Modeling and Optimization of Continuous Viral Vaccine Production

Abstract

A model that captures realistic viral growth dynamics has been developed based on a continuous and semi-continuous production model of an influenza A virus. This model considers viral growth parameters such as viral latency. It also captures the lag observed during the early production of viruses in a culture and explains later-phase growth dynamics. Furthermore, a sensitivity analysis was performed to investigate the effects of each input on each output. This revealed that production of defective interfering particles (DIPs) highly depends on the number of cells introduced to the viral reactor. The rationale for this is, as per the model, that a reduction in number of cells to be infected causes a reduction in DIPs formed as rate of viral infection decreases. Finally, a flowsheet model was created to optimize the continuous platform, including number of cells supplied to the viral reactor. From this, it was observed that the peak number of DIPs formed could be reduced by one-third. Finally, this model is tailorable to different viral particles using parameter estimation. Therefore, the proposed mathematical model provides a versatile, comprehensive platform that can be tailored to various viral cultures with or without a latent phase.

1. Introduction

Mathematical models are enabling technology that can be used to optimize and understand a variety of platforms. Specifically, first-principle modeling is a widely accepted technique for analysis of population dynamics of viruses [1]. Mathematical models allow for study and analysis of potential experimental techniques with less investment in laboratory equipment. First-principle modeling can also be used as a process optimization tool by determining the most likely experiments to be successful [2,3]. Several mathematical models have been presented for the batch-level design of a virus bioreactor [4,5,6]. These models account for growth and infection of susceptible target cells and growth and production of virus particles [1,7]. However, with the paradigm switch from batch-level design to continuous platforms, developing continuous models that can accurately track the growth dynamics within these newer cultures is of paramount importance [2].

Continuous models for production of viruses for viral vaccines have been proposed in the literature. The difference between continuous and batch models is the inclusion of dilution and feed terms [3,8,9]. Including a dilution term is essential for the liquid phase of a continuous bioreactor. The liquid phase operates continuously in a perfusion (or continuous) bioreactor, whereas cells are considered to be in batch mode [10]. The liquid in a perfusion system is continually added to and drained from the reactor, so the dilution term must account for these volume changes [8,10].

Literature models of continuous vaccine production address the continuity of the liquid phase. Still, they do not address the possibilities of an immune response, a latent phase, or other potential changes in the virus that may affect its production [9]. A latent phase is a period during a virus’s life cycle when it remains dormant within the host cell [11]. This period can account for the beginning of replication (transcription, translation, etc.), or it can simply be a period of no activity. Therefore, in the latent phase, the virus is not infectious, nor does it contribute to production of newly infected cells [6,11,12]. Viral latency affects some but not all viruses. Cori et al. showed that influenza exhibits a viral latency period [12]. Other studies have shown the same and provided estimations for the latency period parameter, k [13]. However, these literature models did not capture the experimental results’ early-phase viral growth dynamics because they did not account for a latent phase [12]. Therefore, latent-phase parameters must be added to address this lag and accurately represent the growth dynamics observed in viral cultivation.

Literature models for continuous viral production include several ODEs [8]. The model herein will analyze four separate cell classifications, including uninfected cells (T), infected cells (I_s), cells that contain defective interfering particles (I_d), and cells co-infected with both DIPs and virus (I_c). Viral latency was introduced into the model via implementation of new terms, including a latent-phase parameter and a population parameter that accounted for latent viruses [12]. Furthermore, the versatility of this model was tested through use of parameter estimation [5,14,15,16,17,18,19,20].

The newly developed model will be applied to various virus types that do or do not present viral latent-phase characteristics [1,7]. As viral latency may or may not be present in all viral cultures, it was essential to ensure that the structure of the model would enable latent-phase terms to be canceled out [21,22,23,24]. The following sections will describe how this model was developed and how latent-phase parameters were added, then present an optimization study, a sensitivity analysis, and a flowsheet model.

2. Materials and Methods

The hypothesis for the following model development was that inclusion of latent-phase parameters and equations would enable the model to accurately track growth dynamics exhibited by viruses in continuous cultivation. This hypothesis was based on experimental results presented in the literature, which showed a delay in viral growth [8]. It will be shown that the newly developed “latent-phase” model could more accurately track experimental results found in the literature. The following subsections will outline the methodology used to improve the model, as well as the parameter estimation, sensitivity analysis, optimization procedure, and flowsheet modeling.

