عدم قطعیت و تجزیه و تحلیل حساسیت از تعداد تکثیر اساسی از یک مدل همه گیر واکسینه شده آنفلوانزا

The basic reproduction number and the point of endemic equilibrium are two very important factors in any deterministic compartmental epidemic model as the basic reproduction number and the point of endemic equilibrium represent the nature of disease transmission and disease prevalence respectively. In this article the sensitivity analysis based on mathematical as well as statistical techniques has been performed to determine the importance of the epidemic model parameters. It is observed that 6 out of the 11 input parameters play a prominent role in determining the magnitude of the basic reproduction number. It is shown that the basic reproduction number is the most sensitive to the transmission rate of disease. It is also shown that control of transmission rate and recovery rate of the clinically ill are crucial to stop the spreading of influenza epidemics.

Influenza is an infectious disease caused by a virus commonly known as influenza virus and transmitted among humans mainly in three ways: (i) by direct contact with infected individuals; (ii) by contact with contaminated objects and (iii) by inhalation of aerosols that contain virus particles. There are millions of people who suffer or die annually from influenza worldwide. Although different control and prevention strategies are available to control influenza transmission, influenza has been a major cause of morbidity and mortality among humans all over the world. Comparative knowledge of the effectiveness and efficacy of different control strategies is necessary to design useful influenza control programs. Mathematical modeling of the spread of influenza can play an important role in comparing the effects of different control strategies [1], [2], [3] and [4]. In particular, understanding of the threshold concepts in epidemiology that govern the spread of infection is very important. Among the various threshold concepts, an epidemiologic parameter, R0R0 is called the basic reproduction ratio, or basic reproductive rate, or basic reproduction number. It is one of the most important theoretical concepts in theoretical infectious diseases epidemiology. Sensitivity analysis of model parameters is very important to design control strategies as well as a direction to future research. There are many methods [5] available for conducting sensitivity analysis such as differential analysis, response surface methodology, the Fourier amplitude sensitivity test (FAST) and other variance decomposition procedures, fast probability integration and sampling-based procedures. Chitnis et al. [6] have evaluated the sensitivity indices of the basic reproduction number and the point of endemic equilibrium to the parameters in the model. They have determined the relative importance of different parameters in the transmission and prevalence of malaria. Sanchez and Blower [7] have investigated an uncertainty and sensitivity analysis of the basic reproduction number of tuberculosis. Helton et al. [5] have performed a comparison study of uncertainty and sensitivity analysis using random and Latin hypercube sampling methods. Uncertainty and sensitivity analysis of influenza epidemic model are not new. Several recent studies [8], [9], [10], [11], [12] and [13] have performed the uncertainty and sensitivity analysis of influenza epidemic model to the parameters. Lee et al. [9] and Wu et al. [13] have explored the effects of uncertainty in parameter estimation in the final size of influenza epidemic. Nuno et al. [10] and van den Dool et al. [12] have performed uncertainty and sensitivity analysis of the model parameters. They have determined the relative importance of model parameters in influenza transmission. Seaholm et al. [11] have performed a comparison study of sensitivity analysis based on Latin hypercube sampling and full factorial sampling approaches. All of these studies have determined the relative importance of different parameters in influenza transmission based on sampling-based uncertainty/sensitivity analysis. In this article we analyze an influenza epidemic model proposed by Samsuzzoha et al. [14] based on the sensitivity indices of the basic reproduction number, (Rvac)(Rvac) and the endemic point of equilibrium, (E∗)(E∗) to the parameters. Sensitivity analysis of the basic reproduction number (Rvac)(Rvac) is performed using two different approaches based on: (i) the local derivative and (ii) sampling. In turn, the sampling-based approach consists of two different methods: (i) random and (ii) Latin hypercube sampling. Thus results are compared using (i) local derivative, (ii) random and (iii) Latin hypercube sampling methods. We then numerically calculate the sensitivity indices of the endemic point of equilibrium (E∗)(E∗). This analysis makes this work different to others as it will provide a comprehensive insight into a difficult problem. This will help us to determine the relative importance of different parameters in transmission and prevalence of influenza.

