Experimental data. All animal experiments were performed under institutional guidelines according to an IACUC-approved protocol. NP

_{366-374}-specific CD8 T cells in the whole lung were measured as follows: C57BL/6NCr mice at 10 to 11 weeks of age were anesthetized using 2,2,2-tribromo-ethanol (Avertin) and then infected intranasally with 1 × 10

^{5} H3N2 A/Hong Kong/X31 (X31) influenza virus (

62). On the day of organ harvest, mice (

*n* = 6/day) were euthanized. Spleen and lymph node samples were collected and processed using Dounce homogenizers in complete minimal essential medium (cMEM) (5% fetal bovine serum) to disrupt the organs into single-cell suspensions. Lung samples were processed in a tea strainer using a rubber plunger. The samples were centrifuged at 400 ×

*g* for 5 min. Red blood cells were lysed using buffered ammonium chloride solution (Gey's solution). Lung cell pellets were resuspended in 5 ml of cMEM over 5 ml of Histopaque 1083 (Sigma Diagnostics) and centrifuged for 18 min at 1,800 ×

*g*. After centrifugation, cells at the interface were carefully removed and washed with 10 ml of cMEM and then resuspended in cMEM for the immune assays. For flow cytometry, cells were Fc blocked and then stained with PA-PE and NP-APC tetramers (Trudeau Institute Molecular Core Facility) for 1 h at room temperature. Cells were pelleted and surfaced stained with CD8 APC-Cy7, CD4 PE-Cy5.5, CD90.2-PerCP, CD44-Alexa 700, CD62L-PE-TR, CD11a-PE-Cy7, CD49a-fluorescein isothiocyanate in Hanks' balanced salt solution containing 1% bovine serum albumin. Sample data were collected using a BD LSRII cytometer. Data were analyzed using FlowJo software (TreeStar). According to direct measurement, NP

_{366-374}-specific CD8 T cells account for approximately 10% of the total CD8 response to influenza virus (

11). Therefore, we introduced a scale factor of 10 to extrapolate total influenza virus-specific CD8

^{+} T-cell counts from NP

_{366-374}-specific CD8 T cells.

Viral titers from lung samples were measured by hemagglutination assay, after expansion in embryonated eggs; the 50% egg infectious dose (EID_{50}) was calculated using the Reed-Muensch equation.

G. T. Belz and W. R. Heath (Immunology Division, The Walter and Eliza Hall Institute of Medical Research, Melbourne, Victoria, Australia) kindly provided us with raw data for the kinetics of virus-loaded dendritic cells (DC) in lymph nodes (LNs) (see Fig. in reference

10). Lymphatic compartment DC kinetics were from C57BL/6 mice intranasally infected with 10

^{2} PFU of HKx31 and determined by mediastinal LNs treated with collagenase-DNase to form single-cell suspensions, which were cultured with a

*lacZ*-inducible hybridoma specific for influenza virus NP and enumerated for β-galactosidase-producing cells (

10). T. D. Randall (Trudeau Institute, Saranac Lake, NY) kindly provided raw data of the kinetics of serum antibody (see Fig. in reference

44). Serum antibody kinetics were from C57BL/6 mice intranasally infected with 100 egg infectious units of influenza virus A/PR8/34, and influenza virus-specific immunoglobulin G (IgG) titers were determined by enzyme-linked immunosorbent assay (

44).

Mathematical model. The immune response to respiratory infection with influenza virus involves several lymphoid and nonlymphoid anatomical compartments. Because productive infection is typically restricted to the lung, we chose a simplified, two-compartment structure limited to the lung (as a whole) and single lymphoid compartment analogous to the draining lymph node and spleen. Certain aspects of the innate immune response to influenza virus, such as secretion of type I interferon, have been modeled in the past (

3,

34). We chose to limit the scope of this model to the adaptive cellular and humoral immune response. Explicit functions that are comprised of multiple cell subsets that may interact in complex ways, such as antigen presentation, are summarized using simplified terms familiar to an immunologist. The ability to carry antigen and activate “naïve” T cells and B cells to become effectors is, for example, subsumed under the term “dendritic cells”. In addition to DC, we include the following cell types: infected and uninfected epithelial cells, cytotoxic (CD8) T cells, “helper” T cells (CD4), and long-lived and short-lived antibody-secreting cells (B cells). Thus, cell-mediated elimination of infected epithelial cells is mediated solely by the CD8 T-cell component, while “help” for DC and B-cell activation and class switching is mediated solely by the CD4 T-cell component. A schematic representation of our model of influenza virus infection and the cellular/humoral immune response is presented in Fig. .

