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–Kuttabased forwardbackward 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.
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 [
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 [
Emergence of drug resistance, which has become a global concern, complicates influenza pneumonia coinfection even more. Drug resistance refers to the ability of diseasecausing 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 [
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 [
The model presented in this paper has the total population subdivided into eight compartments. These are susceptible (S), infected with wildtype influenza strain (
Schematic diagram showing population flow between different epidemiological classes for influenza pneumonia coinfection.
Given the dynamics described in Figure
We assume that all the model parameters are positive and the initial conditions of model system (
Table
Description of parameters used.
Parameters  Description 


Recruitment rate 

Transmission rate of wildtype influenza strain 

Transmission rate of resistant influenza strain 

Transmission rate of pneumonia 

Recovery rate of influenza for individuals in 

Recovery rate of influenza for individuals in 

Recovery rate of influenza for individuals in 

Recovery rate of influenza for individuals in 

Recovery rate of pneumonia for individuals in 

Recovery rate of pneumonia for individuals in 

Recovery rate of both influenza and pneumonia for individuals in 

Recovery rate of both influenza and pneumonia for individuals in 

Rate of losing immunity for influenza, pneumonia, and influenza and pneumonia, respectively 

Rate of developing antiviral resistance 

Diseaseinduced death rates in 

Natural death rate 
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 (
Time is specified and is given by
To identify the required level of effort to control the infection, an objective functional to be minimized is given by
The coefficients
The necessary conditions for the existence of an optimal solution come from Pontryagin’s Maximum Principle [
Given optimal control set
Evaluating (
Following the results in [
There exists optimal controls
The Hamiltonian
The following set of optimality conditions is thus obtained:
Next, the optimality system is obtained as
In this section, the optimal solution of optimality system (
The numerical simulations were carried out using the MATLAB software and the parameter values in Table
Description and values of the different parameters used.
Parameter  Description  Value  Reference. 


Recruitment rate  0.0381  Assumed 

Transmission rate of wildtype influenza strain  0.0102 day 
Assumed 

Transmission rate of resistant influenza strain  0.00026 day 
Assumed 

Transmission rate of pneumonia  0.000162 day 
Reference [ 

Recovery rate of influenza for individuals in 
0.07143 day 
Reference [ 

Recovery rate of influenza for individuals in 
0.0333 day 
Assumed 

Recovery rate of influenza for individuals in 
0.04762 day 
Reference [ 

Recovery rate of influenza for individuals in 
0.0222 day 
Assumed 

Recovery rate of pneumonia for individuals in 
0.033 day 
Reference [ 

Recovery rate of pneumonia for individuals in 
0.033 day 
Reference [ 

Recovery rate of both influenza and pneumonia for individuals in 
0.0166 day 
Assumed 

Recovery rate of both influenza and pneumonia for individuals in 
0.0166 day 
Assumed 

Rate of losing immunity for influenza  0.00833 day 
Reference [ 

Rate of losing immunity for pneumonia  0.00833 day 
Assumed 

Rate of losing immunity for influenza pneumonia coinfection  0.00833 day 
Assumed 

Rate of developing antiviral resistance  0.0118  Assumed 

Wildtype influenza straininduced death rate  0.01  Reference [ 

Resistant influenza straininduced death rate  0.021  Assumed 


0.05  Assumed 


0.05  Assumed 

Average human lifespan 

Reference [ 
Most of the parameter values have a range as indicated in the references given in Table
Figures
Individuals coinfected with wildtype influenza and pneumonia.
Individuals coinfected with resistant influenza and pneumonia.
From Figure
Without any controls, it can be observed from Figure
Simulations are done when there is no control strategy in place and when there are controls involving prevention of wildtype influenza strain, prevention of influenza resistant strain, and prevention of pneumonia. Figures
Individuals coinfected with wildtype influenza and pneumonia.
Individuals coinfected with resistant influenza and pneumonia.
It can be observed from Figure
Similarly, from Figure
The prevention measures help to reduce the transmission of the coinfection. Comparing Figures
Simulations are carried out to investigate the effect of implementing control strategies involving the prevention and treatment of influenza. Figures
Individuals coinfected with wildtype influenza and pneumonia.
Individuals coinfected with resistant influenza and pneumonia.
It can be observed from Figure
Figure
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.
When all the control strategies are applied, the number of infected individuals decreases as shown in Figures
Individuals coinfected with wildtype influenza and pneumonia.
Individuals coinfected with resistant influenza and pneumonia.
It can also be observed from Figure
As observed from Figures
The data used to support the findings of this study are included within the article.
The authors declare that they have no conflicts of interest.
The supplementary material is the source code (MATLAB) that was used to do the numerical simulations.