# Simple Mathematical Models Do Not Accurately Predict Early SIV Dynamics

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Extended Standard Mathematical Model for HIV/SIV Dynamics

#### 2.1.1. Mathematical Model

_{E}, and cells that are actively producing the virus, I (Figure 1). Cells in the eclipse phase transition to the state of virus production at a rate, m, and both types of cells die at rates ${\delta}_{{I}_{E}}$ and δ

_{I}, respectively. Cells in the eclipse phase may die because they could be recognized as infected by mediators of innate immunity (e.g., macrophages or NK cells), due to the activation of DNA-dependent protein kinase during integration of viral DNA into host chromosome or due to accumulation of DNA intermediates in the cell’s cytoplasm [33,34]. In fact, it has been argued that at least in vitro most HIV-infected cells die before virus production begins [33]. Virus-producing cells make infectious viruses, V, at a rate Nδ

_{I}where N is the average number of infectious virions released by an infected cell per its lifetime (burst size). It is generally accepted that a majority of virions produced by infected cells are non-infectious [26,35,36]. Since these viruses are not contributing to the infection of new cells, non-infectious viruses are not tracked in this model (e.g., see [22]). Viruses are removed from the cell-free virus population by either an intrinsic clearance rate, c, or by infecting target cells, which occurs at a rate βT, where T is the (constant) number of available target (CD4 T) cells. The model is then given by the following system of ordinary differential equations:

**Figure 1.**Interaction diagram of the extended standard mathematical model of the early dynamics of HIV/SIV. Viruses, V , infect target cells, T at a rate β and are cleared at a rate c. The target cell enters an eclipse phase, I

_{E}, where it does not actively release any virions. Cells in the eclipse phase die at a rate δ

_{IE}. From the eclipse phase, the cell transitions to the state of a productively-infected cell, I, at a rate m. Productively infected cells die at a rate δ

_{I}or release infectious virions at a rate Nδ

_{I}; N is the average number of infectious virions produced by an infected cell.

#### 2.1.2. Relative Duration of the Eclipse Phase

_{IE}) and 1/δ

_{I }, respectively. Therefore, the fraction of total time cells spend in the eclipse phase is

_{I}, and δ

_{IE}, the relative duration of the cellular eclipse phase, T

_{m}, can be set to emulate a cellular eclipse phase of a short, intermediate, or long duration. Subsequently, when the average life-span of infected cells is fixed, this provides a method to continuously vary the mode of virus production from budding-like to bursting-like.

#### 2.1.3. Asymptotic Behavior of the Model

_{0}, is less than one (R

_{0}< 1). As we show in Supplement, in this model the basic reproductive number is given by

_{0}determines the long-term deterministic behavior of the model, it is not directly measurable. The basic reproductive number can be calculated from observable rates of virus growth and death assuming different modes of virus production by infected cells [40,41]. Therefore, to relate R

_{0}to the parameters that can be directly observed in vivo we calculated the rate at which the virus population expands during the phase of exponential growth or decays if virus replication is blocked, for example, during the use of highly active antiretroviral therapy (HAART) shortly after infection.

_{I}, c ≫ m, c ≫ δ

_{IE}) that V is essentially a function of I:

_{E}of the number of virus-producing cells to the number of cells in the eclipse phase are

_{I}+ m − δ

_{IE}> 0, there is one positive root in Equation (8), B = B

^{∗}. The rate of exponential increase of the viral population, r, is then given by

_{I}as the function of all other parameters

_{0}) and I

_{E}(t

_{0}) are the numbers of virus-producing cells and cells in the eclipse phase, respectively, at the time of the start of the therapy t

_{0}. Due to a short half-life of virions, the dynamics of the virus is proportional to that in Equation (11). Thus, in this model, the asymptotic virus decline during HAART is governed by the smaller of the two values, δ

_{I}or m + δ

_{IE}.