Infection begins when influenza viruses (

*V*) enter the host respiratory tract and infect epithelial cells (

*E*_{p}) within the airways and lung parenchyma. The infection then stimulates the immature DC (

*D*) present in the lung parenchyma to take up virus and become virus-loaded DC (

*D**) capable of antigen presentation to T and B cells. A term is introduced to represent migration of influenza virus-loaded DC into the spleen/lymph node. The DC that are competent to “prime” adaptive naïve cytotoxic, “helper,” and antibody-secreting cells become “mature” (

*D*_{M}). The mature DC (

*D*_{M}) activate naïve CD8 T cells (

*T*_{N}), which differentiate into CD8 effector cells (

*T*_{E}). The

*T*_{E} then migrate into the airway/lung compartment and kill infected epithelial cells (

*E*_{P}*). In addition, mature DC also activate naïve CD4 T cells (

*H*_{N}), which differentiate into helper T cells (

*H*_{E}) that can support antibody class switching and DC maturation. Activation of naïve B cells requires binding of viral antigens to the B-cell receptor and help from activated CD4 T cells (

*H*_{E}). We model this encounter by assuming that mature virus-loaded DC (

*D*_{M}) interact with naïve B cells (

*B*_{N}), allowing B-cell receptor-IAV interactions (

64). We consider two separate populations of antibody-secreting cells, short-lived (

*P*_{S}) and long-lived (

*P*_{L}). Based on data from major histocompatibility complex (MHC) class II-deficient mice, which develop only short-lived plasma cells after infection, we assumed that differentiation of long-lived antibody-secreting cells is mediated by cell-to-cell interactions between effector helper T cells (

*H*_{E}) and activated B cells (

*B*_{A}). In contrast, we assumed that short-lived plasma cells can arise from T-cell-independent activation of B cells. The antiviral antibodies (

*A*) produced by short- and long-lived plasma cells diffuse into the lung compartment to bind and remove free influenza virions.

We used delay differential equations (DDE) to account for the time delays between viral infection, immune cell activation, and migration of immune effector cells between tissue and lymphoid compartments. The model equations describing events in the lung are the following:

The variables and parameters for equations

1 to

5 are specified in Tables and .

| **TABLE 1.**Model variable definitions and initial values |

| **TABLE 2.**Model parameter definitions and values |

The change in the numbers of uninfected epithelial cells (

*E*_{p}) in equation

1 is described by a constant death rate (δ

_{E}) and a term assuming a constant rate of regeneration of uninfected epithelial cells, δ

_{E}*E*_{0}, with

*E*_{0} denoting the initial number of uninfected cells at time zero. Note that a constant regeneration rate does not take into account possible changes in the rate as a result of the infection. Viral infection occurs at the rate β

_{E}*E*_{p}*V*. In equation

2, infected epithelial cells are produced by viral infection, β

_{E}*E*_{p}*V*, and removed by cell death (viral induced or apoptotic) with rate δ

_{E} or eliminated by cytotoxic (CD8) T effectors with the rate constant

*k*_{E}. Assuming that the number of antigen-specific CD8 T cells in the lung compartment is proportional to the number in the lymphoid compartment with proportionality constant γ and a time delay of τ

_{T}, we write the number of CD8 T cells in the lung compartment as

*T*_{lun}_{g} = γ

*T*_{E}(

*t* − τ

_{T}), where

*T*_{E}(

*t*) denotes the number of effector CD8 T cells in the lymphoid compartment. This assumption is made based on experimental measurements of CD8 T-cell kinetics in the lung compartment versus the lymphoid compartment (Fig. ). When we compared the kinetics of influenza virus-specific CD8 T cells in the lung compartment with that in the lymphoid compartment, we found that the CD8 T-cell count in the lung compartment was around 15% of that in the lymphoid compartment with a time delay of 0.5 day.

Equation

3 represents the kinetic changes in the virus population. Free influenza virions are produced from infected epithelial cells at a rate of π

_{V} per cell and are cleared nonspecifically at a rate of

*c*_{V} per virus. Free virions are also cleared by virus-specific immunoglobulin,

*A*(

*t*), with a rate constant of

*k*_{v}.

