# Application of Optimal Control to Influenza Pneumonia Coinfection with Antiviral Resistance

Authors: Caroline W. Kanyiri; Livingstone Luboobi; Mark Kimathi
Journal: Computational and Mathematical Methods in Medicine
 (2020)

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## Abstract

Influenza and pneumonia independently lead to high morbidity and mortality annually among the human population globally; however, a glaring fact is that influenza pneumonia coinfection is more vicious and it is a threat to public health. Emergence of antiviral resistance is a major impediment in the control of the coinfection. In this paper, a deterministic mathematical model illustrating the transmission dynamics of influenza pneumonia coinfection is formulated having incorporated antiviral resistance. Optimal control theory is then applied to investigate optimal strategies for controlling the coinfection using prevalence reduction and treatment as the system control variables. Pontryagin’s maximum principle is used to characterize the optimal control. The derived optimality system is solved numerically using the Runge–Kutta-based forward-backward sweep method. Simulation results reveal that implementation of prevention measures is sufficient to eradicate influenza pneumonia coinfection from a given population. The prevention measures could be social distancing, vaccination, curbing mutation and reassortment, and curbing interspecies movement of the influenza virus.


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## Body

## 1. Introduction

Clinical evidence points out that infection with a particular combination of pathogens results in an aggravated infection with more severe clinical outcome compared with infection with either pathogen alone [1]. This is specially true for influenza virus and bacterium Streptococcus pneumoniae [2–4]. Influenza and pneumonia each contributes greatly to the global burden of morbidity and leads to high death toll, typically over a relatively short period of time [5–8]. Streptococcus pneumoniae, Haemophilus influenzae, and Staphylococcus aureus are the most common causes of pneumonia, the chief being Streptococcus pneumoniae [9, 10]. Coinfection resulting from influenza virus and Streptococcus pneumoniae further increases morbidity and mortality and is a major public health concern. These two pathogens rank among the chief pathogens affecting humans, and their ability to work together presents a major threat to world health [11]. Coinfection greatly impairs the host’s immune system, increases antibacterial therapy intolerance, and can be detrimental to the diagnosis of the disease [12]. According to [13], it can be difficult to identify influenza patients experiencing bacterial coinfections due to symptom overlap of influenza and bacterial infections. In [14], it is indicated that a strong index of suspicion and additional diagnostic testing may be required for diagnosis and treatment of the infections.The morbidity, mortality, and economic burden resulting from the lethal synergism that exists between influenza virus and pneumococcus are of major concern globally. The catastrophic 1918 influenza pandemic is an extreme example of the impact that results from this cooperative interaction [11]. Lung tissue samples examined from those who died during this pandemic revealed that the majority of deaths were as a result of secondary bacterial pneumonia. Data from the subsequent 1957, 1968, and 2009 influenza pandemics are consistent with these findings [15, 16]. In addition, during seasonal influenza outbreaks, coinfections resulting from influenza and Streptococcus pneumoniae have been associated with high morbidity and mortality rates [17–19]. According to [11], influenza virus alters the lungs in such a way that predisposes them to invasion by pneumococcus rendering a mild influenza infection severe or even fatal. This could be through several ways such as epithelial damage, changes in airway function, upregulation and exposure of receptors, dampening of the immune response, or amplification of inflammation. Several studies have been carried out to investigate the time course of susceptibility to Streptococcus pneumoniae after influenza virus infection. Results revealed that on average, individuals developed coinfection within 6 days after influenza virus infection [20–23].Emergence of drug resistance, which has become a global concern, complicates influenza pneumonia coinfection even more. Drug resistance refers to the ability of disease-causing agents to resist the effects of drugs, thereby making the conventional treatment procedure ineffective. This leads to persistence of infections in the body, hence increasing the risk of spread to other individuals [24, 25]. The evolution of drug resistance is accelerated by overuse and misuse of antimicrobials, inappropriate use of antimicrobials, subtherapeutic dosing, and patient noncompliance with the recommended course of treatment [26]. There are two classes of antiviral drugs that are approved to treat influenza infections; these are M2 ion-channel inhibitors and neuraminidase (NA) inhibitors. However, due to antiviral drug resistance in influenza virus, neuraminidase (NA) inhibitors are the only class of antiinfluenza drugs currently in use as most of the circulating influenza viruses have acquired resistance to M2 ion-channel inhibitors [27, 28]. Moreover, many circulating influenza viruses have also acquired resistance to neuraminidase (NA) inhibitors [28, 29] raising an alarm in the health sector. Drug resistance continues to threaten effective prevention and treatment of influenza pneumonia coinfection. In addition, the cost of health care for patients with resistant infections is much higher than care for patients with nonresistant infections especially due to longer duration of illness.Strategies such as vaccination, isolation, and treatment among others are necessary in order to curb the spread of various infectious diseases. However, if they are not administered at the right time and in the right amount, curtailing the spread of the infectious diseases remains a difficult task. The application of optimal control is therefore very vital since it is a necessary tool in making decisions of the viable control strategies to be employed in eradicating diseases. Optimal control theory has been applied in the study of influenza, for instance, in [30–34], and in the study of pneumonia, for instance, in [35]. Given that influenza pneumonia coinfection is more disastrous than either of the single infections alone, this paper seeks to investigate optimal control strategies for the influenza pneumonia coinfection and in the emergence of antiviral resistance.