2.1. Experimental Results

The experimental data used to evaluate and train the first-principle model proposed herein is based on literature sources. These literature sources included rabies, measles, and human influenza A/Puerto Rico/8/34 H1N1 [8,12,14,16,17].

2.2. Continuous Vaccine Production Model with Viral Latency

Viral latency has been studied in batch-mode cultures. Given that several literature models appear to miss early-phase growth patterns, it was hypothesized that viral latency mechanisms could be present in continuous cultivation [8,12,14,16,17]. However, to preserve model versatility for non-latent viruses, it was necessary to ensure that these latent-phase terms could be canceled out.

Figure 1 shows a diagram of the proposed model. It was assumed that viruses infected with standard virus particles (STVs) and co-infected cells enter an eclipse or latent phase [1,5,12]. Also shown in the diagram is the addition of the latency time parameter, k. In this figure, if “viruses with an eclipse phase” were to be removed, the remaining figure would represent a model without viral latency, as presented in the literature [8]. However, the inclusion of this portion of the model affects number and concentration of cells with standard viruses and co-infected cells and thus significantly changes the overall model.

The following equations summarize the newly developed latent-phase model. Each parameter is defined in

Table 1. Estimated model parameters and initial conditions.

Parameter	Definition	Value
µ	Specific Growth Rate (1/h)	0.027
D	Dilution Rate (1/h)	0.0396
f	Fraction of Produced DIPs	10−3
k1	Virus Infection Rate (mL/virion/h)	1 × 10−8
k2	Apoptosis Rate of Infected Cells (1/h)	8 × 10−11
k3	Virus Production Rate (virion/mL/cell)	5
k4	Virus Degradation Rate (1/h)	0.035
Tin	Cell Concentration in the Feed (cells)	3 × 106
k	Latent-Phase Constant (mL/virions/h)	8 × 10−13
T (initial)	Target Cell Population (cells)	5 × 106
Is (initial)	Cells Infected with STVs Population (cells)	0
Id (initial)	Cells Infected with DIPs Population (cells)	0
Ic (initial)	Co-infected Cell Population (cells)	0
Vs (initial)	STV Concentration (virions/mL)	1.25 × 105
Vd (initial)	DIP Concentration (virions/mL)	0
E (initial)	Concentration of Viruses in a Latent Phase (virions/mL)	0

.

$$\frac{dT}{dt}=\mu T-k_1\left(V_s+V_d\right)T+D\left(T_{in}-T\right)$$

(1)

$$\frac{d{I}_d}{dt}=k_1{V}_{d}T-\left(k_1{V}_s-\mu \right)I_d-D{I}_d$$

(2)

$$\frac{d{I}_s}{dt}=k_1{V}_{s}T-\left(k_1{V}_d+k_2\right)I_s-D{I}_s+kE{I}_s$$

(3)

$$\frac{d{I}_c}{dt}=k_1\left(V_s{I}_d+V_d{I}_s\right)-k_2{I}_c-D{I}_c+kE{I}_c$$

(4)

$$\frac{d{V}_s}{dt}=k_3{I}_s-k_3\left(T+I_d\right)+k_4{V}_s$$

(5)

$$\frac{d{V}_d}{dt}=k_3{I}_c+f{k}_3{I}_s-k_3\left(T+I_s\right)+k_4{V}_d$$

(6)

$$\frac{dE}{dt}=k_{1}T{V}_s-kE\left(I_c+I_s\right)$$

(7)

The number of uninfected cells, T (cells), was not affected by latency, and Equation (1) summarizes the growth of the cells at the growth rate μ (h⁻¹). The growth of cells was also affected by the virus infection rate k₁ (mL/virion/h), which involved two cell concentrations: standard virus particles, V_s (virions/mL); and DIP virus particles, V_d (virions/mL). Finally, it was necessary to include a dilution term, D (h⁻¹), to account for entrance and removal of cells (T_in) and media [10].

The number of cells containing DIPs, I_d (cells), was not affected by latency, as DIP particles enter no latent phase. Equation (2) is similar to the equation that accounts for the number of infected cells including the use of the dilution term, D, in order to account for the dilution effects of a continuous platform.

In Equation (3), which represents the population of cells infected with STV, the final term that accounts for virus particles that enter a latent phase, I_s (cells), was added. The last term contains the latent-phase constant, k (mL/virions/h); the concentration of viral particles in a latent phase, E (virions/mL); and the number of cells infected with a standard virus, I_s (cells). This final term estimated the number of cells containing viral particles in latent phase per hour. These cells, while still infected, could not contribute to production of more viral particles until the latent-phase period was complete [12]. Therefore, this final term added a time-delay effect to the virus growth curve.

