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 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.
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 . Pneumococcal pneumonia infections may follow a viral infection, like a cold or flu (influenza) , 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) . 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 . 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 , 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 .
Vaccination is a highly efficient means of preventing diseases and death . 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 . 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.  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 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.
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 time , denoted by . The population is subdivided into six mutually exclusive epidemiological classes: susceptible, vaccinated, exposed, carrier, and infected denoted by , , , , and , respectively. The mathematical formulation adopts a mass-action incidence because it is important in deciding the dynamics of epidemic models , where the contact rate depends on the size of the total human population . We assume a continuous vaccination strategy that is received by the recruited susceptible individuals at a rate and that vaccination does not affect the infectious . 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 , where is the proportion of the serotype not covered by vaccine . The increase in the number of susceptible individuals comes from a constant recruitment 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 . The susceptible individuals become infected through a force of infection and move to the latent class .
The latent class, , accounts for a time delay 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 is and exposed individuals transfer to the infectious class at a rate . Individuals in the carrier class become symptomatic and join the infected class at a rate .
The infectious class accounts for a time delay : the time taken by infected individuals to seek medical care. We assume that infected individuals who survive the natural mortality through the infectious period have a survivorship function . Moreover, infected individuals that delay to seek medical care die of pneumococcal pneumonia at a rate . Infectious individuals upon recovery transfer to the susceptible class at a rate . All classes exhibit a per capita natural mortality rate .
The compartmental diagram of the model is shown in Figure 1.
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  and Yang et al. ; we let be a Banach space of continuous real valued functions equipped with the supremum norm, . The initial conditions of system (1) are represented bywhere and , such that . The following Lemma establishes the positivity of the solutions of system (1).
Lemma 1. Any solution of trajectories (1) with ; remains positive whenever it exists.
Proof. Suppose was to lose positivity on some local existence interval for some constant ; there would be a time at such that .
From the first equation of system (1), it follows thatThis implies that for , where is an arbitrary small positive constant. This leads to a contradiction; it thus follows that is always positive. Hence from the fundamental theory of differential equations, it is shown that there exists a unique solution for of system (1) with initial data in as follows:Therefore,Since , then , . This completes the proof.
Lemma 2. The closed setis positively invariant and absorbing with respect to the set of DDEs (1).
Proof. Summing all equations in system (1) yieldsTherefore, which implies that if . Using the standard comparison test in , we get . Particularly, if for all time ; hence is positively invariant. Further, if , then either the solution enters at finite time nor is close to and the infected variables , and tend to zero. Therefore, is attracting implying that all solutions in finally enter . Consequently, in , 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 ratio is computed using an approach in  and is given bywhereprovided the validity of holds.
The quantity measures the expected number of secondary cases generated by an index case for the susceptible individuals and represents new cases arising from the vaccination program.
Remark 3. The control reproduction ratio with no delays () is given by
3. Stability of Equilibria
Let 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 haveHence, from system (15), we obtain the disease-free equilibrium , whereprovided
It should be noted that, for , the disease-free equilibrium is biologically feasible for any epidemiological parameters, whereas in the absence of vaccination strategy, i.e., for , is only feasible for epidemiological parameters in the susceptible class. From system (15) the endemic equilibrium is given aswhere
3.1. Local Stability of the Disease-Free Equilibrium Point
Suppose that is a disease-free equilibrium point of system (1), then the linearization matrix is given by
Clearly is one of the negative roots (eigenvalues) that guarantee local stability of the disease-free equilibrium . The remaining eigenvalues are obtained from the characteristic polynomial given bywhere , , , and .
Thus computing the roots of polynomial (19) givesSince the rest of the roots are negative, root is negative provided holds implying that
Thus we have the result below
Proposition 4. The disease-free equilibrium is locally asymptotically stable if the control reproduction ratio , whenever conditions and are satisfied and unstable otherwise.
The biological implication of Proposition 4 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 , to obtain, , , , , , , , , , , , , , , , , and .