## 2. Model Formulation

The model presented in this paper has the total population subdivided into eight compartments. These are susceptible (S), infected with wild-type influenza strain (Iw), infected with resistant influenza strain (IR), infected with both wild-type influenza strain and pneumonia (Iwp), infected with both resistant influenza strain and pneumonia (IRp), recovered from influenza (Rz), recovered from pneumonia (Rp), and recovered from both influenza and pneumonia (Rzp). In the model, individuals are first infected with influenza virus and then contract bacterial pneumonia. An assumption is made that there is no primary bacterial pneumonia infection. Individuals enter the population via immigration at the rate of π, and all recruited individuals are assumed to be susceptible. The susceptible can either become infected with wild-type influenza strain or resistant influenza strain at the rates of λ1 and λ2, respectively, where λ1=βwIw+η1Iwp, while λ2=βr¯IR+η2IRp, where βr¯=fβr,b, in which parameter b refers to the rate of developing drug resistance while parameters βw and βr refer to the transmission rate of wild-type strain and resistant strain, respectively. Parameters η1 and η2 are modification parameters accounting for the relative infectiousness of individuals in Iwp and IRp classes in comparison with those in Iw and IR, respectively. Those infected with influenza wild-type strain can recover at a rate of α1 while those infected with influenza resistant strain recover at the rate of α2. The model incorporates development of antiviral resistance; hence, individuals infected with influenza wild-type strain progress to the IR class at a rate of b. Those infected with influenza wild-type strain and those infected with influenza resistant strain can contract secondary bacterial pneumonia at the rate of λ3 with the force of infection of pneumonia being βpIwp+IRp. Parameter βp refers to the transmission rate of pneumonia. Individuals in Iw and IR classes also suffer disease-induced death at the rates of δ1 and δ2, respectively. Individuals infected with both the wild-type influenza strain and pneumonia and those infected with both resistant strain and pneumonia can recover from either influenza alone at rates of α3 and α4, respectively, or pneumonia alone at rates of ϕ1 and ϕ2, respectively, or recover from both influenza and pneumonia at rates of ω1 and ω2, respectively. This means that the Rz class contains some individuals from Iwp and IRp classes that have only recovered from influenza but may still have pneumonia. Rz class is however considered noninfectious given that bacterial pneumonia is considered weakly infectious after antimicrobials are administered. On the other hand, the Rp class contains some individuals from Iwp and IRp classes that have only recovered from pneumonia but may still have influenza. Given that the infectious period of influenza is about seven days after symptoms onset, Rp is considered noninfectious. Individuals in Iwp and IRp suffer disease-induced death at the rate of δ3 and δ4, respectively. Those infected with both wild-type influenza strain and pneumonia could also develop antiviral resistance and progress to IRp class at a rate of b. Individuals who have recovered from influenza, pneumonia, and both influenza and pneumonia lose their immunity and become susceptible again at rates of ϑ1, ϑ2, and ϑ3, respectively. Individuals in all the epidemiological compartments suffer natural death at the rate of μ. Figure 1 shows the population flow between the different compartments.Figure 1
Schematic diagram showing population flow between different epidemiological classes for influenza pneumonia coinfection.


### 2.1. Model Equations

Given the dynamics described in Figure1, the following system of nonlinear ordinary differential equations, with nonnegative initial conditions, describes the dynamics of influenza pneumonia coinfection:(1)dSdt=π+ϑ1Rzt+ϑ2Rpt+ϑ3Rzpt−λ1+λ2+μStdIwdt=λ1St−λ3+b+α1+μ+δ1IwtdIRdt=λ2St+bIwt−λ3+α2+μ+δ2IRtdIwpdt=λ3Iwt−α3+ϕ1+ω1+b+μ+δ3IwptdIRpdt=λ3IRt+bIwpt−α4+ϕ2+ω2+μ+δ4IRptdRzdt=α1Iwt+α2IRt+α3Iwpt+α4IRpt−ϑ1+μRztdRpdt=ϕ1Iwpt+ϕ2IRpt−ϑ2+μRptdRzpdt=ω1Iwpt+ω2IRpt−ϑ3+μRzpt,where(2)λ1=βwIw+η1Iwp,λ2=βr¯IR+η2IRp,λ3=βpIwp+IRp.We assume that all the model parameters are positive and the initial conditions of model system (1) are given by(3)S0>0,Iw0≥0,IR0≥0,Iwp0≥0,IRp0≥0,Rz0≥0,Rp0≥0,Rzp0≥0.Table1 gives the description of the various parameters used in the model.Table 1
Description of parameters used.
Parameters
Description
π
Recruitment rate
β
w
Transmission rate of wild-type influenza strain
β
r
Transmission rate of resistant influenza strain
β
p
Transmission rate of pneumonia
α
1
Recovery rate of influenza for individuals inIw class
α
2
Recovery rate of influenza for individuals inIR class
α
3
Recovery rate of influenza for individuals inIwp class
α
4
Recovery rate of influenza for individuals inIRp class
ϕ
1
Recovery rate of pneumonia for individuals inIwp class
ϕ
2
Recovery rate of pneumonia for individuals inIRp class
ω
1
Recovery rate of both influenza and pneumonia for individuals inIwp class
ω
2
Recovery rate of both influenza and pneumonia for individuals inIRp class
ϑ
1, ϑ2, ϑ3
Rate of losing immunity for influenza, pneumonia, and influenza and pneumonia, respectively
b
Rate of developing antiviral resistance
δ
1, δ2, δ3, δ4
Disease-induced death rates inIw,IR,Iwp, and IRp classes, respectively
μ
Natural death rate