Equation (4) represents the number of co-infected cells, I_c (cells). These cells contained both standard viral particles and DIP particles. Due to distribution of the cells, some co-infected cells contained many viral particles and only a few DIP particles [19]. Other cells contained many DIP particles and a few viral particles. Therefore, co-infected cells may contribute slightly to production of virus particles and may contain viral particles in a latent phase. The final term in this equation accounts for viral particles that enter a latent phase from a number of co-infected cells.

The final equations in this model account for concentration of standard virus, V_s (virions/mL); concentration of DIP particles, V_d (virions/mL); and the number of cells containing viruses in a latent phase, E (cells). Equations (5) and (6) were adapted from the revised three-dimensional model presented by Frensing et al. (2013), which simplified the original equations presented in the six ordinary differential equations model. Additionally, within these final equations, fraction of produced DIPs, f (unitless); virus production rate, k₃ (virions/mL/cell); and virus degradation rate, k₄ (1/h) are introduced. The final equation, Equation (7), was a newly introduced equation to the model. This equation accounts for viral particles entering a latent phase. All the parameters mentioned above, their values, and their units are summarized in Table 1. The values of the initial conditions are also given in Table 1 [3,8].

2.3. Transition to Semi-Continuous Modeling

As discussed, the transition to a continuous platform is a significant topic of interest in the pharmaceutical industry. Due to concerns with batch labeling and investment in new equipment, a transition to a semi-continuous platform may be more feasible for some companies. This combination of a partially continuous platform and a partially batch platform would enable pharmaceutical companies to utilize their present, inherent knowledge of their current batch processes while reaping some of the benefits of a continuous platform.

The cell bioreactor would be a continuously operating perfusion bioreactor for this setup, while viral bioreactors would be changed to batch reactors. The platform would consist of the perfusion bioreactor, which would operate continuously while feeding cells to batch-wise wave bags inoculated with viral particles. Those batch wave bags could serve as the virus bioreactor.

To model the described semi-continuous setup, the continuous model needed to be adapted. The equation for uninfected cells was adjusted to adapt the model, as shown below.

$$\frac{dT}{dt}=\mu T+D\left(T_{in}-T\right)$$

(8)

This equation removed the terms accounting for cellular death related to viral particles. The cell bioreactor was operated separately from the virus reactor. Then, the same model equations were used to model batch reactors, as shown in Equations (8)–(14), but the dilution terms were removed, as the viral reactors were then operating in batch mode. These new equations are shown in Equations (8)–(15).

$$\frac{dT}{dt}=\mu T-k_1\left(V_s+V_d\right)T$$

(9)

$$\frac{d{I}_d}{dt}=k_1{V}_{d}T-\left(k_1{V}_s-\mu \right)I_d$$

(10)

$$\frac{d{I}_s}{dt}=k_1{V}_{s}T-\left(k_1{V}_d+k_2\right)I_s+kE{I}_s$$

(11)

$$\frac{d{I}_c}{dt}=k_1\left(V_s{I}_d+V_d{I}_s\right)-k_2{I}_c+kE{I}_c$$

(12)

$$\frac{d{V}_s}{dt}=k_3{I}_s-k_1\left(T+I_d\right)+k_4{V}_s$$

(13)

$$\frac{d{V}_d}{dt}=k_3{I}_c+f{k}_3{I}_s-k_1\left(T+I_s\right)+k_4{V}_d$$

(14)

$$\frac{dE}{dt}=k_{1}T{V}_s-kE\left(I_C+I_s\right)$$

(15)

With these new model equations, the semi-continuous platform could be simulated. First, the model for the cell bioreactor was analyzed. The primary challenge was determination of the number of cells to dilute from the cell bioreactor in order to inoculate the viral wave reactors. In this analysis, 5 million cells were assumed to be diluted from the cell bioreactor at one time to inoculate a 5 L wave bioreactor for virus infection at 1 million cells per milliliter. Therefore, this simulation was performed on a research-lab scale. This simulation also assumed that cells would be able to enter exponential growth after they were bled from the perfusion cell reactor following a brief lag phase.

Finally, once the continuous portion of the semi-continuous platform was simulated, the batch-wise portion could be simulated. This portion of the simulation focused on production of virus particles. As simulated by this model, production of viral particles was comparable to that of other batch models for viral particles as shown by other researchers.

