# Identifying the Conditions Under Which Antibodies Protect Against Infection by Equine Infectious Anemia Virus

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}, which will indicate the threshold below which the infection cannot sustain itself. The effect of mutation on the antibody vaccine is also unknown. Furthermore, since the parameter values are likely to vary within a range, it would be useful to determine the effect of variation of the parameters on R

_{0}.

_{k}, the system is continuous, behaving as a system of ODEs. At the impulse points, there is an instantaneous change in state in some or all of the variables. This instantaneous change can occur when certain spatial, temporal or spatio-temporal conditions are met [21,22,23,24]. This is related to the use of pulse vaccinations [25], seasonal skipping in recurrent epidemics [26], antiretroviral drug treatment [27] and birth pulses in animals [28]. The current study aims to understand the role that neutralizing antibody vaccines can play in the control of lentivirus infection. Conditions are determined under which wild-type infection is eradicated with the antibody vaccine. The conditions that permit viral escape by the mutant strains are also delineated. Unknown infection parameters are determined, including viral growth rates, the carrying capacity and the rate at which antibody neutralizes virus. The effect of varying the effectiveness of antibody infusion and neutralization is explored, as is the role of stochasticity in R

_{0}. Finally, conditions are derived whereby the presence of a highly mutable, but low-replicating, second mutant may in fact result in the persistence of the first mutant, to the exclusion of other strains. This work contributes to the understanding of virus control and potentially provides insights into the development of vaccines that stimulate the immune system to control infection.

## 2. Methods

_{W}, p

_{M}

_{1}and p

_{M}

_{2}, respectively. The effect of vaccinating at times t

_{k}(k = −1, 7, 14) is to increase the antibody level by a fixed amount, A

^{i}.

_{W}, p

_{M}

_{1}and p

_{M}

_{2}, respectively, and that the wild-type mutates to each mutant strain at rates ϵ

_{1}and ϵ

_{2}, respectively. We also assume that the wild-type has higher replication and is more susceptible to the antibody response than Mutant 1; similarly, we assume that Mutant 1 has higher replication and is more susceptible to the antibody response than Mutant 2. However, we do not necessarily assume that mutation of the first mutant is higher than that of the second mutant.

_{W }V

_{W}A − p

_{M}

_{1}V

_{M}

_{1}A − p

_{M}

_{2}V

_{M}

_{2}A t ≠ t

_{k}

^{i}t = t

_{k}

_{W}+ V

_{M}

_{1}+ V

_{M}

_{2}. We assume the following: p

_{W}> p

_{M}

_{1}> p

_{M}

_{2}, r

_{W}> r

_{M}

_{1}> r

_{M}

_{2}.

^{i}on Day −1, with exponential decay at rate q. The value of q is calculated from the half-life of horse IgG [32]. The half-life of virus due to antibody neutralization, t

_{1}

_{/}

_{2}, was estimated using the half-life of SIV in animals that were CD8-depleted [33,34,35]. The viral growth rate was calculated by fitting data from EIAV-infected SCID horses to the model with antibody neutralization set to zero with the viral clearance rate subtracted. The carrying capacity was also fit to the same data and then adjusted to account for growth and clearance effects. Note that each strain has a different carrying capacity. See Appendix A for the details of the calculated parameters.

Parameter | Definition | Value | Range | Units | Reference |
---|---|---|---|---|---|

r_{W} | Virus growth rate for wild-type in the absence of antibodies | 23.60 | 0–46 | day
^{−}^{1} | Calculated (Appendix A) |

r_{M}_{1} | Virus growth rate for first mutant | 23.23 | 0–46 | day
^{−}^{1} | Calculated (Appendix A) |

r_{M}_{2} | Virus growth rate for second mutant | 23.09 | 0–46 | day
^{−}^{1} | Assumed |

K | Virus carrying capacity | 1.14 × 10^{8} | 7.47 × 10^{8}–3.82 × 108 | virus ml
^{−}^{1} | Calculated (Appendix A) |

p_{W} | Wild-type virus neutralization by antibody | 1.462 × 10^{−}^{2} × m | (1.21 × 10^{−}^{2}–2.67 × 10^{−}^{2}) × m | ml mg
^{−}^{1} day^{−}^{1} | Calculated (Appendix A) |

p_{M}_{1} | Mutant 1 virus neutralization by antibody | (varied) | — | ml mg
^{−}^{1} day^{−}^{1} | Assumed |

p_{M}_{2} | Mutant 2 virus neutralization by antibody | (varied) | — | ml mg
^{−}^{1} day^{−}^{1} | Assumed |

q | Antibody decay rate | 0.0315 | 0.0277–0.0365 | day
^{−}^{1} | [ 32] |

ϵ_{1} | Mutation rate from wild-type to first mutant (per base per cycle) | 2.7 × 10^{−}^{5} | 1 × 10^{−}^{5}–3.4 × 10^{−}^{5} | day
^{−}^{1} | [ 36] |