3.3. Delay-Free System
Here, to show the local stability of , we consider a situation where there are no delays during the latent period () and in seeking medical care (). By letting , (23) reduces towith coefficients of polynomial equation in Appendix A.
Proposition 5. The endemic equilibrium is locally asymptotically stable in the absence of delays , iff the following Routh–Hurwitz conditions are satisfied:with , , , , and defined in Appendix A.2.
Since there exists a positive root for model (1), there is a stability change from unstable to stable of the endemic equilibrium point 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  to prove the conditions for continuation of unstable or stable switches at the endemic equilibrium point, by choosing time delay as the bifurcation parameter.
4.1. Delay Only in Latent Period
In such a situation the characteristic equation (23) reduces towhere
Suppose the endemic equilibrium of system (1) is stable in the absence of delay () to seek medical care, implying that . The bifurcation value of occurs when is purely imaginary, for . Hence, defining the eigenvalue , with infection rate oscillation frequency () and making a substitution in (27) and expressing the exponential in terms of trigonometric ratios, we getwhere
By eliminating from (27), squaring and adding these two equations, and putting , we obtain the Hopf frequency below:whereThe two propositions about stability and critical delay in Wesley et al.  are written as lemmas
Lemma 6. If the guarantee the Routh–Hurwitz criteria, then all eigenvalues of (31) have negative real parts for all delay . Thus the endemic equilibrium if it exists is locally asymptotically stable whenever , provided the endemic steady state is stable in the absence of the latent period delay; specifically will not affect the stability of the dynamical system, for (31) without positive real roots.
Lemma 7. If do not satisfy Routh–Hurwitz criteria, thus or implies that (47) has at least one positive root and suppose that it has a pair of imaginary roots say for such a value of .
Consequently to obtain the main results in this paper, we assume (31) has at least one positive root . By squaring and summing together the imaginary and real parts in (29), we getBy denotingthis allows us to defineand state the result as follows.
Lemma 8. If and are defined as (35) and . The endemic equilibrium point is linearly asymptotically stable for and unstable for and undergoes a Hopf-bifurcation at .
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 nonzero velocity.
Proof (transversality condition for Hopf-bifurcation). Differentiating (27) with respect to we obtainwithRemark 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 .
Therefore, we getwhereLet , if conditions , , and hold. Clearlyhas the same sign as . This completes the proof.
Therefore, Proposition 9 implies that given , the eigenvalue of the characteristic equation (27) close to crosses the imaginary axis from the left to the right as continuously changes from a value less than to one greater than .
4.2. Delay Only in Seeking Medical Care by the Infectious
To understand the influence of time delay in seeking medical care, we set in (23) yieldingwhere
Proposition 11. The endemic equilibrium point is locally asymptotically stable (LAS) for where is the minimum positive value of
Proof. Let be a root of (42) to obtainUsing Euler expansion and separating real and imaginary parts, we obtainEliminating from (46), by squaring and adding these two equations and putting , we obtain the Hopf frequency below:with coefficients in (47) in Appendix A.
Let us denote Since and , then (47) has at least one positive root. Assuming (47) has positive roots, given by , denote by , respectively. Then, (47) has positive roots ifFrom (46), the corresponding , for which the characteristic equation (23) has a pair of purely imaginary roots, is derived to haveThus, denotingwhere , then are a pair of purely imaginary roots of (23). This allows us to define the Hopf-bifurcation threshold time delay value asThis completes the proof.
Proof (transversality condition for Hopf-bifurcation). In order to establish whether the endemic equilibrium point actually undergoes a Hopf-bifurcation at , we let be a root of (23) near and , as . Making a substitution into the L.H.S. of (23) and taking a derivative with respect to , we haveComputing the Sign of , by differentiating the characteristic equation (23) with respect to and evaluating (53) at with and expressing and , we obtain ,with coefficients in Appendix A.
By Remark 10, (55) giveswithLetIf , with , then , and hence the transversality condition holds and the system undergoes Hopf-bifurcation.
4.3. Delay in Latent Period and Seeking Medical Care
Making a substitution of in (23), we getwith