SIR Model to Determine Influenza Treatment Using Rimantadine or Amantadine - Coursework Example

Name Instructor’s Name Course Date SIR model to determine influenza treatment or prophylaxis using Rimantadine or Amantadine Antiviral treatment of influenza with rimantadine and amantadine has been associated with influenza A viruses that are drug-resistant (Hayden et al, p1696; Mat et al, p 988; Houck et al, p533). The drug-resistant viruses’ impact and the probability of drug-resistant virus isolates transmission are incompletely defined. However, transmission of these variants has been linked to drug prophylaxis failures in nursing homes and household settings (Hayden et al, p1696; Mat et al, p 988; Houck et al, p533;Degelau et al, p 390). Further, the drug prophylaxis and therapy benefits in most circumstances tend to outweigh emergence of drug resistance risks (Hayden, p 59) Strategies for drug administration which keep drug resistant viruses’ emergence at reasonably low levels as well as minimize the viral variants deleterious effects, are necessary. The strategies should preferably allow illnesses prevention and symptomatic treatment in order to take maximum advantage of the therapeutic and proven prophylactic activities of rimantadine and amantadine A mathematical model would be an appropriate tool to determine if treatment or drug prophylaxis with rimantadine or amantadine would be effective as an intervention method in the event of an influenza epidemic (or pandemic) within a closed population. This is in light of the rapid emergence potential and drug-resistant influenza viruses spread. To address this, a model describing drug resistance effects on the dynamics of transmission of in a closed population influenza outbreak is discussed in this paper. The best example of a closed population modeling is a nursing home, hospital, or school. Drug resistance is included in the SIR model and distinguishes between infected and hence infectious persons (asymptomatic-subclinical infections) and clinical symptoms that are developed (illness). This distinction of infected persons is based on studies that indicate amantadine prophylaxis effects on influenza infections have subclinical infections accounting for approximately one-third of infections (Petterson et al, p 377; Quarles et al, p 149). Not every infection results in an illness and chemoprophylaxis increases chances of subclinical infection (Oker et al, p 676; Nafta et al, p 423; Monto et al, p 1003), chemoprophylaxis or therapy may result in redistribution among infected persons. An infected person having clinical illness sheds more viruses compared to one with subclinical infection and illness manifestations such as coughing and rhinorrhea, that contributes to infectious aerosols generation. However, infected persons that have clinical symptoms could show virus transmissibility that is reduced if they are confined to bed. Thus, differences between the two rates of transmission are dependent on factors such as design of living quarters, age, social behavior, and severity of illness. The assumption made is that an epidemic begins in a susceptible persons’ closed population with a small number of infected persons having clinical symptoms introduced. Although it is possible for the epidemic to have started via individuals having subclinical infection, these are not detectable easily hence ignoring them has minimal impact on the model. Thus, the model in figure 1 considers populations (see table 1 in the Appendix): susceptible persons (S); Susceptible but prophylactically taking drug (Spr); infected and untreated (I) persons; Infected and untreated, clinical symptoms developing (Is); Infected, untreated and asymptomatic, resistant virus shedding (Ir); Infected and untreated, shedding resistant virus with clinical symptoms (Is,r); infected and treated (Itr) persons, Infected and treated; clinical symptoms developing (Is,tr); Infected, treated and asymptomatic, resistant virus shedding (Ir,tr); and Infected and treated, shedding resistant virus with clinical symptoms (Is,r,tr). r1ᵧ1 θ2 δ3 ᵧ1 ᵧ2 r2ᵧ2 β δ1 θ3 θ1 βr q2 ƙ q1 ƙ δ2 θ3 ᵧ1 ᵧ2 ᵧ2 θ2 δ4 ᵧ1 Figure 1. Model populations schematic presentation during treatment. When susceptible (S) persons are infected they shed drug-resistant virus (Ir) or