## 2.1. Model Equations

Given the dynamics described in Figure1, the following system of nonlinear ordinary differential equations, with nonnegative initial conditions, describes the dynamics of influenza pneumonia coinfection:(1)dSdt=π+ϑ1Rzt+ϑ2Rpt+ϑ3Rzpt−λ1+λ2+μStdIwdt=λ1St−λ3+b+α1+μ+δ1IwtdIRdt=λ2St+bIwt−λ3+α2+μ+δ2IRtdIwpdt=λ3Iwt−α3+ϕ1+ω1+b+μ+δ3IwptdIRpdt=λ3IRt+bIwpt−α4+ϕ2+ω2+μ+δ4IRptdRzdt=α1Iwt+α2IRt+α3Iwpt+α4IRpt−ϑ1+μRztdRpdt=ϕ1Iwpt+ϕ2IRpt−ϑ2+μRptdRzpdt=ω1Iwpt+ω2IRpt−ϑ3+μRzpt,where(2)λ1=βwIw+η1Iwp,λ2=βr¯IR+η2IRp,λ3=βpIwp+IRp.We assume that all the model parameters are positive and the initial conditions of model system (1) are given by(3)S0>0,Iw0≥0,IR0≥0,Iwp0≥0,IRp0≥0,Rz0≥0,Rp0≥0,Rzp0≥0.Table1 gives the description of the various parameters used in the model.Table 1
Description of parameters used.
Parameters
Description
π
Recruitment rate
β
w
Transmission rate of wild-type influenza strain
β
r
Transmission rate of resistant influenza strain
β
p
Transmission rate of pneumonia
α
1
Recovery rate of influenza for individuals inIw class
α
2
Recovery rate of influenza for individuals inIR class
α
3
Recovery rate of influenza for individuals inIwp class
α
4
Recovery rate of influenza for individuals inIRp class
ϕ
1
Recovery rate of pneumonia for individuals inIwp class
ϕ
2
Recovery rate of pneumonia for individuals inIRp class
ω
1
Recovery rate of both influenza and pneumonia for individuals inIwp class
ω
2
Recovery rate of both influenza and pneumonia for individuals inIRp class
ϑ
1, ϑ2, ϑ3
Rate of losing immunity for influenza, pneumonia, and influenza and pneumonia, respectively
b
Rate of developing antiviral resistance
δ
1, δ2, δ3, δ4
Disease-induced death rates inIw,IR,Iwp, and IRp classes, respectively
μ
Natural death rate


