### I. Introduction

### II. Methods

*c*are constant coefficients that varied from −1 to 1; and

_{i}s*P*,

*T*, and

*H*are normalized precipitation, temperature, and specific humidity, respectively. Every one, two, or three possible combinations of these climate parameters correspond to a modified compartmental model. Simultaneously, each extended model is denoted by a notation consisting of ‘SIR’, ‘+’, and a set of letters from {

*P*,

*T*,

*H*}. For example, the SIR+T+P model is constructed by combining the SIR model with temperature and precipitation.

*c*in the case of extended model). At each iteration step, all parameters are determined by minimizing the difference between observations and model predictions. At step

_{i}s*n*=

*k*, the number of weekly influenza cases (model output) is calculated as

*S*, and

*I*are given from step

*n*=

*k*− 1, and the integration is done over week

*k*. In the model-fitting methodology, the least-square cost function fits the model parameters

*θ*= (

*S*,

_{0}*I*, β, γ) and, $\stackrel{~}{\theta}$ = (

_{0}*S*,

_{0}*I*, β, γ,

_{0}*c*…), which correspond to the null and extended models, respectively. In summary, if

_{1}*Z*and

_{ti}*Y*are the estimated and real numbers of infected cases, minimizing the least-square function

_{ti}*l*(

*θ*) = ∥

*Z*−

_{t}*Y*∥ will estimate the optimal values of

_{t}*θ*and $\stackrel{~}{\theta}$. It should be mentioned that the number of estimated parameters is varied from 5 to 7 unknown variables depending on the used climate factors.

*l*(

*θ*). The numerical solution of the optimization problem is obtained by using a constrained ‘Trust-Region’ algorithm. Meanwhile, at each iteration the numerical solution of the involved differential equations is calculated. All simulations were conducted in MATLAB. The model comparisons were performed by using the Akaike information criterion (AIC) and root-meansquare error (RMSE) criteria. Finally, for each case, the effective reproductive number,

*R*(

*t*) =

*S*(

*t*)β/γ, was calculated based on the estimated optimal parameters.

### III. Results

*R*= β/γ, is a vital epidemic estimator. On the other hand, the effective reproductive number, which measures the average number of secondary cases for each primary infected individual at time t, can be estimated using

_{0}*R*(

*t*) =

*S*(

*t*)β/γ. As shown in Figure 4, the SIR+P+T-based estimation of

*R*(

*t*) for season 2013/14 lies between 0.36 and 2.10, while for the basic model, it varies from 0.21 to 3.34. Since the SIR+P+T model gives a more accurate prediction of the weekly number of cases, the estimated

*R*(

*t*) based on the extended model is more reliable.

### IV. Discussion

*R*(

*t*) depends on. In other words, the effective reproductive number is an indirect product of the parameter estimation methodology. According to the formulae presented in the results section, we know that

*R*(

*t*) ≤

*R*. In other words,

_{0}*R*is the upper bound of the basic reproductive number. Equality occurs only when the entire population is susceptible. Since the whole population is not susceptible, it is more practical to use

_{0}*R*(

*t*) instead of

*R*. Based on our findings, time-varying

_{0}*R*(

*t*) function, derived from SIR+P+T based estimates, spans a narrower range of 0.36 to 2.10 in comparison to SIR-based estimations.