wild type (I) at β r β, rate respectively. Susceptible persons that get chemoprophylaxis (Spr) at θl rate or become infected treated persons instantaneously shedding or drug-resistant virus (Ir,tr) or drug-sensitive (Itr). Clinical symptoms develop in asymptomatic infected persons in both groups at δ 2 and δ 1 rates, respectively. Rate of treatment is θ2. Persons subclinically infected, shedding drug-resistant (Ir,tr) or drug-sensitive (Itr) viruses, and receiving treatment may end up being symptomatic at δ4 and δ3 rates and receive further treatment at θ3 rates just as infected symptomatic persons (Is, Is,r) do. Symptomatic infected treated persons (Itr ) and asymptomatic infected treated (Is,tr) during treatment develop at rates q2k and q1k, drug resistance respectively, where q1 and q2 is the likelihood of drug resistance developing. They then become asymptomatic infected treated individuals (Ir,tr) and symptomatic infected treated individuals (Is,r,tr). Recovery of all infected persons is at the rates ᵧ1, ᵧ2 (for asymptomatic, and symptomatic) and r1 ᵧ1, r2 ᵧ2 (when treated) and then they become immune In studies done on influenza A virus isolates that were recovered from untreated persons, less than 1% of the isolates were drug-resistant (Ziegler et al, p 1) Although these kind of viruses might be representative of variants that are naturally resistant, it is assumed in this paper that such variants are absent during an epidemic occurrence. The naturally resistant variants existence can be included easily into the model through a positive value inclusion for the Ir group initial condition. Epidemic identification occurs when infection occurs in approx. 1%–5% of the population. Emergence of drug-resistant viruses occurs rapidly after the treatment commences (2–5 days) (Hayden et al, p1696; Mat et al, p 988). Thus, the epidemic is modeled as a continuous process commencing when 5% of total population is infected and detection of illness is done. It is assumed that chemoprophylaxis or treatment can start at this time. Transmission rate β of infection occurs resulting from infected persons coming into contact with susceptible persons. Transition to symptomatic infected from asymptomatic infected occurs with rate δ. Infected persons upon recovery become immune at rate ᵧ (an infectivity period 1/ ᵧ). The drug resistance development rate is ƙ. Finally, chemoprophylaxis or treatment rate is denoted as θ (the population dynamics are indicated in Appendix 2). Model parameters estimates are reflected in table 2. In the complete model, β, ᵧ, δ, ƙ are subscripts denoting distinctions among the subpopulations (tables 1, 2). The rate of transmission from an infected untreated person to a susceptible individual who develops symptoms (Is) is assumed to be higher than from an (I) person to a susceptible person. It is assumed susceptible persons are infected by I and Is persons having a higher rate than individuals that are by Ir and Is,r because viruses that are resistant are likely to have transmission probability that is lower than wild type virus, as seen in the low drug resistant virus prevalence in the selection absence by drug therapy (Ziegler et al, p 1) However, in individuals shedding drug-resistant virus, it is assumed that treatment does not reduce or prolong viral replication in comparison to no treatment. Drug treatment reduces severity of disease and transmission probability of wild type virus. It is assumed reduction through introduction of transmissibility hence risk of infection, when an encounter takes place between susceptible person and an Itr or Is,tr model. Estimates are based on the drug treatment antiviral effects and in secondary cases reduction in household contacts in the case where treatment is given to ill-index cases (Couch et al, p 431) Validity of these estimates are for an epidemic where prior immunity variable exist Upon development of drug resistance, treatment has minimal effect. Hence, it is assumed no difference would occur in the transmission for infected and susceptible persons contacts with clinical symptoms that shed resistant virus in the case of treatment (Is,r,tr) or not (Is,r). The assumptions are made based on avian influenza direct observations (Bean et al, p 1050) For purposes of calculation, it is