Mathematical Analysis of Influenza A Dynamics in the Emergence of Drug Resistance

Every year, influenza causes high morbidity and mortality especially among the immunocompromised persons worldwide. The emergence of drug resistance has been a major challenge in curbing the spread of influenza. In this paper, a mathematical model is formulated and used to analyze the transmission dynamics of influenza A virus having incorporated the aspect of drug resistance. The qualitative analysis of the model is given in terms of the control reproduction number, Rc. The model equilibria are computed and stability analysis carried out. The model is found to exhibit backward bifurcation prompting the need to lower Rc to a critical value Rc∗ for effective disease control. Sensitivity analysis results reveal that vaccine efficacy is the parameter with the most control over the spread of influenza. Numerical simulations reveal that despite vaccination reducing the reproduction number below unity, influenza still persists in the population. Hence, it is essential, in addition to vaccination, to apply other strategies to curb the spread of influenza.


Introduction
Influenza is a contagious respiratory illness caused by influenza viruses. ere are three major types of flu viruses: types A, B, and C. e majority of human infections are caused by types A and B. Of major concern is influenza A virus which is clinically the most vicious. It is a negativesense single-stranded RNA virus with eight gene segments. e segmented nature of influenza A virus genome allows the exchange of gene segments between viruses that coinfect the same cell [1]. is process of genetic exchange is termed reassortment. Reassortment leads to sudden changes in viral genetics and to susceptibility in hosts. Influenza A virus has a wide range of susceptible avian hosts and mammalian hosts such as humans, pigs, horses, seals, and mink. In addition, the virus is able to repeatedly switch hosts to infect multiple avian and mammalian species. e unpredictability of influenza A virus evolution and interspecies movement creates continual public health challenges [2].
Influenza A virus constantly mutates and is able to elude the immune system of an individual. It can mutate in two different ways: antigenic shift and antigenic drift. Antigenic shift is an abrupt, major change in the influenza virus which happens occasionally and results in a new subtype that most people have no protection against. Such a shift occurred in the spring of 2009 in Mexico and United States, when H1N1 virus with a new combination of genes emerged to infect people and quickly spread, causing a pandemic [3]. is antigenic shift was as a result of extensive reassortment in swine that brought together genes from avian, swine, and human flu viruses [4]. On the other hand, antigenic drift refers to small changes in the genes of influenza viruses that occur continually as the virus replicates. Over time, these small genetic changes result in new strains which the antibodies can no longer recognize. e changes in the influenza viruses are the main reason why individuals are infected with the flu more than once. e viruses infect the nose, throat, and lungs. ey usually are spread through the air when the infected people cough, sneeze, or talk making the surrounding air and surfaces to be temporarily contaminated with infected droplets [5,6]. People get infected when they inhale the infected droplets. A person might also get flu by touching the surface or object that has flu virus on it and then touching their own mouth, eyes, or possibly their nose [6].
Influenza can be prevented by getting vaccination each year. However, given that the virus mutates rapidly, a vaccine made for one year may not be useful in the following year. In addition, antigenic drift in the virus may occur after the year's vaccine has been formulated, rendering the vaccine less protective, and hence, outbreaks can easily occur especially among high-risk individuals [7]. According to [8], other preventive actions include staying away from people who are sick, covering coughs and sneezes, and frequent handwashing.
Influenza spreads rapidly around the world during seasonal epidemics and pandemics [9]. It has afflicted the human population for centuries. For instance, the 1918 influenza pandemic infected nearly one quarter of the world's population and resulted in the deaths of about 100 million people [10]. Studies show that this pandemic is especially responsible for the high morbidity and mortality among vulnerable groups such as children, the elderly, and patients with underlying health conditions [11]. Within the past one hundred years, there have been four pandemics resulting from the emergence of a novel influenza strain for which the human population possessed little or no immunity. Table 1 gives a brief summary of the four influenza pandemics.