2.4. Parameter Estimation

The latent-phase model added one constant to the model, estimated from the experimental data. Furthermore, the apoptosis rate of the infected cells needed to be re-estimated. This is because, with a latent phase, fewer cells are killed by viruses per unit of time, as viruses remain in cells longer before causing cell death. Additionally, virus production and degradation rates had to be re-estimated due to the extended period that a latent virus will spend in a cell. The software gPROMS carried out all parameter estimations and model simulations [20]. Parameter estimation enabled this model to be more versatile. Parameter estimation and model fitting were performed on data for the measles and rabies vaccine [16,17].

2.5. Sensitivity Analysis

The sensitivity analysis performed was a “one-at-a-time” type of analysis. In this analysis, one input variable was manipulated while the others were held at standard values [6,14,18]. The effect on the output was analyzed to determine which output parameters were most sensitive to a given input parameter.

2.6. Optimization and Flowsheet Modeling

A preliminary optimization was completed. The objective function for this optimization problem was to minimize concentration of DIPs, V_d, by manipulating the number of cells entering the viral reactor, T_in. The objective function is shown in Equation (16), in which J is the objective function and V_D is concentration of DIP-infected cells. Concentration of DIP-infected cells was subjected to the constraint that it must have been a number greater than or equal to zero.

$$J=\underset{V_D\in \left[0,\infty \right)}{\min }{V}_D$$

(16)

This objective function is based on the sensitivity analysis results, which showed that the concentration of DIPs highly depended on the inlet number of cells. Because DIP particles inhibit the formation of new viral particles, it is desirable to reduce the number of DIP particles formed [3,8,19]. Furthermore, reducing the number of DIPs formed will lead to lower downstream costs if removal of DIP particles is necessary. The optimization study was also completed with gPROMS through a flowsheet model [20].

Flowsheet modeling provided a realistic platform for the simulation to be performed. It is often possible to miss critical aspects of the actual operation of the equipment associated with the model because the equations do not provide a dynamic picture of the actual process that is being simulated [7]. Furthermore, process parameters that are easier to manipulate could be readily observed, such as flow rates between reactors, inlets, outlet flow rates, and other parameters. In this way, the optimization process became more straightforward.

For the latent-phase model, the variables that could be easily manipulated included the flow rates into, out of, and between the reactors. Therefore, in the optimization study, these variables were studied.

3. Results

3.1. Parameter Estimation Results

The methodology used to re-estimate the parameters (as found in the literature) and estimate the latent-phase parameters was model fitting to the original data presented in the literature [8]. A least-squares methodology was employed to minimize the difference between experimental data and model-predicted results.

The apoptosis, virus infection, and virus production rates were estimated. The apoptosis rate; virus infection rate, k₁; and virus production rate, k₃ had to be re-estimated because the viral particles used for our model were hypothesized to remain in the cells for more extended periods due to latency and thus were different from the values predicted by the literature. In the literature, the apoptosis rate of infected cells was 7.13 $\times$ 10⁻³ per hour, whereas the newly estimated apoptosis rate was 8 $\times$ 10⁻¹¹ per hour [8]. The newly estimated viral infection rate in the new latent-phase model was 1 $\times$ 10⁻⁸ mL/virion/h. Finally, the newly estimated virus production rate was 5 virions/cell/mL/h, whereas literature sources estimated this value to be 168 virions/cell/mL/h [3,8].

3.2. Additional Parameter Estimations for Other Viral Vaccines

Parameter estimation was performed for data found in the literature for both a rabies vaccine and a measles vaccine [16,17]. These vaccines were produced using batch-mode production techniques. Figure 2 shows the results of this parameter estimation.

3.3. Model Comparisons

Literature sources were used to compare experimental results with model-predicted results. Data were extracted from literature sources regarding continuous cultivation of an influenza virus. The experimental setup for the literature sources involved two consecutive reactors. The first reactor was a perfusion-based cell cultivation reactor, whereas the second reactor was a perfusion-based viral production reactor.

Comparison of Literature Model, Latent-Phase Model, and Experimental Results

Data from a TCID₅₀ graph from the literature about continuous cultivation of an influenza virus was extracted. [8] Engauge Digitizer software was used. Engauge Digitizer is available on GitHub. Experimental results and model results are shown graphically in Figure 3.