ϵ_{2} | Mutation rate from wild-type to second mutant (per base per cycle) | 2.7 × 10^{−}^{6} | 2 × 10^{−}^{6}–2.7 × 10^{−}^{2} | day
^{−}^{1} | Assumed |

A^{i} | Amount of antibody infusion | 38.4 × m | (25.6–51.2) × m | mg ml
^{−}^{1} | [ 10 , 11 ] |

A_{0} | Antibody on Day 0 | 37.2 | 24.9–49.4 | mg ml
^{−}^{1} | Calculated (Appendix A) |

V_{W}(0) | Number of wild-type viral particles that initiated infection | 224 | 175–350 | virus ml^{−1} | [ 10,37] |

V_{M}_{1}(0) | Number of Mutant 1 viral particles that initiated infection | 9 | — | virus ml^{−1} | [ 10] |

V_{M}_{2}(0) | Number of Mutant 2 viral particles that initiated infection | 1 | — | virus ml^{−1} | Assumed |

t_{1}_{/}_{2} | Half-life of virus due to antibody neutralization | 1.3 | 0.7–1.8 | day | [ 33–35] |

d_{W }, d_{M}_{1}, d_{M}_{2} | Viral clearance rate | 23 | 9.1–36 | day^{−1} | [ 38] |

m | Antibody magnification factor | {1, 10, 50} | — | — | — |

^{i}. We assume the infused antibodies act systemically. The value of V

_{W}(0) was calculated from horse plasma volume and injection inoculum (10

^{6}TCID

_{50}[10]) using a conversion factor between TCID

_{50}and plaque-forming units (PFU) of 0.7 [37]. The range of V

_{W}(0) assumed a horse plasma volume at time of injection of 2–4 L, which is based on a horse weight of 40–80 kg [10]. The value of V

_{M}

_{1}(0) was calculated given that 1 out of 25 single amplicons sequenced from the inoculum showed the first mutant sequence [10]. A second mutant sequence was not identified experimentally; here, one particle per ml was assumed to exist initially.

## 3. Results

#### 3.1. Theoretical Results

_{0}is a composite, consisting of five threshold values (R

_{1}, R

_{2}, R

_{E}

_{1}, R

_{E}

_{2}, R

_{E}

_{3}) that are derived from bifurcation properties of the existence (or otherwise) of endemic equilibria.

_{Mj}> d

_{Mj}for j = 1, 2, so that the line R

_{2}= 1 is an upper bound. Since the growth rate of Mutant 1 is assumed to exceed that of Mutant 2, it follows that the Mutant 2 equilibrium is always unstable (see Condition (1) in Appendix B). All parameters other than r

_{M2}and ϵ

_{2}are set to their sample values in Table 1, with m = 1. For ${\u03f5}_{2}<\frac{{d}_{M1}{r}_{W}}{{r}_{M1}}-{\u03f5}_{1}-{d}_{W}\approx {10}^{-0.5}$, R

_{E}

_{1}> 1 and R

_{E}

_{3}> 1, so all three strains coexist. As ϵ

_{2}increases, the Mutant 1 equilibrium becomes stable, so Mutant 1 persists. See Appendix B.1 for details.

**Figure 1.**Viral persistence landscape diagram as Mutant 2 varies, showing that an increase in the mutation rate of Mutant 2 can stabilize the Mutant 1 equilibrium. The bistability region is included for completeness, but corresponds to unreasonably high mutation rates. Note the log scale on the axes.

_{2}increases past 0.6 (corresponding to R

_{E}

_{3}= 1), we move into a region of bistability: R

_{E}

_{3}< 1 and R

_{E}

_{1}< 1, so both the disease-free and Mutant 1 equilibria are stable. Bistability means that two equilibria are stable, so the ultimate result depends on the choice of initial conditions. Solutions that start near the disease-free equilibrium will approach it, while solutions that start near the Mutant 1 equilibrium will approach this equilibrium. However, this case is only included for completeness, since we expect that the mutation rate will not be this high in reality. Note also that the curve R

_{E}

_{2}= 1 plays no role in the bifurcation, since Condition (1) holds.