Besides the influenza pandemics, there is an outbreak of influenza every year around the world which results in about three to five million cases of severe illness and about 250,000 to 500,000 deaths [14]. According to a report by Centers for Disease Control and Prevention (CDC), as of December 2017, the estimated number of deaths worldwide resulting from seasonal influenza had risen to between 291,000 and 646,000 [15]. is new estimate was from a collaborative study by CDC and global health partners. In the temperate northern hemisphere (i.e., north of the Tropic of Cancer) and temperate southern hemisphere (i.e., south of the Tropic of Capricorn), influenza has been observed to peak in the winter months [16,17]. In tropical regions, influenza seasonality is less obvious and epidemics can occur throughout the year and more specifically during the rainy seasons [18]. According to [19], the mortality rates due to this respiratory disease are much higher in Africa than anywhere else in the world. Poor nutritional status, poor access to healthcare including vaccination and antibiotics, and the presence of other, less measurable factors related to poverty in Africa may be additional risk factors for higher mortality rates. WHO Global Influenza Surveillance and Response System (GISRS) monitors the evolution of influenza viruses.  [20,21].
Influenza-attributable mortality varies across the seasons.
ere is however paucity of published estimates of influenza mortality for low-and middle-income countries. Data from Centers for Disease Control and Prevention (CDC) databases from the 1999-2000 to the 2014-2015 seasons for the U.S. population aged 65 years and above were used to estimate excess deaths per month over that 15-year span [22]. e data are presented in Figure 3.
In addition to pandemics and seasonal epidemics caused by influenza A virus, over the past 20 years, multiple zoonotic influenza A virus outbreaks have occurred causing a great concern to public health [23][24][25][26]. For instance, H5N1 influenza virus from avian hosts poses an ongoing threat to human and animal health due to its high mortality rate [26][27][28]. H7N9 is yet another highly pathogenic subtype of influenza A virus that is of major concern. According to the World Health Organization (WHO), as of January 2018, 1566 laboratory-confirmed cases of human infection with H7N9 virus have been reported in China, including at least 613 deaths [29]. In addition to the ongoing H5N1 and H7N9 influenza A virus outbreaks, other subtypes, such as H5N6, H9N2, H10N8, and H6N1, have sporadically caused serious human infections in China and Taiwan [30][31][32][33]. e death toll from influenza is unacceptably high, given that it is preventable. Efforts to combat it must therefore be accelerated. In view of the catastrophic effects of influenza globally, several models have been proposed and analyzed with the aim of shedding more light in the transmission dynamics of influenza, for instance [34][35][36][37][38][39][40][41]. Among the pioneer mathematical models used to describe influenza dynamics is one developed by [38].
Emergence of drug resistance which is a growing menace globally [42] complicates influenza even more [43,44]. Drug resistance refers to reduction in the effectiveness of a drug in curing a disease. It occurs when microorganisms such as bacteria, viruses, fungi, and parasites change in ways that render the medications used to cure the infections they cause ineffective [45,46]. e microorganisms are therefore able to survive the treatment. According to [47], epidemics with drug-resistant strains and those with drug-sensitive strains are fundamentally different in their growth and dynamics. Drug-sensitive epidemics are fuelled by only one process, that is, transmission; however, drug-resistant epidemics are fuelled by two processes: transmission and the conversion of treated drug-sensitive infections to drug-resistant infections (acquired resistance). erefore, the rate of increase in drugresistant infections can be much faster than the rate of increase in drug-sensitive infections. Studies from [48] show that drug resistance is a function of time and treatment rate. In addition, immunosuppression especially in individuals with compromised immune systems contributes to lack of viral clearance often despite antiviral therapy leading to emergence of antiviral resistance [49]. ere are two classes of antiviral drugs that are used to treat influenza: adamantanes and neuraminidase inhibitors. e adamantanes are only effective against influenza A viruses, as they inhibit the M2 protein, which is not coded by influenza B [50]. ese drugs are associated with several toxic effects and rapid emergence of drug-resistant strains. e neuraminidase inhibitors interfere with the release of progeny influenza virus from infected host cells, a process that prevents infection of new host cells and thereby halts the spread of infection in the respiratory tract [7]. Since these drugs act at the stage of viral replication, they must be administered as early as possible. According to [51],    [52]. With the development of drug-resistant influenza viruses, various models have also been formulated in order to understand this phenomenon better. Among them are [53][54][55][56][57]. e morbidity, mortality, and economic burden of influenza cannot be overlooked. With the emerging menace of drug resistance, this burden becomes even more complicated. In order to curb the spread of influenza, there is a dire need to understand among its many aspects, its transmission dynamics especially in light of the drug resistance aspect. In this paper, a mathematical model that illustrates the transmission dynamics of a wild-type influenza strain and the development and transmission of drug-resistant influenza strain is formulated and analyzed.