## 3. Coinfection Model with Controls

In order to identify optimal control strategies that minimize the number of infected individuals and the cost of implementing the controls, a mathematical optimal control problem is formulated and analysed. Influenza pneumonia coinfection model (1) is extended to include time-dependent control measures. Let uit,i=1,2,3,4,5, represent the time-dependent controls whereby controls u1t,u2t, and u3t relate to prevalence reduction of wild-type influenza strain, resistant influenza strain, and pneumonia, respectively. The infection prevalence reduction could be through social distancing, vaccination, curbing mutation and reassortment, and curbing interspecies movement of the influenza virus. Controls u4t and u5t relate to treatment of the wild-type and resistant influenza strains, respectively.Time is specified and is given byt∈0,T, where T is the final time. Given that there is a limitation on the maximum rate of treatment and prevalence reduction controls, bounded Lebesgue measurable control set is introduced and defined as(4)U=u1,u2,u3,u4,u5,0≤ui≤uimax,i=1,2,3,4,5.To identify the required level of effort to control the infection, an objective functional to be minimized is given by(5)Ju1,u2,u3,u4,u5=∫0TA1Iw+A2IR+A3Iwp+A4IRp+12∑i=15qiui2dt,subject to the differential system:(6)dSdt=π+ϑ1Rzt+ϑ2Rpt+ϑ3Rzpt−1−u1λ1+1−u2λ2+μStdIwdt=1−u1λ1St−1−u3λ3Iwt−α1+u4Iwt−b+μ+δ1IwtdIRdt=1−u2λ2St+bIwt−1−u3λ3IRt−α2+u5IRt−μ+δ2IRtdIwpdt=1−u3λ3Iwt−α3+ϕ1+ω1+b+μ+δ3IwptdIRpdt=1−u3λ3IRt+bIwpt−α4+ϕ2+ω2+μ+δ4IRptdRzdt=α1+u4Iwt+α2+u5IRt+α3Iwpt+α4IRpt−ϑ1+μRztdRpdt=ϕ1Iwpt+ϕ2IRpt−ϑ2+μRptdRzpdt=ω1Iwpt+ω2IRpt−ϑ3+μRzpt.The coefficientsA1,A2,A3, and A4 represent the costs associated with minimizing the infected population. On the other hand, the expression 1/2qiui2 represents costs associated with controls ui,i=1,2,3,4,5. Quadratic expressions of the cost of controls are considered because costs follow a nonlinear representation especially at high intervention levels. The objective of minimizing the infected population and the cost of controls can be achieved through proper implementation of the controls over a time interval given by 0,T. Therefore, we seek an optimal control set u1∗,u2∗,u3∗,u4∗,u5∗ such that(7)Ju1∗,u2∗,u3∗,u4∗,u5∗=minu1,u2,u3,u4,u5Ju1,u2,u3,u4,u5.The necessary conditions for the existence of an optimal solution come from Pontryagin’s Maximum Principle [36]. This principle converts (5)–(6) into a problem of minimizing pointwise Hamiltonian H, with respect to u1,u2,u3,u4,u5, and is obtained as(8)H=A1Iw+A2IR+A3Iwp+A4IRp+12q1u12+12q2u22+12q3u32+12q4u42+12q5u15+p1tπ+ϑ1Rzt+ϑ2Rpt+ϑ3Rzpt−1−u1λ1+1−u2λ2+μSt+p2t1−u1λ1St−1−u3λ3Iwt−α1+u4Iwt−b+μ+δ1Iwt+p3t1−u2λ2St+bIwt−1−u3λ3IRt−α2+u5IRt−μ+δ2IRt+p4t1−u3λ3Iwt−α3+ϕ1+ω1+b+μ+δ3Iwpt+p5t1−u3λ3IRt+bIwpt−α4+ϕ2+ω2+μ+δ4IRpt+p6tα1+u4Iwt+α2+u5IRt+α3Iwpt+α4IRpt−ϑ1+μRzt+p7tϕ1Iwpt+ϕ2IRpt−ϑ2+μRpt+p8tω1Iwpt+ω2IRpt−ϑ3+μRzpt,where pit,i=1,…,8 are the corresponding adjoint or costate variables to be determined by applying Pontryagin’s Maximum Principle as in Theorem 1.Theorem 1.
Given optimal control setu1∗,u2∗,u3∗,u4∗,u5∗ and the corresponding solutions S∗∗,Iw∗∗,IR∗∗,Iwp∗∗,IRp∗∗,Rz∗∗,Rp∗∗,andRzp∗∗ of system (6) that minimizes Ju1,u2,u3,u4,u5 over U, there exist adjoint variables pi,i=1,…,8, such that(9)dp1dt=−∂H∂S,dp2dt=−∂H∂Iw,dp3dt=−∂H∂IR,dp4dt=−∂H∂Iwp,dp5dt=−∂H∂IRp,dp6dt=−∂H∂Rz,dp7dt=−∂H∂Rp,dp8dt=−∂H∂Rzp,with transversality condition piT=0 for i=1,…,8.