#### 2.1.4. Probability of Extinction

_{0}< 1, virus extinction is guaranteed. However, if R

_{0}> 1, virus extinction may occur with some probability depending on the parameters and whether infections start with free virus or virus-infected cells. In the case when the infection is started with one virion, the majority of extinctions will occur because the virion is cleared (c) before infecting a target cell (βT); the probability of this happening is simply $\frac{c}{c+\beta T}$. If infection of a cell does occur, then extinction will occur if the cell in the eclipse phase dies (δ

_{IE}) before maturing (m) into virus-producing cell; the probability of this happening is $\frac{{\delta}_{{I}_{E}}}{{\delta}_{{I}_{E}}+m}$. Finally, the probability of extinction depends on the number of infectious viruses produced by a virus-producing cell during its lifespan. Taken together, the probability that the infection becomes extinct when starting with a single virion (V

_{0}= 1) is given by

_{0}≥ 1 virions are initially present is

_{0}= αD, λ = −α ln q and q is given in Equation (12). This model is identical to “single-hit” models for exposure of tissues to radiation or infection of hosts with a given dose of a pathogen [43,44,45,46]. As we show below, this model is unable to accurately describe the data of Liu et al. [16].

_{0}= αD

^{n}.

_{i}= p

_{0}e

^{−}

^{λi}. In this model, a higher number of viruses leads to a lower probability of established infection, which indirectly implies antagonistic interactions between different viruses. The normalized probability of infection of the animal given initial number of viruses V

_{0}= αD is then

^{2}, $f\left(\lambda \right)=\frac{\lambda}{{\sigma}^{2}}\frac{{(\lambda \overline{\lambda}/{\sigma}^{2})}^{{\overline{\lambda}}^{2}/{\sigma}^{2}-1}}{({\overline{\lambda}}^{2}/{\sigma}^{2}-1)!}{e}^{-\lambda \overline{\lambda}/{\sigma}^{2}}$, then the probability of infection is given by

_{inf}.

#### 2.1.5. Parameter Estimations

^{−}

^{1}(Table 1), although the exact value does not strongly influence model predictions as long as c is large enough, i.e., c ≫ δ

_{I}, m, δ

_{IE}.

_{IE or δI}. This decay rate is approximately 0.5 − 1.5 day

^{−}

^{1}as estimated in many experimental studies (e.g., [51,52,53,54]), although only one study (as far as we are aware) looked at viral decay rates during treatment of acute infections [42]. Therefore, the minimal of the two rates, m + δ

_{I}or δ

_{I}, should be approximately equal to 0.5 − 1.5 day

^{−}

^{1}.

^{−}

^{1}.

**Table 1.**An example of parameter values for the mathematical model used in several deterministic and stochastic simulations. Even though parameters were chosen to guarantee a net rate of increase of the viral load of r = 1.5 day

^{−}

^{1}[16,39], the observed rates of expansion, r

_{o}, were lower due to a finite value for the clearance rate of the virus, c. The relative duration of the eclipse phase, T

_{m}, is given by Equation (4). Higher values of T

_{m}imply a long eclipse phase with burst-like virus production, and smaller values imply a short eclipse phase with budding-like virus production. Note that m = 5 day

^{−}

^{1}and m = 0.7 day

^{−}

^{1}are on the extremes of short and long average times of the duration of the eclipse phase: 5 and 34 h, respectively.

Parameter, Units | Virus Production Mode | Parameter Description | References | |||
---|---|---|---|---|---|---|

Continuous | Intermediate | Burst | ||||

δ_{IE}, day^{−}^{1} | 0.5 | 0.5 | 0.5 | death rate of cells in the eclipse phase | unknown | |

m, day^{−}^{1} | 5.0 | 1.5 | 0.7 | eclipse phase transition rate | [54,55] | |

δ_{I}, day^{−}^{1} | 0.583 | 1.313 | 5.06 | death rate of virus-producing cells | unknown | |

min(m + δ_{IE}, δ_{I}) | 0.583 | 1.313 | 1.2 | virus decay rate during HAART | [51,52,53,54,58] | |

c, day^{−}^{1} | 20 | 20 | 20 | virion clearance rate | [49,50] | |

N | 10 | 10 | 10 | infectious virion burst size | unknown | |

βT, day^{−}^{1} | 20 | 20 | 20 | rate of infection | unknown | |

T_{m} | 0.10 | 0.40 | 0.81 | relative duration of the eclipse phase | unknown | |

r_{o}, day^{−}^{1} | 1.443 | 1.445 | 1.433 | observed net viral growth rate | [16,39] |