assumed that the probability of transmission of viruses that are drug-resistant is equal to or, as drug-resistant is mostly uncommon, lower by 5-fold than wild type virus. It is assumed that chemoprophylaxis reduces the contacts susceptibility (Petterson et al, p 377; Quarles et al, p 149). The transmission efficiencies estimates are from influenza outbreaks studies (Blower et al, p 497) Amantadine prophylactic efficacy against infection was found to be between 19% to 52% in pandemics (Oker et al, p 676; Nafta et al, p 423; Monto et al, p 1003), and involved populations immunized or infected naturally with related viruses. Therefore, prophylactic efficacy for infection and illness prevention is higher. It is assumed in this model that an Spr person encountering an I, Is, Itr, or Is,tr person has better protection (by ≥33%) than shown by other pandemic chemoprophylaxis results. The rate of transmission of I or Is to an Spr individual is approximately two thirds of the rate corresponding to a susceptible person not taking chemoprophylaxis. This assumption is backed by studies results of amantadine prophylaxis during influenza pandemic (Oker et al, p 676; Nafta et al, p 423; Monto et al, p 1003), The epidemic corresponding value due to prior immunity is assumed as being lower (a third of the rate where susceptible person abstains from chemoprophylaxis). A similar assumption is made for Itr or Is,tr individuals in contact with Spr persons. The infection probability of an Itr or an Is,tr individual infecting an Spr is taken to be very minimal (approx. 10% compared to wild type virus relative infectivity with no drug intervention in an epidemic). The prophylactic efficacy in an epidemic for preventing infection estimation is 66%–79% and 85%–91% for preventing illness (Dolin et al, p 580) In a pandemic, efficacy of prophylactic for infection prevention estimation is approx 33% and approx 65% for illness prevention (Oker et al, p 676; Nafta et al, p 423; Monto et al, p 1003; Smorodintsev et al, p 44), Spr individuals contact Ir, Is,r, Ir,tr, or Is,r,tr individuals and are assumed unprotected and possesses same rates of transmission as their counterparts abstaining from chemoprophylaxis. The rate of recovery appears to be similar for illness from drug-resistant and wild type virus (Hayden et al, p 1741; Rose et al, p 619). This model assumes that drug treatment has no effect in enhancing recovery of individuals with resistant virus infection. Individuals with clinical symptoms recover slower than asymptomatic infected individuals. Recovery in a pandemic lasts longer when compared to an epidemic. The rates at which I, Ir, Itr and Ir,tr individuals develop clinical symptoms indicate treatment slows clinical symptoms appearance. In resistant or wild type viruses’ in untreated persons and in treated person shedding resistant virus, symptoms development rate is assumed to be same (see table 2). Because drug-resistant variants in untreated populations occur at very minimal levels, hence it is not considered in the model in the event of a spontaneous drug-resistant variants generation in untreated persons in a single outbreak. Following treatment however, up to 30% of the treated persons develop resistance (Hall et al, p 275; Hayden et al, p 59)) In the model, failure of drug treatment results in direct rise in acquired resistance within infected persons and to indirect rise from contact of susceptible with infected persons who shed resistant virus. Strategies of drug administration can be subdivided as: Itr or of infected persons treatment, susceptible persons’ chemoprophylaxis, or both. Additionally, chemoprophylaxis or treatment may be done at an average rate at the time of the epidemic. The rate θ can be modified to a function of infected population [θ (Is)] that is available. Administration of chemoprophylaxis can be instantaneous at a specific time point, for example in small populations. After treatment, the infected individuals shedding or not drug-resistant virus (Ir),(I), are reclassified along infected treated persons (at some rate). Is and Is,r individuals are treated at a rate. Although Is,r persons will most likely not respond to drug treatment, they receive it because they are prospectively not recognized as shedding resistant virus. The