Model Formulation.
e model subdivides the total population into five compartments: Susceptible (S), Vaccinated (V), Infected with Wild-type strain (I w ), Infected with Resistant strain (I R ), and Recovered (R). Individuals in a given compartment are assumed to have similar characteristics. Parameters vary from compartment to compartment but are identical for all individuals in a given compartment. Individuals enter the population at the rate of π, and all recruited individuals are assumed to be susceptible.
e Susceptible get infected after effective contact with either the Infected with Wild-type strain or the Infected with Resistant strain. e force of infection is given by either λ 1 � β w I w (Infection by Wild-type strain) or λ 2 � β r I R (Infection by Resistant strain), where β r � f(β r , b). Parameters β w and β r refer to the transmission rate of wild-type strain and resistant strain, respectively. Parameter b is the rate of developing drug resistance. e susceptible can only be infected by one strain at a time. e rate of vaccination is ϕ. e vaccinated can also become infected with either the wild-type strain or the resistant strain. is depends on the vaccine efficacy. When the vaccine efficacy is 100%, the vaccinated cannot become infected. Individuals who are infected with the wild-type strain are treated and recover at the rate of α, while those who are infected with the resistant strain recover at the rate of α r . e wild-type strain is assumed to mutate to resistant strain, and hence, those infected with the wild type join those infected with the resistant strain at the rate of b. Individuals with wild-type strain and those with resistant strain suffer disease-induced death at the rates a w and a r , respectively. e recovered lose immunity at the rate of ϑ joins the susceptible class. Individuals in all the epidemiological compartments suffer natural death at the rate of μ. e model diagram is given in Figure 4. Figure 4, the following system of nonlinear ordinary differential equations, with nonnegative initial conditions, describes the dynamics of influenza:

Model Equations. Given the dynamics described in
where We assume that all the model parameters are positive and the initial conditions of the model system (1) are given by  initial conditions will remain nonnegative for all t > 0. us, we have the following theorem:

Model Analysis
Theorem 1. Given that the initial conditions of system (1) are us t > 0, and it follows directly from the first equation Using the integrating factor method to solve inequality (4), we have Integrating both sides yields where C is the constant of integration. Hence, Hence, From the second equation in system (1), we obtain Hence, Similarly, it can be shown that erefore, all the solutions of system (1) with nonnegative initial conditions will remain nonnegative for all time t > 0.

Invariant Region.
We show that the total population is bounded for all time t > 0. e analysis of system (1) will therefore be analyzed in a region Ω of biological interest.
us, we have the following theorem on the region that system (1) is restricted to.
is means that N(t) ≤ max N(0), (Π/μ) . erefore, N(t) is bounded above. Subsequently, S(t), V(t), I w (t), I R (t), and R(t) are bounded above. us, in Ω, system (1) is well posed. Hence, it is sufficient to study the dynamics of the system in Ω.

Existence of Equilibrium Points.
In the absence of influenza (I w � I R � 0), system (1) has a disease-free equilibrium, which is given by , 0, 0, 0 .

e Control Reproduction Number.
e control reproduction number, R c , is a key threshold that determines the behaviour of the system in the presence of vaccination. In order to analyze the stability of system (1), we obtain the threshold condition for the establishment of the disease.
us, we employ next-generation matrix operator method as explained in [60]. e matrices of new infections and transition terms evaluated at the disease-free equilibrium are given by e dominant eigenvalue corresponding to the spectral radius ρ(FV −1 ) of the matrix FV −1 is the control reproduction number, which is given by where R cw is a measure of the average number of secondary wildtype influenza infections caused by a single infected individual introduced into the model population. On the other hand, R cr gives the average number of secondary resistant influenza infections caused by one infected individual introduced into the model population.
From eorem 2 in [60], we have the following results.

Proposition 1.
e disease-free equilibrium is locally asymptotically stable whenever R c is less than unity and unstable otherwise. Proof.
e Jacobian matrix evaluated at E 0 is obtained as For the DFE to be locally stable, the eigenvalues of J(E 0 ) must have negative real parts. e characteristic polynomial of J(E 0 ) is given by

(20)
Clearly, the following eigenvalues with negative real parts can be obtained from the polynomial (20): Other roots can be obtained from the remaining part of the polynomial (20), which is given by
We therefore conclude that the disease-free equilibrium E 0 is locally asymptotically stable whenever R c < 1. e biological implication of Proposition 1 is that if R c < 1, influenza will be eliminated from the model population provided that the initial sizes of the subpopulations in various compartments of model (1) are in the basin of attraction of the influenza-free equilibrium.

Effective Reproduction Number.
e effective reproduction number (R e (t)) is the actual average number of secondary cases per primary case at calendar time t (for t > 0) [61]. R e (t) shows time-dependent variation due to decline in susceptible individuals and the implementation of control measures. e effective reproduction number is therefore used to characterize transmissibility in a population that is not entirely susceptible. It is the basic reproduction number times the fraction of the population that is susceptible to infection at time t. e basic reproduction number (R 0 ) is the average number of secondary infections generated by a single infective individual in a totally susceptible population [60]. From model (1), the basic reproduction number is obtained as us, the effective reproduction number R e (t) � fR 0 , where f is the fraction of population susceptible to infection at a time t.

