One of the simplest epidemiological models is the SIR epidemic model, in which the total population is divided into three epidemiological classes: susceptible (

*S*), infected (and infectious,

*I*), and recovered (

*R*) individuals. Let

*S*=

*S*(

*t*),

*I*=

*I*(

*t*), and

*R*=

*R*(

*t*) denote the numbers of individuals at time

*t* in the corresponding classes, and let

*N*=

*S*+

*I*+

*R* denote the total population size. A deterministic epidemic model using ordinary differential equations consists of the following equations

with initial conditions

where,

*I*_{0}>

0 is a constant. In the initial conditions (

2.2),

*t*_{0}>

0 denotes the time of introduction of the infection into the population, and it will be shown later that

*t*_{0} is a critical parameter of the model when the transmission rate

*β*(

*t*) is a periodic function due to seasonality. Note that

*t*=

0 corresponds to the first day of a calendar year and

*t*_{0} is the number of days from

*t*=

0. The model (

2.1) ignores vital dynamics (births and deaths), which is a reasonable assumption when modeling a pandemic.

If vaccines are available before the epidemic starts, a certain level of population immunity can be achieved via vaccination. Let

*p*_{0} denote the level of population immunity at time

*t*_{0}. Then the disease dynamics can be modeled by the equations in (

2.1) with modified initial conditions:

We do not include vaccinated individuals in the

*R* class because the value of

*R*(

*t*) at the end of the disease outbreak will be used to measure the final epidemic size. In system (

2.1), a standard incidence form is used for new infections and the function

*β*(

*t*) represents the rate at which a susceptible individual becomes infected when contacting an infectious individual. The duration of infection is assumed to follow an exponential distribution with the mean period 1/

*γ*; and thus,

*γ* is the per capita recovery rate.

In most deterministic epidemic models,

*β*(

*t*)

=

*β*_{0} is assumed to be constant. However, for many diseases, seasonal variation in the transmission rate can be important. Any periodic function can be expressed as an infinite sum of sines and cosines. In this analysis, we express the periodic transmission rate as the first-order harmonic with a 1-year period:

where,

*β*_{0} is a constant representing a background transmission rate and

*ε* is a constant related to the magnitude of the seasonal fluctuation. The function

*β*(

*t*) given in (

2.4) has its maximum at the beginning of a calendar year. When the transmission rate is expressed in this form, the time of introduction of the pathogen to the population

*t*_{0} is a crucial parameter of the seasonal model because the final size and shape of the epidemic curve (including how many peaks the epidemic exhibits within one calendar year) can depend quite strongly on this parameter (whereas for models with constant

*β*, the final size and shape are independent of

*t*_{0}). Further, the shape of the epidemic curve also strongly depends on the initial fraction of susceptibles in the population at

*t*_{0} when

*β*(

*t*) is periodic.

All the variables and parameters as well as their definitions are listed in Table .

| **Table I**Definition of Symbols and Parameter Values Used in Simulations |

The model (

2.1) can be extended to include both vaccination and antiviral drug treatment. Let

*f*_{0} denote the fraction of infected individuals who will receive treatment at time

*t*;

*I*_{u} and

*I*_{tr} denote the numbers of untreated and treated infected individuals, respectively; and let the infectious period for an untreated individual is 1/

*γ*_{u}. Assume that treatment reduces infectiousness by a factor

*σ* and reduces the infectious period to 1/

*γ*_{tr}<

1/

*γ*_{u}. Following the approach of Lipsitch

*et al.* (

6) to model drug treatment, we can extend the model (

2.1) as

with initial conditions

where

*I*_{u0} is a positive constant representing the initial number of infected people. A transition diagram between the epidemiological classes is shown in Fig. .

Where there is a delay in the supply of vaccines, we could still use equations in (

2.5) with modifications. For example, let

*t*_{v}>

*t*_{0} denote the time at which the vaccination program starts and assume that a fraction

*p*_{0} of the remaining susceptibles will be vaccinated at the time

*t*_{v}. We can first use equations in (

2.5) and the initial conditions (

2.6) with

*p*_{0}=

0 for simulations in the interval

. For the time after

*t*_{v}, we can continue to use the equations in (

2.5), but with new initial conditions (1−

*p*_{0})

*S*(

*t*_{v}),

*I*(

*t*_{v}), and

*R*(

*t*_{v}).

In the next section, we will present some analysis and simulations of the models discussed above and then use the results to examine the effects of vaccination and treatment on disease mitigation and control.