There are 11 parameters involved in the equation of RvacRvac. In order to study the sensitivity analysis, values of the parameters have been chosen as given in Table 1. Using random and Latin hypercube sampling, sensitivity of the model parameters, involved in the equation of RvacRvac, has been investigated. Using these techniques, estimated values of RvacRvac based on the variation of parameters β,βE,βV,σ,γβ,βE,βV,σ,γ and ϕϕ have been obtained as shown in Table 3. Using random sampling, the distributions of parameters β,βV,σ,γβ,βV,σ,γ and ϕϕ along with fixed values parameters give mean, median, variance and standard deviation as 1.86127, 1.82014, 0.209, 0.457631, respectively for RvacRvac. Minimum and maximum values for the distribution of the basic reproduction number, RvacRvac, are 0.663 and 3.655 respectively, with 98.1% of the distribution grater than one. Using Latin hypercube sampling, the distributions of parameters β,βV,σ,γβ,βV,σ,γ and ϕϕ along with fixed values parameters give mean, median, variance and standard deviation as 1.85397, 1.81140, 0.209, 0.457622, respectively for RvacRvac. Minimum and maximum values for the distribution of the basic reproduction number, RvacRvac, are 0.485 and 4.056, respectively, with 98.5% of the distribution grater than one. Thus value of RvacRvac shows a wide range of estimates. This is because of the uncertainty in estimating the values of six input parameters. Using both random sampling and Latin hypercube sampling, it has been observed that there is little difference in variability of the estimated value for RvacRvac with sample size of 1000. The most influential parameter for RvacRvac is the transmission rate ββ, as shown in Table 2. In order to have 1% decrease in the value of RvacRvac, it is necessary to decrease the value of ββ, to 1%. The second most influential parameter is βIβI. In order to have 1% decrease in the value of RvacRvac, it is necessary to decrease the value of βIβI, to 1.10004%. The third most influential parameter is γγ, and in order to obtain 1% decrease in the value of RvacRvac, it is necessary to increase the value of γγ, to 1.10048%. Other important parameters are ϕ,θ,βV,βE,σ,r,κϕ,θ,βV,βE,σ,r,κ and αα. As shown in Table 4, for both sampling approaches, the transmission rate, ββ and recovery rate of clinically ill, γγ are highly correlated with RvacRvac. Moderate correlation exists between the vaccination rate, ϕϕ and RvacRvac. Weak correlation has been observed between βV,βEβV,βE and γγ with RvacRvac. Using random sampling and Latin hypercube sampling for this study, it has been observed that there is little variability for the partial correlation coefficient between each of input variable and RvacRvac. As shown in the sensitivity index column of Table 4, for both sampling used here, the parameter ββ accounts for the maximum variability in the outcome of basic reproduction number. The parameter γγ is the next to account for the variability in the outcome of basic reproduction. Parameter σσ accounts for the least variability in the outcome of basic reproduction number. Using random sampling and Latin hypercube sampling, it has been observed that there is considerable variability for the sensitivity indices for each of the input variable and RvacRvac as well. It has been observed that different techniques of uncertainty and sensitivity analysis in estimating the value of RvacRvac for influenza have enabled the generation of some important quantitative results. These results are very helpful for a better understanding of initial disease transmission and equilibrium disease prevalence. Conclusions may be summarized as follows: • ββ is the most sensitive parameter for S∗,V∗,E∗S∗,V∗,E∗ and R∗R∗ as shown in Table 5. • γγ is the most sensitive parameter for I∗I∗ as shown in Table 5. • Recovery rate of clinically ill individual, γγ is the most important parameter for the equilibrium disease prevalence as shown in Table 5. • ββ is one of the most influential parameters in determining the value of RvacRvac as shown in Table 2 and Table 4. • Using both random sampling and Latin hypercube sampling, it has been observed that there is little difference in variability of the estimated value for RvacRvac as shown in Table 3. • Using random sampling and Latin hypercube sampling, it has been observed that there is considerable variability for the sensitivity indices for each of the input variable for RvacRvac as shown in Table 4. • Sampling-based procedures provide a range of values of RvacRvac in an interval while methodology based on computation of local derivatives provides only a single value of RvacRvac. Thus sampling-based procedures provide more information as compared to local derivative.