Endemic Equilibria.
e endemic equilibria of model (1) are the steady states where influenza may persist in the population. is happens when at least one of the infected classes of the model is nonempty. e rate of change in populations in each compartment is zero at equilibrium; hence, the right-hand side of (1) is set to zero as follows: Next, S * , V * , I * w , I * R , and R * are solved from (24) in terms of the two forces of infection, λ 1 and λ 2 to obtain where

Computational and Mathematical Methods in Medicine
Upon dividing and simplifying the two expressions for λ 1 and λ 2 , we obtain the following polynomial: Note that if λ 1 � 0 in the equation obtained when polynomial (27) is set to zero, then clearly λ 2 � 0. is gives the disease-free equilibrium previously obtained in (15). e solutions to the remaining part of the polynomial (27), described by (28), define the possible endemic states of system (1).
e existence of the endemic equilibrium points for system (1) depends on the solutions of (28), and the roots of the equation must be real and positive to guarantee existence of the endemic equilibrium point(s). Due to mathematical complexity, we are not able to express explicitly the endemic steady states of system (1). We shall however represent the polynomial in (28) graphically as shown in Figure 5.
From the surface plot in Figure 5, it can be observed that there exist endemic steady states for the two-strain influenza model. e steady states only exist for positive values of p(λ 1 , λ 2 ). e endemic equilibria exist in the case where only the wild-type strain is present, the case where only the resistant strain exists or both strains coexist.

Existence of an Endemic State with Wild-Type Strain
Only.
ere exists an endemic state when the wild-type strain persists and the resistant strain dies out. Solving (1) in terms of λ 1 yields Substituting I * w obtained in (29) into λ * 1 yields polynomial (30) given by It is important to note that when λ 1 � 0, a wild-type strain-free equilibrium is obtained which is given by e remaining part of polynomial (30) can be expressed as where e roots of the quadratic equation obtained when the polynomial in (32) is set to zero can be obtained by the quadratic formula given by (32) have a unique positive solution, and hence, the model system (1) has a unique wild-type influenza persistent equilibrium. If R cw < 1, then D 0 > 0, and by adding the conditions D 1 < 0 and Δ > 0, two positive real equilibria are obtained. If R cw � 1, then D 0 � 0, and there is a unique nonzero solution of (32) which is positive if and only if D 1 < 0. e following theorem summarizes the existence of the wild-type influenza endemic equilibria.

Theorem 3. e model system (1) has
(i) a unique endemic equilibrium if R cw > 1 (ii) two endemic equilibria if R cw < 1, D 1 < 0, and Δ > 0 (iii) one positive equilibrium for R cw � 1 and D 1 < 0 (iv) no wild-type influenza endemic equilibrium otherwise Epidemiologically, eorem 3 item (ii) implies that bringing R cw below unity does not suffice for the eradication  Figure 5: Endemic equilibrium points of the two-strain influenza model. 8 Computational and Mathematical Methods in Medicine of wild-type influenza since system (1) exhibits backward bifurcation when R cw < 1. e existence of backward bifurcation indicates that in the neighbourhood of 1, for R cw < 1, a stable wild-type influenza-free equilibrium coexists with a stable wild-type influenza persistent equilibrium. In order to eradicate the disease, the control reproduction R cw should be decreased below the critical value R * cw . To obtain R * cw , the discriminant in (32) is set to zero and R cw made the subject of the relation. is yields It follows that backward bifurcation occurs for values of R cw such that R * cw < R cw < 1. is is illustrated by Figure 6.

Existence of Resistant Influenza Strain Only Endemic
State. ere exists an endemic state when the resistant strain persists and the wild-type strain dies out. Solving (1) in terms of λ 2 and substituting I * R into λ * 2 yields the following equation: When λ 2 � 0, resistant influenza-free equilibrium is obtained. e remaining part of polynomial (36) can be expressed as where Using the procedure as in Section 3.3.1, it can be shown that the system exhibits a backward bifurcation when R cr < 1.
is is illustrated by Figure 7.

