z5=-0.8212.We observe that there is only one negative real root which does not guarantee stability of model (1) in the presence of time delays τ=τ1=τ2>0; thus by Lemma 7 there exists a pure imaginary root wc such that a critical time delay τc is achieved for which there is death or birth of period oscillations (Hopf-bifurcation).Equation (64) yields (71)τc=14v0arccos⁡g22+g32+2g02+g12-G12+G22g22+g32+jπ2v0;j=0,1,3,…with λ=iv (a purely imaginary root of (59)), if condition g02+g12=G12+G22 and τ∈[0,τc) holds. Without loss of generality, let v0 represent the value v0 corresponding to τc. We thus state the result below.Proposition 13.
If conditiong02+g12=G12+G22 holds, then the chronic steady state P∗ is locally asymptotically stable for τ∈[0,τc) and unstable when τ>τc and undergoes a Hopf-bifurcation.


## 5. Numerical Simulation and Results

In this section, we use MATLAB dde23 function to obtain numerical simulations and graphical representations of model (1) to supplement the analytical solutions in Section 4. Parameter values in Table 2 are used in the simulation.The positive endemic equilibrium isP∗=(S∗,V∗,E∗,C∗,I∗)=(2099,6,54,2,100). In the absence of delays τ1=τ2=0, the characteristic polynomial equation (24) is(72)λ5+0.7364λ4-148.4007λ3-4.9408λ2-0.3965λ-0.0001806.The corresponding eigenvalues are λ1=11.8366, λ2=-0.000472, λ3=-12.5357, and λ4,5=-0.01641 ± 04885i. Therefore, since the eigenvalues have one positive root and four negative roots, the endemic equilibrium changes state of stability from unstable to stable thus and undergoes a Hopf-bifurcation (see Figure 3). This implies that, as time approaches infinity, the partial populations are stable and pneumococcal pneumonia can no longer cause harm to individuals.Figure 3
(a and b) Stability of the endemic equilibrium showing Hopf-bifurcation, with initial variables:S(0)=5586, V(0)=22, C(0)=64, I(0)=11, and E(0)=100. (c and d) The evolution of the infected, the susceptible and corresponding I-S portrait, and 3D phase trajectories, with R0=15.4, R0u=15.14, and R0v=0.271 parameters: μ=2.0547×10-4; ϕ=3.574×10-2. The rest of the parameters remain fixed as in Table 2.
(a) (b) (c) (d)The numerical simulation of (27) yields the characteristic roots as λ1=0,λ2=14.4621i,λ3=-14.4416,λ4=±0.00041+0.3771i,λ5=±0.0579+0.1335i. As τ1 increases from zero, there is a value τ10>0 such that the endemic equilibrium is stable for τ1=[0,τ10] and unstable for τ1>τ10. At this critical value, the endemic equilibrium loses stability and Hopf-bifurcation arises. The real positive root is w10=14.4621 and the critical time delay τ10=0.109 of a day ≈3 hrs.Figure4 shows the evolution of the susceptible and infected population of system (1). The low and high peaks in the number of susceptible and infected individuals indicate the season peak of the disease. If τ1<τ10=0.109 of a day ≈3 hrs, the partial populations of the susceptible and the infected are stable whereas if τ1>τ10=0.109 of a day ≈3hrs, the populations are unstable and it is hard to predict the future pattern of the disease prevalence.Figure 4
Simulation of model (1) for τ2=0 and τ1>0, with initial variable values: (S(0),V(0),E(0),C(0),I(0))=(3280,30,10,10,100). The rest of the parameters are as in Table 2.
(a) (b) (c)The numerical computation of (42) yields eigenvalues λ1=0, λ2=0.06508i, λ3=-0.06522, λ4=-12.038, and λ=-12.3263. The positive root w20=0.06508 and the critical time delay τ20=26 days, and hence system (1) is stable for τ2<26 days and unstable for τ2>τ20. A characteristic polynomial (59) corresponding to two delays is solved to give the eigenvalues as λ1=0, λ2=-17.1963, λ3=-0.6166, λ4=-0.04036, λ5=0.9026i, the real positive root wc=0.9062, and the critical time delay τc=2.069 days.Figure5 depicts the time series solution approaching their equilibrium point as time approaches infinity. This confirms the stability of the system when the value of time delay is less than τc=2.069 days and instability of the system if τ>τc=2.069 days (see Proposition 12).Figure 5
Stability of the endemic equilibriumP∗ for τ1=τ2=2 days. The rest of the parameters are as in Table 2.
(a) (b) (c) (d) (e) (f)To explore the effect of time delayτ2 on pneumococcal pneumonia, we fix time delay τ1=3 days, and the parameter τ2 is varied (Figure 6). The rate of convergence to stability of the endemic equilibrium point is attained with a reduction in the delay and a divergence is due to an increase in the delay that results into instability of the system. This gives rise to Hopf-bifurcation phenomenon.Figure 6
The effect of varyingτ2 on the dynamics of model (1). The delay τ2 was chosen as τ2=2.5,3,3.5. All other parameters remain as stated in Table 2.
(a) (b) (c) (d) (e)In Figure7, time delay τ2 is fixed at 2 days in order to study the effect of time delay τ1 on model (1). We observe an increase in the magnitude of the amplitude of oscillations as τ1 increases; thus divergence from the endemic equilibrium occurs leading to unstable state. This implies that the disease will persist in the population with increased delays if there is no intervention instituted to reduce the delays. On the other hand a decrease in τ1 guarantees the asymptotic stability of the endemic equilibrium which implies the disease can be eradicated from the population.Figure 7
The effect of varying time delayτ1 on the dynamics of model (1). The delay τ1 was chosen as τ1=0.5,2,8.5. All the parameter values are the same as in Table 2.
(a) (b) (c) (d) (e)