_{IE}. Previous models assumed that cells in the eclipse phase have a death rate that is zero or very small [24,30]. However, cells in the mucosal tissues may have an intrinsically high rate of removal due to their proximity to gut microbiota, the non-optimal (nonlymphoid) host tissue for survival, or recognition of virus-infected cells by tissue phagocytes. Recently, it has been observed that in vitro cell cultures infected with HIV die prior to reproducing the virus [33]. The proposed mechanism for this high death rate is the activation of cellular apoptotic mechanisms, triggered by the double strand breaks in DNA preceding integration of viral DNA into the chromosome. Alternatively, the induction of pyroptosis due to accumulation of DNA intermediates can also lead to self-induced death of HIV-infected cells prior to virus production [34]. Although it is not known whether such processes occur in vivo, it is possible that most of cell death occurring during virus replication occurs during the eclipse phase, prior to production of virions. In most of our simulations we assume that cells in the eclipse phase survive for about 2 days (δ

_{IE}= 0.5 day

^{−}

^{1}, Table 1), but also check robustness of our results when δ

_{IE}has lower values. In general, the value of δ

_{IE}is not known.

_{0}and the average number of infectious viruses produced per infected cell N. Although many studies report a particular dose used to infect animals with SIV, it is not known how many viruses actually penetrate the tissue and initiate the infection [13,14,15,16]. However, the fraction of animals that do get infected when exposed to a certain viral dose is generally known; therefore, for a given experiment, we can know p

_{inf}(Equation (13)). To obtain further insights into how N and V

_{0}may be related we first analyze a simplification of our model when the duration of the eclipse phase is small, i.e., m → ∞. At this limit the extended model approaches the standard model. Since lim

_{m}

_{→∞}B = ∞ in Equation (9), by using basic algebraic manipulations and the formulas for Equations (9) and (13) we find the desired relationship

_{0}= − ln(1 − p

_{inf}) and the approximation is found when N ≫ 1. This result suggests that for a fixed probability of infection, increasing the average number of infectious virions produced by an infected cell, N , increases the initial dose of the virus needed to produce the observed p

_{inf}. This relationship can alternatively be viewed as the virus burst size needed to explain the observed virus dynamics for a given initial virus density. For example, in the study by Liu et al. [16], 2 out of 6 macaques were infected with SIV at the lowest dose of 10

^{6}viral particles and only a single transmitted/founder virus initiated the infection. Assuming that V

_{0}= 1, r = 1.5 day

^{−}

^{1}and δ

_{I}= 1 day

^{−}

^{1}[16,39,51,52,53,54], the burst size must be around

_{inf}= 0.33 and a rapid net rate of viral growth, r. This is surprisingly low given that it has been estimated that an SIV-infected cell produces around 5 × 10

^{4}viral particles [57]. Several studies have suggested that SIV/HIV infectivity is relatively low and only 1 in 1000 viral particles are infectious [26,35]; however, a more recent study argued that the fraction of infectious viruses could be much higher [36]. We found in our simulations that if we adjust V

_{0}and the level at which the virus population becomes detectable to match experimental data of Liu et al. [16], our modeling results are not very sensitive to the actual value of N (Figure S2 in Supplement).

_{I}. Because the extended standard model (Equations (1)–(3)) has 2 additional parameters (m and δ

_{IE}), no simple analytical relationship exists between V

_{0}and N in this model. Therefore, to run deterministic and stochastic simulations in the extended model we use the following algorithm. First, we choose values for a set of parameters, for example, m, δ

_{IE}, and N within biologically reasonable range (as discussed above). Then, we calculate the values of the remaining parameters (e.g., βT and δ

_{I}) given the constraints set by the calculations for r, min(δ

_{I}, m + δ

_{IE}), and p

_{inf}, and run simulations using the resulting parameters. This procedure is repeated for various sets of parameter values to determine the overall behavior of the model. An example of the parameter values used in some simulations is shown in Table 1.

_{det}we use experimental data from Liu et al. [16]. In these experiments, monkeys were exposed to variable doses of SIVmac251 intrarectally and the virus kinetics in infected animals were followed over time. The authors found that at the lowest viral dose (10

^{6}viral particles) the virus became detectable in the blood on average 8.5 days post infection [16]. Therefore, for a given set of model parameters, V

_{det}is determined empirically by running deterministic simulations of the model (Equations (1)–(3)) and defining V

_{det}as the virus density at time t = 8.5 days since infection. Given that we have bounded the parameters of the model to satisfy the measured constraints (see Table 1 and Equation (18)), both V

_{det}and V

_{0}are directly proportional to the burst size, N, (Figure S2 in Supplement). As an example, for the parameters shown in Table 1 and for V

_{0}= 1, the detection limit is V

_{det}≈ 10

^{4}(Figure 2 in Supplement).

