Hopf-bifurcation analysis of pneumococcal pneumonia with time delays

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 ratio R0 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 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. Keywords: Time delay, Pneumococcal pneumonia, Vaccination, Stability, Hopf-bifurcation


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 [2,10]. Pneumococci are different, with 90 recognized serotypes; several of these serotypes are capable of causing invasive disease [4].
Pneumococcal pneumonia infections may follow a viral infection, like a cold or flu (influenza) [27], 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 2 respiratory tract infections (e.g., pneumonia), and upper respiratory tract infections (e.g., otitis media and sinusitis) [3]. The wide spread of the disease may be promoted by potentially asymptomatic persons (incubation individuals) [15,16] and an individual remains in the exposed class for a certain latent period prior to becoming infective [13,14].
Diseases exhibit a lot of economic burden including productivity loss, health care related expenses, losses due to disease related mortality and loss of employment [47]. 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 [5], most of which occur in low and middle-income countries [6,7]. 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 [8].
Vaccination is a highly efficient means of preventing diseases and death [34]. 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 [41]. 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 [22,23].
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 [9,12,21]. 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 Hopfbifurcations [9,17]. Work done by researchers for example [13,42,43,44,45] demonstrate the role played by time delays in different capacities in controlling the spread of infectious diseases. Sharma et al. [1] 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 increments 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 3 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 Section 2, 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.

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 time t, denoted by N(t). The population is sub-divided 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's important in deciding the dynamics of epidemic models [39], where the contact rate depends on the size of the total human population [19]. We assume a continuous vaccination strategy that is received by the recruited susceptible individuals at a rate ν, and that vaccination doesn't affect the infectious [18]. 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 sero-type not covered by vaccine [20]. 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 and exposed individuals transfer to 1 e   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 description of model variables and parameters is summarized in Table 1 and Table 2 below.  The compartmental diagram of the model is shown in Figure.1.

Positivity of solutions
System (1) is a representation of the dynamics of the human populations, thus it is required that all solutions are non-negative. We use the approach of Bodna [35] and Yang et al. [36], we let C be a where and such that .
The following Lemma establishes the positivity of the solutions of system (1). 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 as follows Therefore, ( , , , , ) , Since S(t1) > 0, then S(t) > 0, t ≥ 0. This completes the proof.
Similarly, it can be shown that and (7) Therefore, from the above integral forms of equations (3) to (7) all solution trajectories are positive for all time t > 0 on [0, +∞].

Boundedness
For boundedness of system (1) with initial condition (2), we consider the following Lemma: is positively invariant and absorbing with respect to the set of DDE's (1).

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 ratio R0 is computed using an approach in [24] and is 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 and , such that at equilibrium, we have (8), we obtain the disease-free equilibrium 00 0 provided It should be noted that for ν > 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 (8) the endemic where .

Local stability of the disease-free equilibrium point
Suppose that P0 = (S 0 , V 0 ,0, 0, 0) is a disease -free equilibrium point of system (1), then the linearization matrix JP0 is given by Clearly y1 = −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 (11) where . ** * To illustrate the stability of disease-free equilibrium, we use parameter values in Table 2   The biological implication of Proposition 3.1, means that in the long run the vaccinated and susceptible populations will be stable and pneumococcal pneumonia will be under control.

The transcendental equation
We obtain the expression for the transcendental equation by linearizing system (1) around , to obtain (12) and .
The variational matrix of (12) is given by Then, we obtain the transcendental equation of the linearized system at P * : with coefficients of the transcendental equation (13) given in Appendix A.1.

Delay-free system
Here, to show the local stability of P * , 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, equation (13) reduces to: (14) with coefficients of polynomial equation in Appendix A.2. Numerically, using parameter values in Table 2 the characteristic equation (14) is given as The resulting eigenvalues are given by: 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 Hopfbifurcation.

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 [38] to prove the conditions for continuation of unstable or stable switches at the endemic equilibrium point. By choosing time delay as the bifurcation parameter.