model for influenza epidemic and the model of influenza epidemic which has drug intervention as well as their corresponding calculations is shown in Appendix 3 In summary, if administration of drugs strongly reduces transmission of influenza and protects persons that are susceptible that take chemoprophylaxis, then the model predicts chemoprophylaxis as being very beneficial, while at the same time drug resistance can be maintained at low levels. Symptomatic infected persons’ treatment with drugs and chemoprophylaxis will be positive in lowering an epidemic. Drug treatment carried out on symptomatic infected persons will only benefit this group and will have little impact on the epidemic. The predictions of the model point out the necessity of getting better drug efficacy estimates. Drug administration has no effect in drug-resistant influenza isolates infections (Bean et al, p 1050) and considering the partial treatment efficacy of wild type virus infections, the model suggests that drug resistance low probability of emergence and transmission may offer an explanation to the drug resistance emergence relatively low frequency in closed population outbreaks Mast et al, p 988; Houck et al, p533). Otherwise, development of drug-resistant isolates enforcement that results in secondary epidemics within the original type is the end result. References Bean WJ, Threlkeld SC, Webster RG. Biologic potential of amantadine resistant influenza A virus in an avian model. J Infect Dis 1989;159:1050–6. Blower SM, Small PM, Hopewell PC. Control strategies for tuberculosis epidemics: new models for old problems. Science 1996;273:497–500. Couch RB, Kasel JA, Glezen WP, et al. Influenza: its control in persons and populations. J Infect Dis 1986; 153:431–40. Degelau J, Somani SK, Cooper SL, Guay DRP, Crossley KB. Amantadine resistant influenza A in a nursing facility. Arch Intern Med 1992;152:390–2. Dolin R, Reichman RC, Madore HP, Maynard R, Linton PN, Webber Jones J. A controlled trial of amantadine and rimantadine in the prophylaxis of influenza A infection. N Engl J Med 1982;307:580–4. Hall CB, Dolin R, Gala CL, et al. Children with influenza A infection: treatment with rimantadine. Pediatrics 1987;80:275–82. Hayden FG, Belshe RB, Clover RD, Hay AJ, Oakes MG, Soo W. Emergence and apparent transmission of rimantadine-resistant influenza A virus in families. N Engl J Med 1989;321:1696–702. Hayden FG, Sperber SJ, Belshe RB, Clover RD, Hay AJ, Pyke S. Recovery of drug-resistant influenza A virus during therapeutic use of rimantadine. Antimicrob Agents Chemother 1991;35:1741–7. Hayden FG. Amantadine and rimantadine-clinical aspects. In: Richman DD, ed. Antiviral drug resistance. Chichester, UK: John Wiley and Sons, 1996:59–77. Houck P, Hemphill M, LaCroix S, Hirsh D, Cox N. Amantadine-resistant influenza A in nursing homes: identification of a resistant virus prior to drug use. Arch Intern Med 1995;155:533–7. Mast EE, Harmon MW, Gravenstein S, et al. Emergence and possible to transmission of amantadine-resistant viruses during nursing home out breaks of influenza A (H3N2). Am J Epidemiol 1991;134:988–97. Monto AS, Gunn RA, Bandyk MG, King CL. Prevention of Russian influenza by amantadine. JAMA 1979;241:1003–7. Nafta I, Turcanu AG, Braun I, et al. Administration of amantadine for the prevention of Hong Kong influenza. Bull WHO 1970;42:423–7. Oker-Blom N, Hovi T, Leinikki P, Palosuo T, Pettersson R, Suni J. Protection of man from natural infection with influenza A2 Hong Kong virus by amantadine: a controlled field trial. Br Med J 1970;3:676–8. Pettersson RF, Hellstrom PE, Penttinen K, et al. Evaluation of amantadine in the prophylaxis of influenza A (H1N1) virus infection: a controlled field trial among young adults and high-risk patients. J Infect Dis 1980; 142:377–83. Quarles JM, Couch RB, Cate TR, Goswick CB. Comparison of amantadine and rimantadine for prevention of type A (Russian) influenza. Antiviral Res 1981;1:149–55. Rose HJ. The use of amantadine and influenza vaccine in a type A influenza epidemic in a boarding school. J R Coll Gen Pract 1980;30:619–21 Smorodintsev AA, Karpuchin GI, Zlydnikov DM, et al. The prospects of amantadine for prevention of influenza A2 in humans (effectiveness of amantadine during influenza A2/Hong Kong epidemics in January–February 