#### 2.2. Simulating Virus Dynamics

#### 2.2.1. In Deterministic Simulations, the Mode of Virus Production by Infected Cells Does Not Strongly Impact the Time to Virus Detection

**Figure 2.**For biologically reasonable parameter values, the mode of virus production by infected cells has little impact on the time to virus detection in deterministic simulations. We solve the mathematical model (Equations (1)–(3)) numerically and plot predicted values for the infectious virus (V), infected cells in the eclipse phase (I

_{E}), and virus-producing cells (I). Time courses are shown for the model with continuous-mode virus production (short eclipse phase, left panel), intermediate mode virus production (eclipse phase takes approximately half of the life-span of infected cells, middle panel), and bursting mode of virus production (long eclipse phase, right panel). Parameters used are given in Table 1. All infections are started with V

_{0}= 1. The time of virus detection, t

_{det}, is the time when the virus population reaches V

_{det}= 10

^{4}infectious viruses and is indicated on individual panels.

#### 2.2.2. In Stochastic Simulations, the Initial Viral Dose Impacts the Time to Virus Detection but the Mode of Virus Production Does Not

^{4}infectious viruses was prohibitively slow, we gained initial insights into the impact of parameter values on virus dynamics by calculating the time to 100 infectious viruses, t

_{100}. There was a wide distribution of times to 100 infectious viruses in our stochastic simulations (Figure 3). We characterized these distributions by calculating the mean and mode of the distribution of t

_{100}for all simulations. Because the t

_{100}distributions were not highly skewed, the mean and median time to detection were nearly identical (results not shown). A number of interesting results emerged from these simulations.

_{100}between different runs in the model with bursting virus production is expected, since some cells do not survive to make viruses. Lastly, and somewhat unexpectedly given previous results [22], a burst-like mode of virus production led to a lower probability of established infection, compared to a continuous/budding production mode (0.19 vs. 0.35, when starting with a single infectious virion for the parameters in Table 1, see also Figure 3 and Figure 4). This stems from the assumption that cells in the eclipse phase have a non-zero chance of dying, and thus, an increase in the duration of the eclipse phase reduces the chance that the cell will transition into a state of virus production.

**Figure 3.**Distributions of times to 100 infectious viruses obtained using the Gillespie algorithm for the extended standard mathematical model of HIV/SIV dynamics (see Equations (1)–(3)). Simulations are for continuous/budding viral production mode (short eclipse phase, left panel), intermediate viral production mode (middle panel), and burst-like viral production mode (long eclipse phase, right panel). All infections start with one infectious virion, V

_{0}= 1. We performed 20,000 simulations for every parameter set (Table 1). Note that the time to 100 infectious viruses is shorter than the average time to virus detection as observed in experiments of Liu et al. [16] (see also Figure 6).

_{inf}, and for the relative duration of the eclipse phase, T

_{m}, we explored whether our predictions from simulations hold true for a wider range of parameters. We observed that the results were dependent on how changes in a relative duration of the eclipse phase were achieved. When only the transition rate, m, or only the death rate of cells in the eclipse phase, δ

_{IE}, was changed, the probability of established infection always decreases with a longer duration of the eclipse phase as long as δ

_{IE}> 0 (Figure 5A). However, if m and δ

_{IE}were correlated, the opposite trend may be observed (i.e., increasing the duration of the eclipse phase increases the probability of established infection if it also increases the death rate in the eclipse phase; Figure 5B). Thus, we conclude that whether the mode of virus production leads to either a higher or lower probability of established infection in general depends on the parameters of the model; however, if the parameters are uncorrelated, the bursting mode of virus production leads to a lower probability than the continuous/budding production mode as long as δ

_{IE}> 0. If δ

_{IE}= 0, then the mode of virus production has no influence on the probability that an infection becomes established. The latter result is in contrast with the conclusion found in simulations with the standard model [22].