Delay only in latent period ( )
In such a situation the characteristic equation (13) reduces to (15) where Suppose the endemic equilibrium of system (1) is stable in the absence of delay ( ) to seek medical care, implying that Re(λ) = ξ (0) < 0. The bifurcation value of occurs when is purely imaginary, for . Hence, defining the eigenvalue with infection rate oscillation frequency ( ) and making a substitution in (15) and expressing the exponential in terms of trigonometric ratios, we get, Where By eliminating from equation (15), squaring and adding these two equations and putting we obtain the Hopf frequency below: where The two Propositions about stability and critical delay in Wesley et al. [37] are written as lemmas 12    Consequently, to obtain the main results in this paper, we assume equation (17) has at least one positive root w10. By squaring and summing together the imaginary and real parts in equation (16)   To ensure the occurrence of the Hopf bifurcation, it is desirable to verify the transversality condition. Without loss of generality, the delay 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 non-zero velocity. Therefore, Proposition 4.1 implies that given m > 0, the eigenvalue of the characteristic equation (15) close to crosses the imaginary axis from the left to the right as continuously changes from a value less than to one greater than

), arctan c w k w k c k w c w w h w h c w h w h c c c c D c c c c c c c c D c w h w c c c c
where,

Proposition 4.2. The endemic equilibrium point P * is locally asymptotically stable (LAS)
for where, is the minimum positive value of

Proof.
Let be a root of equation (22) to obtain .
Then, equation (24) p w p w p q w q w q q w q w p w w p w w q w q w q q w q w q w q w p n n n n n n n n n n n n n n n n p w p w p q w q w q q w q w p w w p w w q w q w q q w q w q w q w p

Proof (Transversality condition for Hopf-bifurcation)
In order to establish whether the endemic equilibrium point P * actually under goes a Hopfbifurcation at we let be a root of equation (13) near and as Making a substitution into the L.H.S of equation (13) and taking a derivative with respect to λ, we have (27) Computing the Sign of by differentiating the characteristic equation (13)   .
If with then hence the transversality condition holds and the system undergoes Hopf-bifurcation.

Delay in latent period and seeking medical care ( )
Making a substitution of in equation (13), we get (30) with In order to examine whether or not the endemic equilibrium loses stability and undergoes Hopfbifurcation as an outcome with inclusion of the time delays, a pair of purely imaginary root of the transcendental equation (30) is found. Suppose the pair of the imaginary root is given as with infection rate oscillation frequency ( ), using Euler's expansion and making a substitution into equation (30), separating real and imaginary parts, we obtain: where Squaring and adding equation (31) and (32), we get following equation:   3  2  4  2  3  5  3  0  1  3  1  2  4  0  2  2  3  3  1  1  3  3  3  1  1   24  2  2  2  4  0  0 , , ,

g s v s v g s v s v s g s v g s v s v G v k s s v k s
which reduces to (35) Let such that we obtain equation (35) in terms of : with Since equation (36), has a high degree polynomial we compute the eigenvalues numerically by using parameter values in

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 is P * = (S * , V * , E * , C * , I * ) = (2099,6,54,2,100). In the absence of delays , the characteristic polynomial equation (14) is The corresponding eigenvalues are; Therefore, since the eigenvalues have one positive root and four negative roots, the endemic equilibrium changes state of stability from unstable to stable thus under goes 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 individual  The rest of the parameters remain fixed as in Table 2 The numerical simulation of equation (15) yields the characteristic roots as; As increase from zero, there is a value such that the endemic equilibrium is stable for and unstable for At this critical value, the endemic equilibrium loses stability and Hopf bifurcation arises. The real positive root is and the critical time delay of a day ≈ 3 hrs. The numerical computation of equation (22) Table  2.  (e) (f) Figure 5: Stability of the endemic equilibrium P * for days. The rest of the parameters are as in Table 2.  Table 2 To explore the effect of time delay on pneumococcal pneumonia, we fix time delay days, and the parameter 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.  Table 2 In Figure 7, time delay is fixed at 2 days in order to study the effect of time delay on model (1).
We observe an increase in the magnitude of the amplitude of oscillations as 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 We investigated the effect of two delays and on the stability of model (1). Basing on the numerical simulations obtained in this paper, we found out that when , are below the critical values 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 value of the delays , are greater than the critical values and τ20 respectively, a Hopf-bifurcation arises 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  a a  a  a a a a  a a a a  a a a a  a a a a  a a a a a  a a a a  a a a   l  a a a a a a  a a a a a a  a a a a a a  a a a a a a . m a a a a a a n  a a a  a a a a a a n  a a a  a a a a   n a a a