1969 in Leningrad). Ann NY Acad Sci 1970;173:44–61. Ziegler T, Hemphill M, Ziegler ML, Klimov A, Cox N. Rimantadine resistance of influenza A viruses: an international surveillance. Pre sented at the 7th International Society for Antiviral Research Confer ence, Charleston, SC, 1994. APPENDIX 1 TABLE 1 POPULATIONS IN THE MODEL SYMBOL DEFINITIONS S Susceptible Spr Susceptible but prophylactically taking drug I Infected and untreated Is Infected and untreated, clinical symptoms developing Ir Infected, untreated and asymptomatic, resistant virus shedding I s,r Infected and untreated, shedding resistant virus with clinical symptoms I tr Infected and treated I s,tr Infected and treated, clinical symptoms developing I r,tr Infected, treated and asymptomatic, resistant virus shedding I s,r,tr Infected and treated, shedding resistant virus with clinical symptoms APPENDIX 2 TABLE 2 PARAMETERS STANDARD VALUES β 1, β 1,r Rate of transmission between I and S sufficiently close for allowing wild drug-resistant/wild type virus transmission and cause subclinical infection β 1 = β 1,r = 6 × 10-4 per day or β1,r = β 1/5= 1.2 × 10-4 per day β 2, β 2,r Rate of transmission between I and S sufficiently close for allowing wild drug-resistant/wild type virus transmission and cause subclinical infection β 2 = β 2,r = 6 × 10-3 per day or β2,r = β 2/5= 1.2 × 10-3 per day Pi Wild type virus relative infectivity during treatment/chemoprophylaxis compared with wild type virus lacking drug intervention for the following cotacts Itr or I s,tr and S; p1, p2 (p1 = p2 epi/pan) I or Is and Spr; p3, p4 (p3 = p4 epi/pan Itr or Is,tr and Spr; p5, p6 (p5 = p6 epi) 0.67 epi/pan (Couch et al, p431) 0.33 epi; 0.67 pan (Quarles, p 1490 0.10 epi (Hayden et al, p 1741); 0.35, 0.67 pan (Smorodintsev et al, p 44) δj Rate of transition where I becomes Is (δ1), Ir becomes Is,r (δ2), Ir,tr becomes Is,r,tr (δ4), (δ1 = δ2 = δ4) Itr becomes Is,tr (δ3) 0.5 per day 0.10 per day epi; 0.17 pan θl Rate of treatment/Chemoprophylaxis of: S; (θ1); I and Ir; (θ2), and Is and Is,r; (θ3) 0.0 per day 0.70 per day ᵧ1, ᵧ 2 Rate of recovery from subclinical infection (ᵧ 1) or infection alongside clinical symptoms (ᵧ 2) 0.50 per day 0.25 per day (Rose et al, p 619) r1, r2 Relative recovery of I tr; (r1) (in comparison to I); r2 relative recovery of Is,tr infection (in comparison to Is) 2.0 epi and 1.60 pan 1.33 epi and 1.05 pan ƙ drug resistance rate of development during treatment of I s,tr which then becomes I s,r,tr 0.25 per day (Houck, p 533) q1, q2 Emergence probability of drug resistance that is acquired (as a result of failure in treatment ) from I tr which then becomes I r,tr; (q1) or from I s,tr which then becomes I s,r,tr; (q2) 0.02 0.20 (Hall et al, p 275) APPENDIX 3 Model for Influenza epidemic The nonlinear differential equations system giving the model dynamics is as shown below dS/ dt = -(β1I + β 2Is)S, (A1) dI/ dt = (β1I + β2Is)S 0 -(δ1 / ᵧ1)I, (A2) dIs/ dt = δ1I - ᵧ2Is, (A3) and S(t0) = S0, I(t0) = 0, Is(t0)= Is0. The model is not inclusive of death occurring as a result of influenza, hence all persons that are infected recover at the rate ᵧ1 (from infection that is asymptomatic) and ᵧ2 (from infection that is symptomatic). In addition, N, which is the total population, is assumed as constant. Therefore, calculation of the fraction of individuals who are removed (R) is R = N- (S + I + Is). Table 2 shows the corresponding values of the parameter values (β1, β2, ᵧ1 ᵧ2). Model of Influenza epidemic which has drug intervention The model described above is further expanded to include drug intervention in the event of an influenza epidemic and covers the table 1 populations. The population dynamics equations are as follows: dS/ dt = -(β1I +β 2Is /+ β 1,rIr + β 2,rI s,r + p1 β1Itr +p2 β 2I s,tr + β 1,rIr,tr + β 2,rI s,r,tr)S- θ1S, (A4) dSpr/dt = -(p3 β 1I + p4 β2I s + β1,rIr + β 2,rIs,r + p5 β1Itr + p6 β 2Is,tr + β 1,rIr,tr + β 2,rIs,r,tr)Spr+ θ1S, (A5) dI/dt = (β 1I / p1 β 1Itr + β 2Is + p2 β 2Is,tr)S- (ᵧ1 / δ1 / θ2)I, (A6) dIs/dt = δ1I - (ᵧ2 / θ3)Is, (A7) dIr/dt = (β1,rIr + β1,rIr,tr +β2,rIs,r + β2,rIs,r,tr)S+ (β1,rIr +β2,rIs,r)Spr - (ᵧ1 +δ2 +θ2)Ir, (A8) dIs,r/ dt= δ2Ir - (ᵧ2 +θ3)Is,r, (A9) dItr/dt = (p3β1I + p4β2Is + p5β1Itr + p6β2Is,tr)Spr- (r1ᵧ1 / δ3 / q1ƙ)Itr +θ2I, (A10) dIs,tr /dt = δ3Itr - (r2ᵧ2 + q2ƙ)Is,tr + θ3Is, (A11) dIr,tr/dt = (β1,rIr,tr + β 2,rIs,r,tr)Spr - (ᵧ1 +δ4)Ir,tr + q1ƙItr +θ2Ir, (A12) dIs,r,tr/ dt = δ4Ir,tr - ᵧ2Is,r,tr + q2ƙIs,tr + θ3Is,r, (A13) and S(t1) = S0,1, Is(t1) = Is0,1, I(t1) = I0,1, Ir(t1) = Is,r(t1) = Itr(t1) = Is,tr(t1) = Ir,tr(t1) = Is,r,tr(t1) = 0, and t1 > t0, where t1 represents the time point on which intervention commences. As described in the simple model, the recovered persons and who are removed (Rtot) is calculated as follows: Rtot = N - (S + Spr + I +Is + Ir + Is,r + Itr + Is,tr + Ir,tr + Is,r,tr). (A14) Read More

