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The Impact of Neglecting Vaccine Unwillingness in Epidemiology Models

Glenn Ledder

TL;DR

This study uses a simple generic disease model to address two questions: how much error is introduced in key model outcomes by neglecting vaccine unwillingness, and can the error be reduced by incorporating vaccine unwillingness into the vaccination rate constant rather than the rate diagram.

Abstract

With significant population fractions in many societies who refuse vaccines, it is important to reconsider how vaccination is incorporated into compartmental epidemiology models. It is still most common to apply the vaccination rate to the entire class of susceptibles, rather than to use the more realistic assumption that the vaccination rate function should depend only on the population of susceptibles who are willing and able to receive a vaccination. This study uses a simple generic disease model to address two questions: (1) How much error is introduced in key model outcomes by neglecting vaccine unwillingness?, and (2) Can the error be reduced by incorporating vaccine unwillingness into the vaccination rate constant rather than the rate diagram? The answers depend greatly on the time scale of interest. For the endemic time scale, where longterm behavior is studied with equilibrium point analysis, the error in neglecting unwillingess is large and cannot be improved upon by decreasing the vaccination rate constant. For the epidemic time scale, where the first big epidemic wave is studied with numerical simulations, the error can still be significant, particularly for diseases that are relatively less infectious and vaccination programs that are relatively slow.

The Impact of Neglecting Vaccine Unwillingness in Epidemiology Models

TL;DR

This study uses a simple generic disease model to address two questions: how much error is introduced in key model outcomes by neglecting vaccine unwillingness, and can the error be reduced by incorporating vaccine unwillingness into the vaccination rate constant rather than the rate diagram.

Abstract

With significant population fractions in many societies who refuse vaccines, it is important to reconsider how vaccination is incorporated into compartmental epidemiology models. It is still most common to apply the vaccination rate to the entire class of susceptibles, rather than to use the more realistic assumption that the vaccination rate function should depend only on the population of susceptibles who are willing and able to receive a vaccination. This study uses a simple generic disease model to address two questions: (1) How much error is introduced in key model outcomes by neglecting vaccine unwillingness?, and (2) Can the error be reduced by incorporating vaccine unwillingness into the vaccination rate constant rather than the rate diagram? The answers depend greatly on the time scale of interest. For the endemic time scale, where longterm behavior is studied with equilibrium point analysis, the error in neglecting unwillingess is large and cannot be improved upon by decreasing the vaccination rate constant. For the epidemic time scale, where the first big epidemic wave is studied with numerical simulations, the error can still be significant, particularly for diseases that are relatively less infectious and vaccination programs that are relatively slow.
Paper Structure (12 sections, 2 theorems, 52 equations, 5 figures)

This paper contains 12 sections, 2 theorems, 52 equations, 5 figures.

Table of Contents

  1. Introduction
  2. An SIR Model with Vaccination
  3. Scaling on an Epidemic Time Scale
  4. Scaling on an Endemic Time Scale
  5. Parameter Estimates
  6. Endemic Analysis
  7. The Disease-Free Equilibrium (DFE) and Basic Reproduction Number
  8. The Endemic Disease Equilibrium (EDE)
  9. Results
  10. The Endemic Scenario
  11. The Epidemic Scenario
  12. Discussion

Key Result

Theorem 1

Let $J$ be the Jacobian matrix for a four-component dynamical system. Let $c_1=-\operatorname{tr}(J)$, $c_2$ be the sum of all $2 \times 2$ subdeterminants of $J$, $c_3$ be the negative of the sum of all $3 \times 3$ subdeterminants of $J$, $c_4=\det{J}$, and combine these quantities as

Figures (5)

  • Figure 1: An SVIR model with vaccination applied only to a willing subclass of susceptibles.
  • Figure 2: Left: Reduction factor from ${\cal{R}}_0$ to ${\cal{R}}_v$; Right: Regions in disease-vaccine space where the DFE is stable (below and left of the curves)
  • Figure 3: The rate of new infections at equilibrium
  • Figure 4: Population fractions that escape infection in the epidemic scenario, with $\sigma=0$ (left) and $\sigma=0.7$ (right). The nominal vaccination rate constants are $\theta=0.1$ (blue) and $\theta=0.5$ (orange). Solid curves have $W=0.7$, and dash-dot curves have $W=1$. Dashed curves are for the model that assumes full willingness but deprecates the vaccination rate constant by a factor of $W=0.7$.
  • Figure 5: Total population infectiousness $\widehat{I}$ (red) and vaccinated fraction $V$ (blue), with solid curves for the model \ref{['epidemic']} and dash-dot curves for the simplified model that accounts for unwillingness by decreasing the vaccination rate constant to $W \theta$ rather than modifying the standard SVIR compartment model; ${\cal{R}}_0=3$ (left panels), ${\cal{R}}_0=6$ (right panels), $\theta=0.1$ (top panels), $\theta=0.5$ (bottom panels); all panels have $\sigma=0.5$ and $W=0.7$.

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Theorem 2