**Figure 4.**Changes in the probability of established infection (p

_{inf}, panels A and C) and the time to 100 infectious viruses (t

_{100}, panels C and D) with the initial number of viruses (V

_{0}) and the relative duration of the eclipse phase (T

_{m}, Equation (4)) as predicted by stochastic and deterministic simulations of the mathematical model (Equations (1)–(3)). In panels A and B, points represent values from stochastic simulations and dashed lines are analytical predictions (Equations (13)). Black triangles in panel B show the probability of infection for budding and bursting modes of virus production by infected cells as calculated in Pearson et al. [22]. In panels C and D, points are the results from the simulations and solid lines are the predictions from deterministic solutions of the model. Parameters for the simulations are given in Table 1 with values for the maturation rate m varied between 0.7 day

^{−}

^{1}and 5 day

^{−}

^{1}with δ

_{I}being adjusted to satisfy constrains as described in the text.

**Figure 5.**Variable dependence of the probability of infection p

_{inf}on the relative duration of the eclipse phase, T

_{m}. Values for c, βT, and N are fixed according to Table 1 and the maturation rate, m, is varied for four different fixed values of the death rate of cells in the eclipse phase δ

_{I}(panel A) or when m and δ

_{I}are correlated (δ

_{I}= θm

^{2}, panel B). For a given pair of these parameters, the value of the death rate of virus-producing cells, δ

_{I}, is computed using Equation (10). The probability of established infection, p

_{inf}, and the relative duration of eclipse phase, T

_{m}, are found using Equations (4) and (13) , respectively.

_{100}(Figure 4C).

#### 2.3. Comparing Model Predictions with Experimental Data

#### 2.3.1. The Model Does Not Accurately Predict the Change in the Virus Detection Time with Increasing Viral Dose

^{6}viral particles) the time to infection was on average 8.5 days (for 2 infected animals the detection times were 7 and 10 days, Figure 6). Increasing the initial viral dose 10-, 10

^{2}-, or 10

^{3}-fold in these experiments shortened the time to virus detection to 6 days but did not change the variance in virus detection time between different animals ([16], Figure 6). Our initial simulations (see Figure 4), however, differ from these experimental results, as we predict much shorter times of virus detection, t

_{100}(by about 3–4 days, see Figure 4C). This is likely because our defined threshold of virus detection (100 infectious viruses) may be lower than the actual number of viruses that must accumulate in the host for the infection to be detectable with standard assays.

^{3}(t

_{1000}) and then ran the model deterministically for 1, 2, or 3 extra days. Effectively, this approach allowed us to to determine the time to virus detection defined as V

_{det}= 4 × 10

^{3}, 1.8 × 10

^{4}or 7.6 × 10

^{4}infectious viruses, respectively. To explain detection times at the lowest challenge dose under this model in stochastic simulations, the number of infectious viruses in the body needs to be approximately 2 × 10

^{4}(Figure 6B). Importantly, the model is unable to fully explain the initially large decrease in detection time with increasing viral dose and could not accurately predict the average time as observed in the data for all doses (Figure 6). Furthermore, this model predicted small variance in the time to virus detection when animals are infected with large initial doses; while in the data, the variance in the time to virus detection was approximately independent of the initial dose (results not shown). We thus conclude that the extended standard mathematical model for the SIV/HIV dynamics does not accurately predict the change in virus detection time with the initial viral dose in experimental infections of monkeys [16]. This inability of the extended standard model to accurately predict the change in time to virus detection was not due to the specific set of parameters used for simulations, since using higher values for the burst size, N , virus clearance rate, c, and the death rate of virus-producing cells, δ

_{I}, as well as varying the rate of transition, m, resulted in similar disagreements between model predictions and the data (see Figure S6 in Supplement). In many of these additional simulations, the initial viral dose was higher than one and was determined using Equation (18).

#### 2.3.2. The Model Does Not Accurately Predict the Change in the Probability of Established Infection with Increasing Viral Dose

^{6}viral particles), 2 out of 6 animals became infected while at 100 fold higher doses, only 4 out of 6 animals became infected (Figure 7). The standard mathematical model for SIV/HIV dynamics assumes that individual viruses do not compete within the host, and therefore, the probability of established infection of the animal should be a monotonically increasing function of the dose (Equation (13)). As discussed previously, this was the single-hit model case. When our model was fit to the experimental data using maximum likelihood, it was clear that the model is not consistent with the data (single-hit curve in Figure 7; p = 0.006, χ

^{2}test). Therefore, we investigated whether three alternative models (power law model (Equation (14)), competition model (Equation (15)), and gamma model (Equation (17)) were able to explain these data.