Evaluating (9) leads to the following adjoint system:(10)dp1dt=−p1−μ−1−u2βR¯IR+η2IRp−1−u1βwIw+η1Iwp−p31−u2βR¯IR+η2IRp−p21−u1βwIw+η1Iwp,dp2dt=−A1−p2−α1−b−δ1−μ−1−u3βpIwp+IRp+1−u1βwS−u4−bp3+p11−u1βwS−p6α1+u4−p41−u3βpIwp+IRp,dp3dt=−A2−p3−α2−δ2−μ−1−u3βpIwp+IRp+1−u2βR¯S−u5+p11−u2βR¯S−p51−u3βpIwp+IRp−p6α2+u5,dp4dt=−A3−p5b+1−u3βpIR−p4−α3−b−δ3−μ+1−u3βpIw−ω1−ϕ1−α3p6+p31−u3βpIR+p1βwη1S1−u1−p2βwη1S1−u1−1−u3βpIw−p8ω1−p7ϕ1dp5dt=−A4−α4p6+βR¯η2p1S1−u2−p3βR¯η2S1−u2−1−u3βpIR+p21−u3βpIw−p5−α4−δ4−μ+1−u3βpIR−ω2−ϕ2−p41−u3βpIw−p8ω2−p7ϕ2dp6dt=−p6−μ−ϑ1−p1ϑ1dp7dt=−p7−μ−ϑ2−p1ϑ2dp8dt=−p8−μ−ϑ3−p1ϑ3.
Following the results in [37, 38], the existence of optimal control is stated and proved using Theorem 2.Theorem 2.
There exists optimal controlsu1∗,u2∗,u3∗,u4∗,u5∗ which minimizes J over the region U satisfying the optimality condition:(11)u1∗=minmax0,u1¯,u1max,u2∗=minmax0,u2¯,u2max,u3∗=minmax0,u3¯,u3max,u4∗=minmax0,u4¯,u4max,u5∗=minmax0,u5¯,u5max,where(12)u¯1=βwS∗∗p2−p1Iw∗∗+η1Iwp∗∗q1,u¯2=βR¯S∗∗p3−p1IR∗∗+η2IRp∗∗q2,u¯3=βpIwp∗∗+IRp∗∗Iw∗∗p4−p2+p5−p3βpIwp∗∗+IRp∗∗IR∗∗q3,u¯4=p2−p6Iw∗∗q4,u¯5=p3−p6IR∗∗q5.Proof.
The HamiltonianH is minimized with respect to the controls u1,u2,u3,u4, and u5 at the optimal control functions. This is done by differentiating the Hamiltonian function H with respect to each of the control variables on the set U; that is,(13)∂H∂ui=0.
The following set of optimality conditions is thus obtained:(14)∂H∂u1=p1S∗∗βwIw∗∗+η1Iwp∗∗−p2S∗∗βwIw∗∗+η1Iwp∗∗+q1u1=0,∂H∂u2=p1S∗∗βR¯IR∗∗+η2IRp∗∗−p3S∗∗βR¯IR∗∗+η2IRp∗∗+q2u2=0,∂H∂u3=p3βpIR∗∗Iwp∗∗+IRp∗∗−p5βpIR∗∗Iwp∗∗+IRp∗∗+p2βpIw∗∗Iwp∗∗+IRp∗∗−p4βpIw∗∗Iwp∗∗+IRp∗∗+q3u3=0,∂H∂u4=−p2Iw∗∗+p6Iw∗∗+q4u4=0,∂H∂u5=−p3IR∗∗+p6IR∗∗+q5u5=0,at u1=u1∗,u2=u2∗,u3=u3∗,u4=u4∗, and u5=u5∗, respectively. Solving for u1∗,u2∗,u3∗,u4∗, and u5∗ and using the bounds for the controls in U, that is,(15)ui∗=0,ifui¯≤0,ui¯,if0<ui¯<uimaxuimax,ifui¯≥uimaxin compact notation yields,(16)u1∗=minmax0,βwS∗∗p2−p1Iw∗∗+η1Iwp∗∗q1,u1max,u2∗=minmax0,βR¯S∗∗p3−p1IR∗∗+η2IRp∗∗q2,u2max,u3∗=minmax0,βpIwp∗∗+IRp∗∗Iw∗∗p4−p2+p5−p3βpIwp∗∗+IRp∗∗IR∗∗q3,u3max,u4∗=minmax0,p2−p6Iw∗∗q4,u4max,u5∗=minmax0,p3−p6IR∗∗q5,u5max.
Next, the optimality system is obtained as(17)dSdt=π+ϑ1Rz+ϑ2Rp+ϑ3Rzp−1−u1∗λ1+1−u2∗λ2+μS,dIwdt=1−u1∗λ1S−1−u3∗λ3Iw−α1+u4∗Iw−b+μ+δ1Iw,dIRdt=1−u2∗λ2S+bIw−1−u3∗λ4IR−α2+u5∗IR−μ+δ2IR,dIwpdt=1−u3∗λ3Iw−α3+ϕ1+ω1+b+μ+δ3Iwp,dIRpdt=1−u3∗λ4IR+bIwp−α4+ϕ2+ω2+μ+δ4IRp,dRzdt=α1+u4∗Iw+α2+u5∗IR+α3Iwp+α4IRp−ϑ1+μRz,dRpdt=ϕ1Iwp+ϕ2IRp−ϑ2+μRp,dRzpdt=ω1Iwp+ω2IRp−ϑ3+μRzp,dp1dt=−p1−μ−1−u2βR¯IR+η2IRp−1−u1βwIw+η1Iwp−p31−u2βR¯IR+η2IRp−p21−u1βwIw+η1Iwp,dp2dt=−A1−p2−α1−b−δ1−μ−1−u3βpIwp+IRp+1−u1βwS−u4−bp3+p11−u1βwS−p6α1+u4−p41−u3βpIwp+IRp,dp3dt=−A2−p3−α2−δ2−μ−1−u3βpIwp+IRp+1−u2βR¯S−u5+p11−u2βR¯S−p51−u3βpIwp+IRp−p6α2+u5,dp4dt=−A3−p5b+1−u3βpIR−p4−α3−b−δ3−μ+1−u3βpIw−ω1−ϕ1−α3p6+p31−u3βpIR+p1βwη1S1−u1−p2βwη1S1−u1−1−u3βpIw−p8ω1−p7ϕ1,dp5dt=−A4−α4p6+βR¯η2p1S1−u2−p3βR¯η2S1−u2−1−u3βpIR+p21−u3βpIw−p5−α4−δ4−μ+1−u3βpIR−ω2−ϕ2−p41−u3βpIw−p8ω2−p7ϕ2,dp6dt=−p6−μ−ϑ1−p1ϑ1,dp7dt=−p7−μ−ϑ2−p1ϑ2,dp8dt=−p8−μ−ϑ3−p1ϑ3.with piT=0 for i=1,…,8 and S0=S0,Iw0=Iw0,IR0=IR0,Iwp0=Iwp0,IRp0=IRp0,Rz0=Rz0,Rp0=Rp0,andRzp0=Rzp0.