*D* in the equations refers to the function of APC that can become “professional” APC and activate “naïve” T and B cells to become effectors in the model. Out of conventional and plasmacytoid DC subsets, we consider the DC population which plays a role of presenting antigen (

32,

81). Equation 4 describes the kinetics of immature DC,

*D*, with the initial number of immature DC in the lymphatic compartment as

*D*_{0}. We assume that in the absence of infection, this level of DC is maintained as a constant,

*D*_{0}. Immature dendritic cells die either at a rate of δ

_{D} or become virus-loaded DC at a rate constant of β

_{D}. Virus-loaded DC,

*D**, are cleared at the rate δ

_{D}_{*}. Here, the clearance of the virus-loaded DC includes the death of the cells and their migration into the lymphoid compartment.

In the lymphoid compartment, we describe the kinetics of immune cells as follows:

Dendritic cells are the major antigen-presenting cells in the lymphatic compartment (

77). A recent study supported our model assumption that antigen presentation is mediated by DC subsets in the lymphatic compartment. Ingulli et al. observed that lymph node DC, not lung DC, presented OVA influenza virus antigen to naive CD8 T cells during the first 72 h after infection (

41). Equation

6 describes the kinetics of DC maturation to become competent to activate naïve precursors to become cytotoxic, helper, or antibody-secreting cells. The parameter

*k*_{D} denotes the rate of maturation from virus-loaded APC to mature APC, which encompasses the rates of migration of virus-loaded DC to the lymphatic compartment, antigen processing, DC maturation, and antigen transfer to non-lung-derived DC. In addition, we denote τ

_{D} as a time delay for the migration of antigen-loaded DC from the lung compartment, and the clearance rate of

*D*_{M} is denoted by δ

_{DM}. It should also be noted that certain aspects of antigen presentation are not explicitly depicted in the model. For instance, it has been demonstrated that DC can transfer antigen to lymph node and spleen resident DC (

10,

15,

45,

46), thereby increasing the number virus-loaded DC in the lymph nodes to a level greater than can be accounted for from the lung. Quantitatively, DC migrating from the respiratory tract to the draining lymph node represent only 1% to 3% of the total lymph node DC (

45), with interaction between non-airway-derived DC and CD8 T cells reported (

10). These aspects are effectively subsumed under the terms for DC maturation.

*D*_{M} is therefore a simplification designed to represent what is biologically a much more complicated and multicellular process.

Another assumption is that DC maturation and function are not dependent on feedback from CD4 T cells. Biologically, bidirectional CD4 T-cell-DC interactions may be important. For this reason, we explicitly modeled CD4 T-cell-mediated “licensing” of DC later. We denote τ_{D} as a time delay for the migration of antigen-loaded DC from the lung compartment. The clearance rate of *D*_{M} is denoted by δ_{DM}.

Equation

7 describes the activation of naïve helper T cells, CD4 T cells, as

*H*_{N}, with a rate of π

_{H}(

*D*_{M}) and their clearance at a rate of δ

_{HN}. The rate of the constant source of the naïve cells is described with δ

_{HN}*H*_{N}_{0}. The mechanisms through which DC prime naïve T cells have been extensively studied (

46). Since the priming of naïve CD4 T cells is initiated with contact between the naïve CD4 T cell and the antigen-presenting cell, we impose an activation profile, π

_{H}(

*D*_{M}*) = π*_{H1}*D*_{M}/(

*D*_{M} + π

_{H}_{2}). The constant π

_{H}_{1} denotes the maximum activation rate, and the constant π

_{H}_{2} denotes the level of antigen-presenting cells which provides the half of the maximum activation rate. We introduce this nonlinear activation profile to reflect the limitation in the activation rate as the level of mature DC increases. The activated helper CD4 T cells,