**Figure 6.**Stochastic simulations of the mathematical model do not accurately predict the change in the time to virus detection with increasing viral dose. In experiments, monkeys were challenged intrarectally with different doses of SIVmac251 and the time for virus detection in the blood was recorded [16]. We plot the time to virus detection for individual animals per dose (dots), the average of these times (bars), and the predictions from stochastic simulations of the mathematical model (solid lines) along with the 95% confidence intervals of the model predictions (dashed lines). In these simulations, we assume that exposure to 10

^{6}particles leads to an infection with one infectious virus (V

_{0}= 1), since in these experiments only a single founder virus was detected [16]. Simulations were run with parameters for an intermediate mode of virus production (Table 1). Simulations were run stochastically until 1000 infectious viruses were generated (t

_{1000}) and then deterministically for 1 (panel A), 2 (panel B), or 3 (panel C) additional days. Given the growth rate of the virus population, r

_{o}, (Table 1) the virus population will expand approximately 4, 18, or 76 fold in 1, 2, or 3 days, respectively.

**Figure 7.**The probability of infection of monkeys with increasing SIV dose is not described well by the simple, single-hit model. We fit four different models to the experimental data of Liu et al. [16] using maximum likelihood. The different models are a single hit model (Equation (13)), the power law model (Equation (14)), the competition model (Equation (15)), and the gamma model (Equation (17)). The points represent the data with 95% confidence intervals calculated as Jeffreys intervals for binomial distributions [61]. Predictions of the models are given by lines. Estimated parameters of the models are: single-hit (λ = 5.2 × 10

^{−}

^{8}), powerlaw (λ = 6.7 × 10

^{−}

^{3}and n = 0.32), competition (λ = 0.08, α = 5.7 × 10

^{−}

^{7}), and gamma (λ = 7.3 × 10

^{−}

^{7}, σ = 1.0 × 10

^{−}

^{6}). All models with the exception of the single-hit model describe the data with good quality as judged by the χ

^{2}test. The powerlaw and gamma models describe the data significantly better than the single-hit model based on likelihood ratio test (p < 0.01). The powerlaw, competition, and gamma models describe the data with similar quality as judged by AIC [62].

## 3. Discussion

_{inf}, with the burst-like mode of production resulting in a lower p

_{inf}. The latter result arises because under burst-like virus production, a cell may die before starting virus production. This aspect was different in previous models [22,23], where the death of the infected cell in the bursting model always led to virus release. In the special case when there is no death in the eclipse phase (i.e., δ

_{IE}= 0), there appears to be a stronger dependence of the time to virus detection on the mode of virus production, with the bursting mode resulting in a longer time to detection; however, in that case, the mode of virus production then had no influence on the probability of infection (see Equation (12) and Figure S5 in Supplement). Our results also demonstrate that the type of initiating infectious agent (virus particles or infected cells) and the initial load of the agent has a major influence on the probability of infection. In particular, in our simulations, starting with 10 infected cells nearly always resulted in an established infection (Figure S7 and [23]).

_{det}as the virus density reached in our simulations with the lowest initial viral dose by t

_{det}= 8.5 days since infection as was observed in experimental data ([16], Figure 6). The actual value of V

_{det}was dependent on the model parameters, and in particular, V

_{det}was directly proportional to the value of the viral burst size N (Figure S2 in Supplement). If N = 5 × 10

^{4}[57], V

_{det}≈ 5 × 10

^{7}. It is interesting to compare this empirical estimate to a value calculated using basic anatomical properties of monkeys. Current conventional sequencing methods allow detection of a SIV/HIV infection when there is about 200 copies of viral RNA per mL of plasma, which implies V = 100 viral particles per mL of plasma (since each virion contains 2 RNA molecules). Assuming that the same virus concentration occurs in total body fluids and that during the first 10 days of infection there is very little cell-associated virus, the total amount of virus is V w where w is the extracellular fluid volume (EFV). We could not locate a good estimate for w in rhesus monkeys, but observed that monkeys weigh about 5–7 kg or 10 times less than humans. In humans, an estimate for EFV is 13.5 L [73], so we estimate w = 1.4 L for the monkey. Thus, the total number of viral particles in the whole body at the limit of detection in the blood is about V

_{T}= 100 viruses/mL × 1.4 × 10

^{3}mL = 1.4 × 10

^{5}viruses. This is lower than our theoretically calculated value (V

_{det }≈ 5 × 10

^{7}viruses) suggesting that the actual burst size N ≈ 350 during early virus dynamics is smaller than was suggested by the in vivo study [57], or that only a small fraction of viruses is infectious (350/(5 × 10

^{4}) = 0.7%).