## 4. Numerical Simulation

In this section, the optimal solution of optimality system (17) is investigated. Given that there are initial conditions for the state variables and terminal conditions for the adjoints, the optimality system is a two-point boundary value problem with separated boundary conditions at times t=0 and t=T. The forward-backward sweep method described in [39, 40] is hence used for the numerical solution of the optimal control problem. The method is named based on how the algorithm solves the problem’s state and adjoint equations. The state variables are solved using the forward difference scheme while the adjoint variables are solved using the backward difference scheme. The solution iterative scheme makes an initial guess of the controls and using that guess solves the state system forward in time using a 4th-order Runge–Kutta scheme. Next, using the stored values of the controls and the solution of the state system, the adjoint system is solved backward in time using a 4th-order Runge–Kutta scheme. The RK4 method has to be adapted to account for solving backward in time. The controls are then updated using a convex combination of the previous controls and the values obtained using the characterizations. The updated controls are then used to repeat the solution of the state and adjoint systems. This process is repeated with values in the current iteration being tested for convergence against a user provided tolerance and depending on that, the algorithm either starts the process over again using the updated control or the algorithm terminates. The final approximations for the control, state, and adjoint systems are considered to be the solution to the optimal control problem.


### 4.1. Simulation Results

The numerical simulations were carried out using the MATLAB software and the parameter values in Table2. There is however paucity of published estimates of morbidity and mortality rates resulting from influenza pneumonia coinfection in the middle- and low-income countries. On the other hand, in high-income, temperate countries where influenza surveillance has been done for years, these rates are well documented [47]; hence the data used in this research study are from these countries. The population under consideration is children <5 years and adults >60 years. Studies show that influenza pneumonia coinfection affects people of all ages; however, the morbidity and mortality rates of the high risk group which includes the immunocompromised, children <5 years, and adults >60 years are disproportionately high [9, 47, 48].Table 2
Description and values of the different parameters used.
Parameter
Description
Value
Reference.
π
Recruitment rate
0.0381
Assumed
β
w
Transmission rate of wild-type influenza strain
0.0102 day−1
Assumed
β
r
Transmission rate of resistant influenza strain
0.00026 day−1
Assumed
β
p
Transmission rate of pneumonia
0.000162 day−1
Reference [41]
α
1
Recovery rate of influenza for individuals inIw class
0.07143 day−1
Reference [42]
α
2
Recovery rate of influenza for individuals inIR class
0.0333 day−1
Assumed
α
3
Recovery rate of influenza for individuals inIwp class
0.04762 day−1
Reference [43].
α
4
Recovery rate of influenza for individuals inIRp class
0.0222 day−1
Assumed
ϕ
1
Recovery rate of pneumonia for individuals inIwp class
0.033 day−1
Reference [41]
ϕ
2
Recovery rate of pneumonia for individuals inIRp class
0.033 day−1
Reference [41]
ω
1
Recovery rate of both influenza and pneumonia for individuals inIwp class
0.0166 day−1
Assumed
ω
2
Recovery rate of both influenza and pneumonia for individuals inIRp class
0.0166 day−1
Assumed
ϑ
1
Rate of losing immunity for influenza
0.00833 day−1
Reference [44]
ϑ
2
Rate of losing immunity for pneumonia
0.00833 day−1
Assumed
ϑ
3
Rate of losing immunity for influenza pneumonia coinfection
0.00833 day−1
Assumed
b
Rate of developing antiviral resistance
0.0118
Assumed
δ
1
Wild-type influenza strain-induced death rate
0.01
Reference [45]
δ
2
Resistant influenza strain-induced death rate
0.021
Assumed
δ
3
I
w
p class disease-induced death rate
0.05
Assumed
δ
4
I
R
p class disease-induced death rates
0.05
Assumed
1
μ
Average human lifespan
70
×
365 days
Reference [46]Most of the parameter values have a range as indicated in the references given in Table2. Given that this research study focuses on a deterministic mathematical model to give an indication of the likely dynamics of influenza pneumonia coinfection, baseline parameter values were used for the simulations.


#### 4.1.1. Control by Treatment Only

Figures2 and 3 show the effect of using treatment efforts only in an attempt to curb the spread of influenza pneumonia coinfection.Figure 2
Individuals coinfected with wild-type influenza and pneumonia.Figure 3
Individuals coinfected with resistant influenza and pneumonia.From Figure2, it can be observed that with and without the treatment efforts, the wild-type influenza and pneumonia coinfection persists in the population.Without any controls, it can be observed from Figure3 that the number of individuals coinfected with resistant influenza and pneumonia increases initially and then decreases slowly but not to zero. With the treatment control strategies in place, the number of the infected individuals is lower but not significant and does not decrease to zero by day 30. This shows that use of treatment alone as a control strategy is not effective in curbing the spread of influenza pneumonia coinfection. As explained in details in [4, 49–51], treatment is often rendered ineffective because it is not always possible to administer the drugs at the right time and the diagnosis of the coinfection can be challenging because of timing of sample collection and false negative results when viruses replicate in the lower respiratory tract. In addition, treatment often aggravates development of resistance.


#### 4.1.2. Control by Prevention Measures Only

Simulations are done when there is no control strategy in place and when there are controls involving prevention of wild-type influenza strain, prevention of influenza resistant strain, and prevention of pneumonia. Figures4 and 5 show the results.Figure 4
Individuals coinfected with wild-type influenza and pneumonia.Figure 5
Individuals coinfected with resistant influenza and pneumonia.It can be observed from Figure4 that when preventive efforts are implemented as control strategies, the number of individuals coinfected with wild-type influenza and pneumonia decreases to zero.Similarly, from Figure5, it can be observed that with preventive efforts as control strategies, the number of individuals coinfected with resistant influenza strain and pneumonia decreases drastically right from the beginning to zero by day 30.The prevention measures help to reduce the transmission of the coinfection. Comparing Figures2 and 3 with Figures 4 and 5, it can be observed that the preventive control strategies are more effective in curbing the spread of the coinfection compared to treatment control strategies.


#### 4.1.3. Control with Prevention and Treatment of Influenza

Simulations are carried out to investigate the effect of implementing control strategies involving the prevention and treatment of influenza. Figures6 and 7 show the results.Figure 6
Individuals coinfected with wild-type influenza and pneumonia.Figure 7
Individuals coinfected with resistant influenza and pneumonia.It can be observed from Figure6 that with the prevention and treatment of influenza as control strategies, the number of individuals coinfected with wild-type influenza and pneumonia decreases right from the beginning and it is at zero by about day 25.Figure7 shows that with the implementation of these control strategies, the number of individuals coinfected with resistant influenza and pneumonia significantly decreases and it is inconsequential at day 30 implying that it is unlikely for the coinfection to persist in the population.The prevention and treatment of influenza as control strategies aid in reducing the transmission and in treatment of those who are already infected; however, the treatment poses a danger of development of drug resistance. Therefore, caution should be taken during drug administration.