3.4. Semi-Continuous Flowsheet Modeling

Through a semi-continuous setup, the findings of current platforms could be used to optimize batch setup of reactors. The semi-continuous model combines the benefits of the fully continuous platform and the semi-continuous platform and provides a complete picture of viral production dynamics in cell culture. In the desired semi-continuous setup, a cell culture reactor would be operated continuously as a perfusion reactor. Viral production reactors would be operated as batch-wise wave bag reactors. Figure 4 shows a simulated result of bleeding of cells from the cell cultivation reactor to batch-wise wave reactors for viral production.

3.5. Sensitivity Analysis and Optimization

Two processes were completed to further analyze the latency model created in this paper. First, a sensitivity analysis was performed to assess input and output parameter correlations. Second, an optimization analysis was performed to improve the platform process.

Sensitivity Analysis Results

A sensitivity analysis was performed for the continuous latent-phase model. The sensitivity analysis results are summarized in Figure 5. The coloration in Figure 4 indicates whether or not a 50% change in a given input had a positive or negative impact on a given output. Different shades of red indicate adverse effects, and different shades of green indicate positive impacts. Darkness of a shade indicates magnitude of impact; darker shades indicate more significant impacts.

3.6. Optimization and Flowsheet Modeling Results

The optimization study aimed to reduce the number of defective interfering particles formed. The rationale for performing this study was that defective interfering particles interfere with viral reproduction. Therefore, more viral particles per cell could be produced if the number of DIPs could be reduced. This study was performed utilizing a flowsheet model built with gPROMS software [20].

The flowsheet model created for the “latent-phase” model is shown in Figure 6. The flowsheet model utilized the model equations given in the previous section and built-in functions and equations provided by gPROMS [20]. These built-in functions enabled a connection between the model within the cell bioreactor and the model within the viral reactor. In this way, the model could be simulated closer to reality. The flowsheet model enabled variation of the number of cells introduced into the viral reactor. The number of DIPs formed could be minimized without significant impact to the viral titer.

The results of the optimization study are shown in Figure 7. There was a reduction of one-third in number of DIPs formed, which was accomplished through reduction of the number of cells introduced to the viral reactor by 10%. This was also met with a slight reduction of approximately 7.4% in the peak number of STV (viral particles) formed. Finally, there was a reduction of 42% in the number of co-infected cells.

4. Discussion

The results of the analyses presented herein indicate that the model developed is a versatile model with enhanced accuracy. Several conclusions could be drawn through parameter estimation, model fitting, and optimization.

4.1. Parameter Estimation for Other Viral Vaccines

Figure 2 shows the parameter estimation results for the rabies and the measles vaccine. The gap seen between the modeled results and the experimental results is explained by the fact that both of these experimental results were derived from batch-mode production techniques. The model used was continuous; therefore, there were terms in the model that caused a gap between the modeled and the experimental results. In Figure 2A, the tailing observed was not captured by the model because the dilution term present in the model caused a reduction in the modeled viral titer that was faster than the reduction observed during the experiment. Overall, however, it can be clearly seen that the model can capture general trends observed in different viral cultures. A similar phenomenon, which used a single model of population balance modeling to track the production of viruses by cell lines that are altered differently (via genetic engineering), has been published [25]. In general, these universal models are commonly based on population balance models and can be very useful in process optimization [26,27,28,29,30].

4.2. Comparison of Literature Model, Latent-Phase Model, and Experimental Results

As shown in Figure 3, the plaque-forming units from the experimental data peaked around 4.4 × 10⁶ PFUs. The latency model peaked at 4.56 × 10⁶ PFUs. Furthermore, there were several later peaks in the experimental data around days 7 and 15. Only the latent-phase model can capture these late-phase dynamics.

As shown in Figure 3, neither model had the same peak plaque-forming units as the experimental results. The experimental data showed a slight lag period before viruses begin to accumulate. The latent-phase model can capture this lag time. This model peaked after the experimental data but is closer in value. Therefore, the latent-phase model can better capture experimental data trends.

Finally, under the curve, the latent-phase model has a more comparable area to the experimental data. The approximate area under the curve for the experimental data from the first two days was 3.3 × 10^6, and the area under the curve for the latent-phase model from the first two days was approximately 3.4 × 10⁶. Therefore, the latent-phase model accurately captures PFU-AUC (plaque-forming units—area under the curve in units of plaque-forming units—days).