*H*_{E}, proliferate at a rate of ρ

_{H}(

*D*_{M}) = ρ

_{H}_{1}*D*_{M}/(

*D*_{M} + ρ

_{H}_{2}), with the maximum proliferation rate, ρ

_{H}_{1}, and the level of mature DC needed for half-maximum activation, ρ

_{H}_{2}.To allow for memory effects, the clearance profile of

*H*_{E} is defined as δ

_{H}(

*D*_{M}) = δ

_{H1}*D*_{M}/(

*D*_{M} + δ

_{H}_{2}); when antigen is not present, δ

_{H}(

*D*_{M}) becomes zero, reflecting the persistence of

*H*_{E}. Effector CD4 T cells are cleared by death and migration out of the lymphoid compartment. The profiles of CD4 T-cell activation, proliferation, and clearance are assumed to be regulated by the number of mature antigen-presenting DC, rather than chosen as constants throughout the primary response. The kinetics of activation, proliferation, and clearance of CD8 T cells are modeled in a similar manner in equations

9 and

10. In particular, the clearance of antigen-specific CD8 cells includes a fraction of these cells that migrate from the lymphoid to the lung compartment with a time delay (see equation

2). This was done with the view that T-cell activation, expansion, and death are more dependent upon the availability of mature antigen-presenting cells than the amount of free virus present. Thus, we introduced nonlinear proliferation and clearance functions that depend on the number of mature DC. In addition, because of the difficulty in distinguishing activated and effector T cells from memory T cells by experimental measurements, a formal separation of the T cells that survive to form memory cells has been deliberately ignored. The decreased death rate of the memory T-cell population is modeled by using a DC-dependent death rate in our model. Note that when

*D*_{M} decreases, so does the death rate of T cells giving rise to a long-lived memory component.

Recently, activation of B cells via interaction with antigen-presenting cells has been described as a pathway for T-cell-independent B-cell activation (

10,

44,

46,

64). In the model, mature DC can carry antigen to the lymph nodes for presentation to the naïve B cells. Activation of B cells hence depends on the level of mature DC and is described as similar to the case of CD4 and CD8 T-cell activation, by π

_{B}(

*D*_{M}) = π

_{B}_{1}*D*_{M}/(

*D*_{M} + π

_{B}_{2}). We assume that the proliferation of activated B cells depends both on the numbers of mature DC and effector CD4 T cells, as in ρ

_{B}(

*D*_{M} +

*hH*_{E}) = ρ

_{B}_{1}(

*D*_{M} +

*hH*_{E})/(

*D*_{M} +

*hH*_{E} + ρ

_{B}_{2}) (

42). This choice is driven by the observations that although the initiation of B-cell activation is mediated by soluble antigen with other viruses (

59,

71), it is difficult to find substantial amounts of free virus in the lymph node during an infection with low-pathogenicity influenza A virus, such as the mouse-adapted X31 virus. This may change in other situations, and alternative model scenarios, such as direct activation of B cells by virus, are explicitly addressed below. We assume a constant clearance rate of activated B cells, δ

_{BA}. In equation

12, the activated B-cell differentiates into short-lived plasma (antibody-secreting) cells,

*P*_{S}, at a rate of π

_{S}. Interactions between effector CD4 T cells and activated B cells, mediated in vivo by CD40-CD40L (

42), cause the activated B cells to differentiate into long-lived plasma cells,

*P*_{L}, at a rate of π

_{L}. The short-lived plasma and long-lived plasma cells have clearance rates of δ

_{S} and δ

_{L}, respectively. We assume different antibody secretion efficiencies for short- and long-lived plasma cells, given by π

_{AS} and π

_{AL}, respectively. The clearance rate of antibody is δ

_{A}.

Parameter selection. The rate of infection of epithelial cells by unit influenza virus was chosen as 7 × 10

^{−5} day

^{−1} (EID

_{50}/ml)

^{−1}, which is comparable to the average infection rate estimated from the viral kinetics of experimentally infected adults (

3). We assumed the death rate of infected epithelial cells, δ

_{E}_{*}, as 1.2 day

^{−1}, which is comparable to the estimated average life span of an infected epithelial cell, 1 day (

83).