#### 3.2. Numerical Simulations

^{i}(and A

_{0}), and the neutralization ability, p

_{j}(j = W,M1, M2), by a magnification factor, m, where m = 1, 10, 50. (Note that all three virus neutralization rates were multiplied by m, regardless of their relative effect.) The value m = 10 means antibodies are ten times greater when infused and are 10 times more effective at neutralizing the virus. The magnification factor thus accounts for theoretical improvements on the vaccine.

Relative effectiveness | Antibody magnification factor m | Figure | |||
---|---|---|---|---|---|

p_{M}_{1} | p_{M}_{2} | 1 | 10 | 50 | |

p_{W} | p_{W} | Wild-type dominates (coexistence) | Eradication (wild-type last) | Eradication (exponentially fast) | 3 |

0.1p_{W} | 0.01p_{W} | Wild-type dominates (coexistence) | Mutant 2 escape (others eradicated) | Eradication (Mutant 2 last) | 4 |

0.01p_{W} | 0.01p_{W} | Wild-type dominates (coexistence) | Mutant 1 escape (Wild-type eradicated) | Mutant 1 escape or eradication | 5 |

**Figure 2.**The antibody count for the case when both mutants have 100-fold resistance and m = 10. Antibody boosts occur on Day 7 and Day 14. This figure looks similar for other values of m.

_{M}

_{1}= p

_{M}

_{2}= p

_{W}); see Figure 3. When antibody magnification is m = 1, all three strains coexist, but the wild-type dominates (note the log scale on the axes). Both 10-fold and 50-fold antibody magnifications eventually control all three strains of the virus. The sharp drop-off after seven days in the case of 10-fold antibody magnification corresponds to the first antibody boost on Day 7, which accelerates the eradication process. Eradication occurs exponentially quickly in the case of 50-fold magnification.

**Figure 3.**The long-term outcome for virus strains using the sample values in Table 1 for the case of equal virus neutralization rates (p

_{M}

_{1}= p

_{M}

_{2}= p

_{W}) as the antibody magnification factor m varies. (

**A**) m = 1; (

**B**) m = 10; (

**C**) m = 50.

_{M}

_{1}= 0.1p

_{W}) and Mutant 2 had 100-fold resistance (i.e., p

_{M}

_{2}= 0.01p

_{W}); see Figure 4. When antibody magnification is m = 1, all three strains coexist, but the wild-type dominates. Tenfold antibody magnification controls the wild-type and Mutant 1, but allows Mutant 2 to escape; note the accelerated decreases at Days 7 and 14 as antibodies are infused. However, 50-fold antibody magnification eventually controls all three strains of the virus; note that Mutant 2 is eradicated before the antibodies have decayed to zero.

**Figure 4.**The long-term outcome using the sample values in Table 1 for the case when Mutant 1 has 10-fold resistance and Mutant 2 has 100-fold resistance (p

_{M}

_{1}= 0.1p

_{W}, p

_{M}

_{2}= 0.01p

_{W}) as the antibody magnification factor m varies. (

**A**) m = 1; (

**B**) m = 10; (

**C**) m = 50.

_{M}

_{1}= p

_{M}

_{2}= 0.01p

_{W}); see Figure 5. When antibody magnification is m = 1, all three strains coexist, but the wild-type dominates. Tenfold antibody magnification controls the wild-type and reduces Mutant 2, but allows Mutant 1 to escape (reversing the outcome from the previous case); note that there are no antibodies after about 40 days, so Mutant 2 will eventually be out-competed by Mutant 1. Fifty-fold antibody magnification still allows Mutant 1 to escape, but controls Mutant 2; in this case, Mutant 1 is reduced, but not eradicated, when the antibodies decay to zero, allowing it to bounce back (Figure 5C). This is a different outcome from the two previous cases.

_{M}

_{1}(0) = V

_{M}

_{2}(0) = 0. In this case, Mutant 1 still emerges (due to the mutation rate, ϵ

_{1}), but quickly decays below the eradication threshold. We examined the issue of no initial mutants for all other cases, and the results were qualitatively unchanged in all figures (results not shown), except for Figure 5C. This suggests that initial fluctuations in Mutant 1 may affect the outcome when the antibody magnification rate is sufficiently high.