4.3. Semi-Continuous Modeling

Based on Figure 4, the bleeding from the perfusion cell reactor took approximately 2.5 days to return to the initial viable cell density. This enabled multiple wave bag reactors to be inoculated in a short period. Additionally, each wave-bag viral reactor could be designated as a batch, thus simplifying batch designation, which can be difficult with continuous cultivation.

4.4. Sensitivity Analysis

The sensitivity analysis showed that most of the outputs were not dramatically affected by an increase and/or a perturbation in the input parameters. It was discovered that population and concentration of DIPs were affected by an increase in specific growth rate, μ; virus production rate, k₃; and cell concentration in the feed, T_in. The rationale is that if the specific growth rate increases for cells, or the number of cells added to the cell feed increases, then the number of cells will increase, and thus the number of cells to be infected will also increase. This, in turn, will enable the number of DIPs formed to increase, as more defective viral particles can be formed due to the increased amount of infection occurring. Furthermore, if the viral production rate increases, the number of DIPs formed must decrease, as viral particles cannot form in the presence of DIPs, and thus these two must be inversely proportional.

In addition to number of DIPs formed being affected by changes in input parameters, number of standard viruses formed was also more affected if viral production rate, k3, was increased. This is rational, as number of viruses formed and viral production rate are directly proportional.

In general, most of the output parameters were not sensitive to changes in the input parameters. However, the revelation that production of DIPs is affected by specific growth rate of cells is interesting. This suggests that cells with slower growth rates may not be as affected by DIPs as are other cells, as concentration of DIPs formed tended to increase with increasing specific growth rates per this simulation. Therefore, this information could determine what cells to use to produce viral vaccines.

4.5. Optimization and Flowsheet Modeling

This optimization study successfully determined a more optimal process. This simulation demonstrated that the number of DIPs could be dramatically reduced through reduction of the number of cells introduced into the viral reactor. Furthermore, if fewer cells are required to be transferred from the cell bioreactor to the viral reactor, it may be possible to reduce the size of the cell bioreactor. This reduction in the size of the cell bioreactor could cause a smaller capital investment requirement if a bioreactor must be purchased, cause a minor footprint requirement, and cause a smaller amount of media to be used during the cultivation. Therefore, potential benefits of this optimization study include multiple benefits for the proposed continuous process.

5. Conclusions

Modeling influenza A growth dynamics for vaccine production is critical to understanding and optimizing the process. This model’s goal was to better explain realistic viral growth dynamics observed in continuous cultivation. The experimental data in the literature showed a noticeable delay in viral reproduction in the early phases of viral production [3,8]. Adding latent-phase effects to the continuous model improved the model’s accuracy in tracking experimental results. Furthermore, it proved effective in capturing fed-batch dynamics that literature sources explored.

Continuous production enables various benefits, including increased titer, faster production time, and less downtime. The viral titer obtained via a continuous platform can be 100-fold greater than batch-wise derived titers [1,2,7,10]. The modeled titer for influenza A studied in this report reached approximately 10⁸ TCID₅₀/mL, whereas batch-wise viral titers for influenza have been reported in the literature at approximately 10⁶ TCID₅₀/mL. Therefore, the obtained titer has a clear advantage when a continuous platform is used.

Finally, the developed latent-phase model was adapted to a flowsheet model used in optimization studies. It was demonstrated that use of a flowsheet model enabled optimization studies, which showed that there could be a reduction in the number of DIPs through manipulation of the number of cells fed to the viral reactor. Reducing the number of cells needed to inoculate the viral reactor makes it possible to require a smaller reactor for the cell cultivation reactor, thus reducing overall capital and costs of consumables. Additionally, reduction of DIP particles formed could positively impact downstream purification. The model presented in this paper also provides a methodology for modeling various viral vaccines. This is possible through parameter estimation, which enables the model to be tailored to different viral vaccine data.

6. Future Directions

The model presented in this paper provides a robust model for continuous production of a viral vaccine. Newly formulated influenza vaccines are developed to ensure that “flu season” can be handled without widespread infection. Therefore, accurate models could be critical in developing new formulations of the flu vaccine and could significantly impact future research endeavors. Additionally, this model has proven versatile, so it may be used to develop platforms for various viral vaccines.

Development of universal mathematical models for important production platforms such as viral vaccines cannot be ignored. Universal models enable users to tailor each model to suit their desired platform, enabling faster development. The model presented herein is part of the first steps toward creating a universal vaccine-production model. Future efforts to improve this model’s usefulness will be invaluable to the pharmaceutical industry as a whole.