The initial numbers of naïve CD8 T cells, CD4 T cells, and B cells in the lymphatic compartment were each chosen as 10

^{3}. The maximum proliferation rate of CD4 T cells (ρ

_{H}_{1}) was chosen as 1.51 day

^{−1}, which corresponds to a doubling time of 11 h (

18), and the maximum proliferation rate of CD8 T cells was chosen as 2.6 day

^{−1}, which corresponds to a doubling time of 6.40 h. The constants π

_{H}_{1} and π

_{T}_{1} denote the maximum activation rates of naïve CD4 and CD8 T cells, respectively, and the constants π

_{H}_{2} and π

_{T}_{2} denote the levels of DC which provide the half-maximum activation rates of CD4 and CD8 T cells, respectively. We chose π

_{H}_{1} of 1.5 day

^{−1}, π

_{H}_{2} of 100 cells, π

_{T}_{1} of 3 day

^{−1}, and π

_{T}_{2} of 100 cells. The half-maximal rate was set as 100, based on the expected precursor frequencies of CD4 and CD8 T cells and that activation/expansion should begin 48 to 72 h after infection. Using these rates, a value of 100 cells for

*D*_{M} corresponds to initiation activation of CD4 and CD8 at around 2 days postinfection. With these choices, the peak number of the influenza virus-specific CD8 T cells is 4.1 × 10

^{6} and that of CD4 T cells is 1.4 × 10

^{5}. This is consistent with the observation that during primary influenza virus infection, the magnitude of the virus-specific CD4

^{+} T-cell response is approximately 10-fold lower in both frequency and number than that of the CD8

^{+} response (

16). Similar results have been reported in a

*Listeria* spp. infection model, which suggests intrinsic differences in programming of CD4

^{+} and CD8

^{+} T-cell activation, proliferation, and effector function (

29,

72).

We introduce the nonlinear activation profile to reflect the limitation in the activation rate as the level of mature DC increases. The clearance rate of

*H*_{E} is defined as δ

_{H}(

*D*_{M}) = δ

_{H}_{1}*D*_{M}/(

*D*_{M} + δ

_{H}_{2}), with the maximum death rate, δ

_{H}_{1}, of 0.4 day

^{−1} and the level of mature DC needed for half-maximum death, δ

_{H}_{2}, of 1. The clearance rate of

*T*_{E} is defined as δ

_{T}(

*D*_{M}) = δ

_{T}_{1}*D*_{M}/(

*D*_{M} + δ

_{T}_{2}), with the maximum death rate, δ

_{T1}, of 0.75 day

^{−1} and the level of mature DC needed for half-maximum death, δ

_{T}_{2}, of 1. The clearance parameters for CD8 T cells used are the same ones as we used for helper T cells. The killing rate of CD8 T cells has been estimated as follows: a half-life of 1.4 h for lymphocytic choriomeningitis virus-positive target cells by CD8 T-cell-mediated elimination in vivo (

5) suggests that

. During the primary infection, if we chose an overall number of effector CD8 T cells as 10

^{4} in the lung compartment, we estimated

*k*_{E} as 1.19 × 10

^{−3} day

^{−1}. Although other estimates of the cytotoxic T-lymphocyte (CTL) killing rate of infected cells have been reported using different approaches under different viral infections (

31,

66,

82), no estimate of the CTL killing rate for influenza virus infection is available. We adopted the earlier estimate in our simulation and then performed the sensitivity analysis for this parameter (along with other key parameters) (see Fig. , below). The clearance rate of antibodies was chosen as 0.04 day

^{−1}, which corresponds to a half-life of 17.3 days, which is around two times longer than the estimates of the serum IgG half-life in mice, 4 to 8 days (

79,

80), under the assumption that decay would be slower in lung tissue.

The CD8 T-cell migration delay from lymphoid compartment to lung (τ

_{T}) and the migration factor (γ) were chosen as 0.5 day and 0.15, respectively, by comparing experimental kinetics of antigen-specific CD8 T cells in the lung with that in the lymphoid compartment (Fig. ). We chose a viral clearance rate,

*c*_{v}, of 1 day

^{−1}, which is comparable to that from experiments suggesting that nonspecific physical removal (expulsion, phagocytosis, uptake by target cells, etc.) of infective virions takes 4 to 24 h (

13).

All the other parameters were set to match the experimental kinetics of viral load (

62), DC in the lymphoid compartment (

10), and antibody (

44) as well as to describe the deletion experiments of CD8 T cells, B cells, both CD8 and B cells, and CD4 T cells simultaneously with one set of parameters. Where parameter values were not available from the literature, values were selected to match reported data by first limiting values to a biologically plausible range and then fine-tuning values to achieve a good match to the data. Rigorous statistical estimates of kinetic parameters based on experimental data are needed. We are pursing these based on recently collected time series data by our laboratories, and these results will be reported in a future paper.