**Figure 5.**The long-term outcome using the sample values in Table 1 for the case when both mutants have 100-fold resistance (p

_{M}

_{1}= 0.01p

_{W}, p

_{M}

_{2}= 0.01p

_{W}) as the antibody magnification factor m varies. (

**A**) m = 1; (

**B**) m = 10; (

**C**) m = 50; (

**D**) the same as (C), except with no initial mutants.

_{2}) was significantly higher than the mutation rate of Mutant 1 (ϵ

_{1}) for the case when Mutant 1 had 10-fold resistance and Mutant 2 had 100-fold resistance. Figure 6 is the analogue of Figure 3B, Figure 4B and Figure 5B (i.e., when m = 10), but with a high mutation rate of Mutant 2. Figure 6A shows that Mutant 2 can escape if its mutation rate is sufficiently high, which is not surprising. Figure 6C is qualitatively unchanged from Figure 5B.

**Figure 6.**The case when the mutation rate of Mutant 2 is high (ϵ

_{2}= 2.7 × 10

^{1}). Here, Mutant 1 has 10-fold resistance to the antibodies, Mutant 2 has 100-fold resistance to the antibodies and m = 10, recreating the conditions of Figure 3B, Figure 4B and Figure 5B (i.e., the cases of 10-fold magnification), except for the high mutation rate of Mutant 2. (

**A**) Unlike Figure 3B, Mutant 2 escapes; (

**B**) unlike Figure 4B, Mutant 1 persists; conversely, Mutant 2 is eradicated, despite its extremely high mutation rate; (

**C**) the persistence of Mutant 1, similar to Figure 5B.

#### 3.2.1. Sensitivity Analysis

_{0}determines whether the virus will persist or be eradicated, we are interested in the ability of variations in parameter values to affect R

_{0}. To examine this sensitivity of R

_{0}to variations in parameters, we used Latin hypercube sampling and partial rank correlation coefficients (PRCCs) with 1000 Monte Carlo simulations per run. Latin hypercube sampling is a statistical sampling method that allows for an efficient analysis of parameter variations across simultaneous uncertainty ranges in each parameter by using repeated Monte Carlo simulations [41]. PRCCs illustrate the degree of the effect that each parameter has on the outcome. Parameters with positive PRCCs will increase R

_{0}when they are increased, whereas parameters with negative PRCCs will decrease R

_{0}when they are increased. However, the magnitude of the PRCC is critical, since it indicates the strength of the effect the parameter has, regardless of sign. These methods have been used in studies for the spread of viral infection in order to elucidate trends about parameter dependence [42,43,44].

**Figure 7.**Tornado plot showing partial rank correlation coefficients (PRCCs) of R

_{0}to its dependent parameters. The parameter with the largest impact on R

_{0}is the viral growth rate, r

_{W}.

**Figure 8.**Landscape plots of R

_{0}vs. r

_{W}, d

_{W}, ϵ

_{1}and ϵ

_{2}using Monte Carlo simulations. If the viral growth rate can be reduced below 10, then eradication is likely, regardless of variations in the other parameters.

_{0}, which is the difference between eradication and persistence. The Monte Carlo simulations are illustrated in Figure 8. Each dot is one of 1000 simulations. The mutation rates have very little effect on R

_{0}when they are varied, but the virus growth and clearance rates have a noticeable trend.

_{0}values for r

_{W}peak around r

_{W}= 20 and are thus higher than if the growth rate was maximized (e.g., r

_{W}= 45). This is because the viral growth and clearance rates are linked (see Appendix A), so if r

_{W}reaches its extreme, then the viral clearance rate has to be adjusted accordingly. The result is a narrowing of the range of R

_{0}values.

_{0}as all parameters are varied is illustrated in Figure 9. This illustrates the median, interquartile range and extreme values of R

_{0}across all parameter ranges. The median value is almost exactly one.

**Figure 9.**Box plot of distribution of R

_{0}values as all parameters are varied throughout their ranges in Table 1 with m = 1. The median value of R

_{0}is almost exactly one, suggesting that small variations in parameters can result in either viral persistence or eradication.

## 4. Discussion

_{2}is low enough and the Mutant 2 viral growth rate is in a specified range (Figure 1). See Table 2 for a summary.

_{0}; and we demonstrated the stability of the steady states, giving the regions of long-term viral persistence, for ranges of values of mutation rate ϵ

_{2}and the growth rate of Mutant 2. Uncertainty and sensitivity analyses showed that the model parameter with the greatest effect on the outcome was the viral growth rate, r

_{W}(Figure 7 and Figure 8), which can lower R

_{0}below one and eradicate the virus if it is sufficiently low.

## 5. Conclusions

^{i}and A

_{0}), resulting in eradication of wild-type and mutant strains, is important for developing an effective vaccine.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

## A. Calculating Parameters

#### A.1. Calculating p

_{0}e

^{−}

^{q}

^{t}

_{1}

_{/}

_{2}, the time it takes for virus to be reduced to 50%. Thus:

_{0}and t

_{1}

_{/}

_{2}.

