# Unexpected Infection Spikes in a Model of Respiratory Syncytial Virus Vaccination

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## Abstract

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## 1. Introduction

## 2. The Nonimpulsive Model

## 3. Analysis

#### Constant Transmission

- ${\nu}_{V}^{*}>\nu $
- $f\left({\nu}_{V}^{*}\right)<0$ and
- ${\nu}_{V}^{*}$ is a local minimum.

## 4. The Impulsive Model

#### 4.1. Impulsive Analysis

#### 4.2. Susceptible Individuals

#### 4.3. Vaccinated Individuals

#### 4.4. Infected Individuals

## 5. Numerical Simulations

#### 5.1. The Nonimpulsive Model

#### 5.2. The Impulsive Model

## 6. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Possible sketch of the form of $f\left({\nu}_{V}\right)$ with a negative minimum between two positive extremes.

**Figure 3.**Results from the nonimpulsive model. (

**A**) There is an outbreak, and the infectious population oscillates, eventually approaching an endemic equilibrium. (

**B**) A small proportion of individuals are (and remain) vaccinated, with a low-level outbreak among vaccinated individuals. Note the log scale in this figure.

**Figure 4.**Without vaccination, the disease infects up to 7% of the population. (

**A**) The total infected population, including vaccinated individuals; (

**B**) The final size in each population.

**Figure 5.**50% coverage with a vaccine that reduced transmissibility by half and waned after two years resulted in a substantial reduction in the disease compared to no vaccination. (

**A**) The total infected population, including vaccinated individuals; (

**B**) The final size in each population.

**Figure 6.**75% coverage with a vaccine that reduced transmissibility by half and waned after two years resulted in theoretical eradication of the disease. (

**A**) The total infected population, including vaccinated individuals; (

**B**) The final size in each population.

**Figure 7.**Population dynamics for 50% vaccination coverage for a vaccine that reduced transmissibility by half and waned after two years. Note the low-level oscillations in both infected classes.

**Figure 8.**Extreme parameters show that perfect vaccination can induce unexpected infection spikes. (

**A**) With no vaccine ($r=0$), the result is that the infection clears and the entire population remains susceptible. (Note that the timescale is given for only 0.5 years to show the decline but was run for 15 years). (

**B**) The final size of each compartment in the case of no vaccine after 15 years. (

**C**) When an imperfect vaccine is given to the entire population ($r=1$), the result is a series of disease spikes in the vaccinated population. Note that the transmission rate is not oscillating in this example. (

**D**) The final size of each compartment in the case of full vaccination after 15 years. Vaccination thus destabilises the DFE.

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**MDPI and ACS Style**

Smith, R.J.; Hogan, A.B.; Mercer, G.N.
Unexpected Infection Spikes in a Model of Respiratory Syncytial Virus Vaccination. *Vaccines* **2017**, *5*, 12.
https://doi.org/10.3390/vaccines5020012

**AMA Style**

Smith RJ, Hogan AB, Mercer GN.
Unexpected Infection Spikes in a Model of Respiratory Syncytial Virus Vaccination. *Vaccines*. 2017; 5(2):12.
https://doi.org/10.3390/vaccines5020012

**Chicago/Turabian Style**

Smith, Robert J., Alexandra B. Hogan, and Geoffry N. Mercer.
2017. "Unexpected Infection Spikes in a Model of Respiratory Syncytial Virus Vaccination" *Vaccines* 5, no. 2: 12.
https://doi.org/10.3390/vaccines5020012