Sensitivity Analysis
In order to curb the spread of influenza in a given population, it is essential to know the relative importance of the different parameters responsible for its transmission and prevalence. Influenza transmission and endemicity are directly related to R c . As in [62,63], the normalized forward sensitivity analysis is used for this model. e normalized sensitivity index which measures the relative change in a parameter k, with respect to the reproduction number R c is given by P q � (k/R c )(zR c /zk), [64]. e sign of P q determines the direction of changes, increasing (for positive P q ) and decreasing (for negative P q ) [65]. e sensitivity indices of the model reproduction number to the parameters in the model at the parameter values described in Table 2 are calculated. ese indices reveal how crucial each parameter is to disease transmission and spread making it possible to discover parameters that have a high impact on R c and should be targeted by intervention strategies. e calculated sensitivity indices of R c are given in Table 3.
Small variations in a highly sensitive parameter lead to large quantitative changes; hence, caution should be taken when handling such a parameter. A positive sensitivity index indicates that R c is an increasing function of the corresponding parameter, and hence, an increase in the parameter while other factors are held constant leads to an increase in the reproduction number and could lead to disease spread [65]. On the other hand, a negative sensitivity index shows that an increase in the parameter while other factors are held constant leads to a decrease in the reproduction number, which could then lead to disease   Table 2, as the drug resistance increases, the changes in the reproduction numbers can be observed as shown in Figure 8.

Effects of Drug Resistance. For the parameter values in
In conformity with the expectation, increased drug resistance leads to an increase in R cr . It can also be observed that R cw decreases with increased drug resistance. e implication of increased drug resistance on infected population is discussed in the next section.

Effects of Drug Resistance on Infected Population.
e rate of drug resistance is varied holding all the other parameter values constant. Figures 9 and 10 are obtained.
It can be observed from Figure 9 that when there is no development of drug resistance (b � 0), the number of individuals infected with resistant strain decreases to zero. An increase in the rate of drug resistance leads to an increase in the number of individuals infected with resistant strain.
Next, the effect of drug resistance on individuals infected with wild-type strain is investigated.
From Figure 10, it can be observed that an increase in the rate of drug resistance leads to a decrease in the number of individuals infected with wild-type strain. For instance, when b � 1, the number of individuals infected with wildtype strain decrease to zero. is could be attributed to the mutation of the wild-type strain to resistant strain.

Effect of Vaccination on Reproduction Number and on
Influenza Prevalence in the Model Population. Figures 11 and  12 show the population dynamics of the infected individuals in a case where there is no vaccination. e reproduction number of the resistant strain is obtained as 2.7762, while that of the wild-type strain is obtained as 3.0288.
Note that the reproduction number for the two cases is greater than one. It can be observed from Figures 11 and 12 that the resistant strain and the wild-type strain persist in the population.
Next, numerical simulation is done in the case where there is vaccination. Using the parameter values in Table 2, Figures 13 and 14 are obtained. e control reproduction number (17), R cr , is obtained as 0.9059, and R cw is obtained as 0.9883.
Note that the reproduction number in this case is less than one. Vaccination reduces the reproduction number. However, from Figures 13 and 14, it can be observed that  is shows that bringing the reproduction number below unity does not describe the necessary effort to curb the spread of influenza. erefore, the intervention strategies should be carefully implemented to bring the reproduction number below the critical value. It can also be observed from Figures 11-14 that the level of persistence of the resistant strain is higher than that of the wild-type strain.  higher the number of infected individuals. e number of infected individuals drastically decreases to zero within a short period of time but then starts to increase again shortly after and the disease does not completely die out after that (this is when β w � 0.002 and 0.0015). When β w � 0.00095, it can be observed that the number of infected individuals declines to zero and the disease completely dies out. It should be noted that for this case, the R cw � 0.9205 which is below the critical value R * cw � 0.9351.

Case 2:
Effect of β r on I R Individuals. It is observed from Figure 16 that the higher the transmission rate, the higher the number of infected individuals. It is also interesting to note that when β r � 0, there still exist individuals infected with the resistant strain and the strain persists in the population. is shows that curbing the spread of the resistant strain is quite difficult. is could be due to the fact that the spread of the resistant strain is fuelled by two processes: transmission and mutation of the wild-type strain to resistant strain.

Conclusion
To completely wipe out influenza from a population continues to prove difficult. is is because the virus evolves very rapidly and is able to change from one season to the other. is is extensively explained in [3,7,9]. Results from our model show that vaccination reduces the reproduction number, and hence, it could be used as a control strategy. However, caution should be taken because influenza can still persist in case there is backward bifurcation. Results also show that it is easier to curtail the spread of the wild-type strain especially in a given season than the resistant strain.
is could be through social distancing and issuing travel bans to areas affected with the virus. For the resistant strain, social distancing could also be used as a control strategy in addition to reducing the mutation of the wild-type strain.

Data Availability
e data used to support the findings of this study are included within the article.