#### 4.1.4. Control with All Strategies

When all the control strategies are applied, the number of infected individuals decreases as shown in Figures8 and 9. From Figure 8, it can be observed that with all the control strategies in place, the number of individuals coinfected with wild-type influenza and pneumonia decreases to zero by day 30.Figure 8
Individuals coinfected with wild-type influenza and pneumonia.Figure 9
Individuals coinfected with resistant influenza and pneumonia.It can also be observed from Figure9 that when all the control strategies are applied, the number of individuals coinfected with resistant influenza and pneumonia drastically decreases, and by about day 28, the number is already at zero.


## 4.1. Simulation Results

The numerical simulations were carried out using the MATLAB software and the parameter values in Table2. There is however paucity of published estimates of morbidity and mortality rates resulting from influenza pneumonia coinfection in the middle- and low-income countries. On the other hand, in high-income, temperate countries where influenza surveillance has been done for years, these rates are well documented [47]; hence the data used in this research study are from these countries. The population under consideration is children <5 years and adults >60 years. Studies show that influenza pneumonia coinfection affects people of all ages; however, the morbidity and mortality rates of the high risk group which includes the immunocompromised, children <5 years, and adults >60 years are disproportionately high [9, 47, 48].Table 2
Description and values of the different parameters used.
Parameter
Description
Value
Reference.
π
Recruitment rate
0.0381
Assumed
β
w
Transmission rate of wild-type influenza strain
0.0102 day−1
Assumed
β
r
Transmission rate of resistant influenza strain
0.00026 day−1
Assumed
β
p
Transmission rate of pneumonia
0.000162 day−1
Reference [41]
α
1
Recovery rate of influenza for individuals inIw class
0.07143 day−1
Reference [42]
α
2
Recovery rate of influenza for individuals inIR class
0.0333 day−1
Assumed
α
3
Recovery rate of influenza for individuals inIwp class
0.04762 day−1
Reference [43].
α
4
Recovery rate of influenza for individuals inIRp class
0.0222 day−1
Assumed
ϕ
1
Recovery rate of pneumonia for individuals inIwp class
0.033 day−1
Reference [41]
ϕ
2
Recovery rate of pneumonia for individuals inIRp class
0.033 day−1
Reference [41]
ω
1
Recovery rate of both influenza and pneumonia for individuals inIwp class
0.0166 day−1
Assumed
ω
2
Recovery rate of both influenza and pneumonia for individuals inIRp class
0.0166 day−1
Assumed
ϑ
1
Rate of losing immunity for influenza
0.00833 day−1
Reference [44]
ϑ
2
Rate of losing immunity for pneumonia
0.00833 day−1
Assumed
ϑ
3
Rate of losing immunity for influenza pneumonia coinfection
0.00833 day−1
Assumed
b
Rate of developing antiviral resistance
0.0118
Assumed
δ
1
Wild-type influenza strain-induced death rate
0.01
Reference [45]
δ
2
Resistant influenza strain-induced death rate
0.021
Assumed
δ
3
I
w
p class disease-induced death rate
0.05
Assumed
δ
4
I
R
p class disease-induced death rates
0.05
Assumed
1
μ
Average human lifespan
70
×
365 days
Reference [46]Most of the parameter values have a range as indicated in the references given in Table2. Given that this research study focuses on a deterministic mathematical model to give an indication of the likely dynamics of influenza pneumonia coinfection, baseline parameter values were used for the simulations.


### 4.1.1. Control by Treatment Only

Figures2 and 3 show the effect of using treatment efforts only in an attempt to curb the spread of influenza pneumonia coinfection.Figure 2
Individuals coinfected with wild-type influenza and pneumonia.Figure 3
Individuals coinfected with resistant influenza and pneumonia.From Figure2, it can be observed that with and without the treatment efforts, the wild-type influenza and pneumonia coinfection persists in the population.Without any controls, it can be observed from Figure3 that the number of individuals coinfected with resistant influenza and pneumonia increases initially and then decreases slowly but not to zero. With the treatment control strategies in place, the number of the infected individuals is lower but not significant and does not decrease to zero by day 30. This shows that use of treatment alone as a control strategy is not effective in curbing the spread of influenza pneumonia coinfection. As explained in details in [4, 49–51], treatment is often rendered ineffective because it is not always possible to administer the drugs at the right time and the diagnosis of the coinfection can be challenging because of timing of sample collection and false negative results when viruses replicate in the lower respiratory tract. In addition, treatment often aggravates development of resistance.


### 4.1.2. Control by Prevention Measures Only

Simulations are done when there is no control strategy in place and when there are controls involving prevention of wild-type influenza strain, prevention of influenza resistant strain, and prevention of pneumonia. Figures4 and 5 show the results.Figure 4
Individuals coinfected with wild-type influenza and pneumonia.Figure 5
Individuals coinfected with resistant influenza and pneumonia.It can be observed from Figure4 that when preventive efforts are implemented as control strategies, the number of individuals coinfected with wild-type influenza and pneumonia decreases to zero.Similarly, from Figure5, it can be observed that with preventive efforts as control strategies, the number of individuals coinfected with resistant influenza strain and pneumonia decreases drastically right from the beginning to zero by day 30.The prevention measures help to reduce the transmission of the coinfection. Comparing Figures2 and 3 with Figures 4 and 5, it can be observed that the preventive control strategies are more effective in curbing the spread of the coinfection compared to treatment control strategies.