Not every infection results in an illness and chemoprophylaxis increases chances of subclinical infection (Oker et al, p 676; Nafta et al, p 423; Monto et al, p 1003), chemoprophylaxis or therapy may result in redistribution among infected persons. An infected person having clinical illness sheds more viruses compared to one with subclinical infection and illness manifestations such as coughing and rhinorrhea, that contributes to infectious aerosols generation. However, infected persons that have clinical symptoms could show virus transmissibility that is reduced if they are confined to bed.

Thus, differences between the two rates of transmission are dependent on factors such as design of living quarters, age, social behavior, and severity of illness. The assumption made is that an epidemic begins in a susceptible persons’ closed population with a small number of infected persons having clinical symptoms introduced. Although it is possible for the epidemic to have started via individuals having subclinical infection, these are not detectable easily hence ignoring them has minimal impact on the model.

Thus, the model in figure 1 considers populations (see table 1 in the Appendix): susceptible persons (S); Susceptible but prophylactically taking drug (Spr); infected and untreated (I) persons; Infected and untreated, clinical symptoms developing (Is); Infected, untreated and asymptomatic, resistant virus shedding (Ir); Infected and untreated, shedding resistant virus with clinical symptoms (Is,r); infected and treated (Itr) persons, Infected and treated; clinical symptoms developing (Is,tr); Infected, treated and asymptomatic, resistant virus shedding (Ir,tr); and Infected and treated, shedding resistant virus with clinical symptoms (Is,r,tr).

r1ᵧ1 θ2 δ3 ᵧ1 ᵧ2 r2ᵧ2 β δ1 θ3 θ1 βr q2 ƙ q1 ƙ δ2 θ3 ᵧ1 ᵧ2 ᵧ2 θ2 δ4 ᵧ1 Figure 1. Model populations schematic presentation during treatment. When susceptible (S) persons are infected they shed drug-resistant virus (Ir) or wild type (I) at β r β, rate respectively.

Susceptible persons that get chemoprophylaxis (Spr) at θl rate or become infected treated persons instantaneously shedding or drug-resistant virus (Ir,tr) or drug-sensitive (Itr). Clinical symptoms develop in asymptomatic infected persons in both groups at δ 2 and δ 1 rates, respectively. Rate of treatment is θ2. Persons subclinically infected, shedding drug-resistant (Ir,tr) or drug-sensitive (Itr) viruses, and receiving treatment may end up being symptomatic at δ4 and δ3 rates and receive further treatment at θ3 rates just as infected symptomatic persons (Is, Is,r) do.

Symptomatic infected treated persons (Itr ) and asymptomatic infected treated (Is,tr) during treatment develop at rates q2k and q1k, drug resistance respectively, where q1 and q2 is the likelihood of drug resistance developing. They then become asymptomatic infected treated individuals (Ir,tr) and symptomatic infected treated individuals (Is,r,tr). Recovery of all infected persons is at the rates ᵧ1, ᵧ2 (for asymptomatic, and symptomatic) and r1 ᵧ1, r2 ᵧ2 (when treated) and then they become immune In studies done on influenza A virus isolates that were recovered from untreated persons, less than 1% of the isolates were drug-resistant (Ziegler et al, p 1) Although these kind of viruses might be representative of variants that are naturally resistant, it is assumed in this paper that such variants are absent during an epidemic occurrence.

The naturally resistant variants existence can be included easily into the model through a positive value inclusion for the Ir group initial condition. Epidemic identification occurs when infection occurs in approx.

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