^{6}) initial viral doses (Figure S11 in Supplement).

_{I}and death rate δ

_{I }. These dynamics can be then modeled as a simple linear random birth-death process [79,80]. Our analysis demonstrates that the extended standard mathematical model is too simple to accurately explain the change in the time to virus detection with increasing viral doses (Figure 6). It is possible that the discrepancy between the model and the data is due to relatively poor sampling of the viral load during early infection or because the initial number of viruses, starting the infection, is not proportional to the initial viral dose. Alternatively, this discrepancy could be due to some important biological components that are missing from the model. In particular, the model assumes unlimited supply of target cells for virus replication, while several studies have highlighted the spatial heterogeneity of uninfected and infected cells, and the loss of many CD4 T cells during early infection [5,81,82,83,84]. Furthermore, it has been suggested that systemic virus infection occurs not from the gut tissues but from lymph nodes [85], and viral spread from the gut to lymph nodes was not included in our model. Finally, we assumed that parameters for virus dynamics are identical between different animals, but the observed variability in the time to virus detection even at high initial viral doses could be a function of differences in susceptibility between individual animals. Indeed, a model in which the probability of virus extinction is varied between animals allowed for a good description of the data on the probability of established infection with increasing viral doses (Figure 7). Alternatively, there may be competition between viruses replicating at different sites of the gut. Such competition may impact the change in the probability of infection with increasing viral doses (Figure 7) and could potentially influence the change in the time to virus detection with increasing viral doses (Figure 6). Future studies should investigate these alternative hypotheses.

_{log10r}= 0.07). The analysis shows that variability in the virus replication rates translates into variability in the time to virus detection with larger variability in times to detection at smaller initial doses (Figure S12 in Supplement). However, this experimentally observed variability in the virus replication rates was still unable to accurately predict a relatively constant time to virus detection at high initial viral doses. We expect that combining stochastic simulations with variability in model parameters will lead to even greater variance in the time to virus detection at low initial viral doses which is inconsistent with experimental data of Liu et al. [16].

## 4. Materials and Methods

#### 4.1. Implementing Stochastic Simulations

`GillespieSSA`package in R (

`www.r-project.org`). In the Gillespie algorithm, the system of differential equations is organized into a state-change matrix that describes each of the possible occurrences in the system and the rate at which they occur: a virus can infect a target cell, thereby becoming an infected cell in the eclipse phase, such a cell can become a productively infected cell, a productively infected cell can release viruses, or any of the species can die. For every time step, two random numbers are generated and used to determine (1) which event will occur next and (2) the amount of time that will pass until it occurs.

_{det}= 100 or 1000 infectious viruses because at this value the dynamics of the populations in the model became relatively deterministic. At V ∼ 10

^{3}, the dynamics of the populations in our model are nearly deterministic since the coefficient of variation, for example, in virus number is only $1/\sqrt{{10}^{3}}=0.032$ or 3%, which is fairly small. Dynamics of cell populations at later times were simulated deterministically using the ode routine in library

`deSolve`in

`R`.

#### 4.2. Statistical Tests

## 5. Conclusions

## Supplementary Files

Supplementary File 1## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Noecker, C.; Schaefer, K.; Zaccheo, K.; Yang, Y.; Day, J.; Ganusov, V.V. Simple Mathematical Models Do Not Accurately Predict Early SIV Dynamics. *Viruses* **2015**, *7*, 1189-1217.
https://doi.org/10.3390/v7031189

**AMA Style**

Noecker C, Schaefer K, Zaccheo K, Yang Y, Day J, Ganusov VV. Simple Mathematical Models Do Not Accurately Predict Early SIV Dynamics. *Viruses*. 2015; 7(3):1189-1217.
https://doi.org/10.3390/v7031189

**Chicago/Turabian Style**

Noecker, Cecilia, Krista Schaefer, Kelly Zaccheo, Yiding Yang, Judy Day, and Vitaly V. Ganusov. 2015. "Simple Mathematical Models Do Not Accurately Predict Early SIV Dynamics" *Viruses* 7, no. 3: 1189-1217.
https://doi.org/10.3390/v7031189