## 6. Discussion

In this paper, we propose and analyze a mathematical model of pneumococcal pneumonia with time delays. We derive the control reproductive ratioR0. The results show that, without delays (τ1=τ2=0), the disease-free equilibrium P0 is locally asymptotically stable if the control reproductive ratio R0<1, whenever conditions (μ+ζ)(μ+ν)>ζν and R0v<1 hold, and unstable if R0>1.The analysis of model (1) is done. The results show that the endemic equilibrium is locally stable without delays and stable if the delays are under conditions. The transversality conditions for the existence of Hopf-bifurcation are stated and proved for three cases: (1) τ1=0, τ2=τ>0, (2) τ1=τ>0, τ2=0, and (3) τ1=τ2=τ>0. Critical values at which Hopf-bifurcation occur have been obtained. The results show that, at critical values τ10=0.109≈3 hrs, τ20=26 days, and τc=2.069 days, the endemic equilibrium losses stability.We investigated the effect of two delaysτ1 and τ2 on the stability of model (1). Based on the numerical simulations obtained in this paper, we found out that when τ1, τ2 are below the critical values τ10 and τ20, respectively, model (1) is asymptotically stable, which implies that the number of individuals in the five subpopulations will be in ideal equilibrium and prevalence of pneumococcal pneumonia can easily be controlled. Conversely, if the values of the delays τ1, τ2 are greater than the critical values τ10 and τ20, respectively, a Hopf-bifurcation arises and this phenomenon suggests persistent of pneumococcal pneumonia in the population. The number of individuals in the five subpopulations of model (1) will fluctuate periodically; this is not helpful; effort should be put to control such a phenomenon.Longer time delays destabilize the system and give rise to Hopf-bifurcations. This explains the oscillatory seasonal change of pneumococcal pneumonia disease in human population whose immune systems are weak. Therefore, measures to reduce delays in latent and seeking medical care during pneumococcal pneumonia epidemic should be prioritized. The results herein could be helpful to direct future research of bacterial infections that become severe in individuals that have history of exposure to viral infections such as influenza A virus.



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