### 4.1.3. Control with Prevention and Treatment of Influenza

Simulations are carried out to investigate the effect of implementing control strategies involving the prevention and treatment of influenza. Figures6 and 7 show the results.Figure 6
Individuals coinfected with wild-type influenza and pneumonia.Figure 7
Individuals coinfected with resistant influenza and pneumonia.It can be observed from Figure6 that with the prevention and treatment of influenza as control strategies, the number of individuals coinfected with wild-type influenza and pneumonia decreases right from the beginning and it is at zero by about day 25.Figure7 shows that with the implementation of these control strategies, the number of individuals coinfected with resistant influenza and pneumonia significantly decreases and it is inconsequential at day 30 implying that it is unlikely for the coinfection to persist in the population.The prevention and treatment of influenza as control strategies aid in reducing the transmission and in treatment of those who are already infected; however, the treatment poses a danger of development of drug resistance. Therefore, caution should be taken during drug administration.


### 4.1.4. Control with All Strategies

When all the control strategies are applied, the number of infected individuals decreases as shown in Figures8 and 9. From Figure 8, it can be observed that with all the control strategies in place, the number of individuals coinfected with wild-type influenza and pneumonia decreases to zero by day 30.Figure 8
Individuals coinfected with wild-type influenza and pneumonia.Figure 9
Individuals coinfected with resistant influenza and pneumonia.It can also be observed from Figure9 that when all the control strategies are applied, the number of individuals coinfected with resistant influenza and pneumonia drastically decreases, and by about day 28, the number is already at zero.


## 4.1.1. Control by Treatment Only

Figures2 and 3 show the effect of using treatment efforts only in an attempt to curb the spread of influenza pneumonia coinfection.Figure 2
Individuals coinfected with wild-type influenza and pneumonia.Figure 3
Individuals coinfected with resistant influenza and pneumonia.From Figure2, it can be observed that with and without the treatment efforts, the wild-type influenza and pneumonia coinfection persists in the population.Without any controls, it can be observed from Figure3 that the number of individuals coinfected with resistant influenza and pneumonia increases initially and then decreases slowly but not to zero. With the treatment control strategies in place, the number of the infected individuals is lower but not significant and does not decrease to zero by day 30. This shows that use of treatment alone as a control strategy is not effective in curbing the spread of influenza pneumonia coinfection. As explained in details in [4, 49–51], treatment is often rendered ineffective because it is not always possible to administer the drugs at the right time and the diagnosis of the coinfection can be challenging because of timing of sample collection and false negative results when viruses replicate in the lower respiratory tract. In addition, treatment often aggravates development of resistance.


## 4.1.2. Control by Prevention Measures Only

Simulations are done when there is no control strategy in place and when there are controls involving prevention of wild-type influenza strain, prevention of influenza resistant strain, and prevention of pneumonia. Figures4 and 5 show the results.Figure 4
Individuals coinfected with wild-type influenza and pneumonia.Figure 5
Individuals coinfected with resistant influenza and pneumonia.It can be observed from Figure4 that when preventive efforts are implemented as control strategies, the number of individuals coinfected with wild-type influenza and pneumonia decreases to zero.Similarly, from Figure5, it can be observed that with preventive efforts as control strategies, the number of individuals coinfected with resistant influenza strain and pneumonia decreases drastically right from the beginning to zero by day 30.The prevention measures help to reduce the transmission of the coinfection. Comparing Figures2 and 3 with Figures 4 and 5, it can be observed that the preventive control strategies are more effective in curbing the spread of the coinfection compared to treatment control strategies.


## 4.1.3. Control with Prevention and Treatment of Influenza

Simulations are carried out to investigate the effect of implementing control strategies involving the prevention and treatment of influenza. Figures6 and 7 show the results.Figure 6
Individuals coinfected with wild-type influenza and pneumonia.Figure 7
Individuals coinfected with resistant influenza and pneumonia.It can be observed from Figure6 that with the prevention and treatment of influenza as control strategies, the number of individuals coinfected with wild-type influenza and pneumonia decreases right from the beginning and it is at zero by about day 25.Figure7 shows that with the implementation of these control strategies, the number of individuals coinfected with resistant influenza and pneumonia significantly decreases and it is inconsequential at day 30 implying that it is unlikely for the coinfection to persist in the population.The prevention and treatment of influenza as control strategies aid in reducing the transmission and in treatment of those who are already infected; however, the treatment poses a danger of development of drug resistance. Therefore, caution should be taken during drug administration.


## 4.1.4. Control with All Strategies

When all the control strategies are applied, the number of infected individuals decreases as shown in Figures8 and 9. From Figure 8, it can be observed that with all the control strategies in place, the number of individuals coinfected with wild-type influenza and pneumonia decreases to zero by day 30.Figure 8
Individuals coinfected with wild-type influenza and pneumonia.Figure 9
Individuals coinfected with resistant influenza and pneumonia.It can also be observed from Figure9 that when all the control strategies are applied, the number of individuals coinfected with resistant influenza and pneumonia drastically decreases, and by about day 28, the number is already at zero.


## 5. Conclusion

As observed from Figures2–9, different intervention mechanisms produce different results. Of great importance to public health is the control strategy that will help eradicate the diseases. Given that availability of resources is always a factor to consider when implementing control strategies, it is paramount to have a strategy with maximum benefit. From the results of this study, with the implementation of prevention measures only, which include and not limited to social distancing, vaccination, curbing mutation and reassortment, and curbing interspecies movement of the influenza virus, influenza pneumonia coinfection can be eradicated from a given population. Compared to the other control strategies investigated and discussed in Section 4.1, these preventive control strategies are more effective in curbing the spread of influenza pneumonia coinfection; hence, the public health sector and other stake holders could apply them to eradicate the coinfection within a given population.



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*Source: 5984095-2020-03-11.xml*