#### A.2. Calculating A_{0}

_{0}was determined from the initial condition A

_{−}

_{1}by calculating the exponential decay after a single day had elapsed. Thus:

_{0}= A

_{−}

_{1}e

^{−}

^{q}

#### A.3. Calculating r_{W} and K

_{W}. The virus growth rate for Mutant 1, r

_{M}

_{1}, was determined equivalently by fitting data from infected EIAV-specific antibody-infused SCID horses, A2239 and A2240 [10], and subtracting the clearance rate.

_{W}, we allowed the difference (r

_{W}− d

_{W}) to range from −9.1 to 9.1 and then recovered r from the viral clearance rate in each Monte Carlo simulation. This is because the viral growth and clearance rates are linked, and the viral clearance rate ranges from 9.1 to 36 [38]. It follows that the practical outcome of this is that r ranges from zero to 46, but in such a way that r

_{W}> d

_{W}.

## B. The Non-Impulsive System

^{i}= 0.

#### B.1. Equilibria

_{W}V

_{W}+ p

_{M}

_{1}V

_{M}

_{1}+ p

_{M}

_{2}V

_{M}

_{2}= −q < 0. It follows that any equilibrium has A = 0.

_{W}= 0, then either V

_{M1}= 0 or ${r}_{M1}(1-\frac{V}{K})={d}_{M1}$. In the former case, either V

_{M}

_{2}= 0 or:

_{M}

_{2}= 0

_{M}

_{1}= 0.

_{W}≠ 0, then:

_{W}satisfies:

_{W}is as above. These are, respectively, the disease-free equilibrium, the Mutant 1 equilibrium, the Mutant 2 equilibrium and the coexistence equilibrium.

#### B.2. Jacobian

_{1}|J

_{2}], where:

_{E}

_{3}> 1, or R

_{1}> 1 or R

_{2}> 1. Note that, for our parameter choices, r

_{j}> d

_{j}(j = W, M1, M2), so R

_{1}> 1 and R

_{2}> 1. It follows that the disease-free equilibrium is always unstable.

_{M}

_{1}, 0, 0), we have:

_{E}

_{1}> 1 or if:

_{M}

_{1}r

_{M}

_{2}− d

_{M}

_{2}r

_{M}

_{1}> 0

_{M}

_{1}= d

_{M}

_{2}and r

_{M}

_{1}> r

_{M}

_{2}).

_{M}

_{2}, 0), we have:

_{E}

_{2}> 1 or if Condition (1) fails. As noted above, Condition (1) usually fails in practice, suggesting that this equilibrium is usually unstable.

#### B.3. Calculating R_{0}

_{1}, R

_{2}, R

_{E}

_{1}, R

_{E}

_{2}, R

_{E}

_{3}} > 1

_{E}

_{3}> R

_{1}> R

_{2}and R

_{E}

_{3}> R

_{E}

_{2}> R

_{E}

_{1}. Thus, the condition for the disease to persist is:

_{0}= R

_{E}

_{3}> 1

## C. Stabilizing the Mutant 1 Equilibrium

_{W}= d

_{W}+ γ, where γ ≈ 0.5. Then:

_{E}

_{3}is likely to be greater than one in all realistic situations.

_{E}

_{1}< 1.

_{M}

_{1}= r

_{W}− 0.01. Then we can write:

_{M}

_{1}extremely close to r

_{W}. Larger variations will make this worse. If the value is negative, then this can never equal one.)

_{M1}> r

_{M2}, but ${\u03f5}_{2}$ is very large indeed). That is, the wild-type would mutate easily to a mutant that did not replicate very well.

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**MDPI and ACS Style**

Schwartz, E.J.; Smith?, R.J.
Identifying the Conditions Under Which Antibodies Protect Against Infection by Equine Infectious Anemia Virus. *Vaccines* **2014**, *2*, 397-421.
https://doi.org/10.3390/vaccines2020397

**AMA Style**

Schwartz EJ, Smith? RJ.
Identifying the Conditions Under Which Antibodies Protect Against Infection by Equine Infectious Anemia Virus. *Vaccines*. 2014; 2(2):397-421.
https://doi.org/10.3390/vaccines2020397

**Chicago/Turabian Style**

Schwartz, Elissa J., and Robert J. Smith?.
2014. "Identifying the Conditions Under Which Antibodies Protect Against Infection by Equine Infectious Anemia Virus" *Vaccines* 2, no. 2: 397-421.
https://doi.org/10.3390/vaccines2020397