Lotka-Euler Equation with Truncated Gaussian

• Lotka-Euler theory is defined as the relationship between epidemic growth and generation time distributions, using a truncated Gaussian to account for negative intervals.
• The method employs a closed-form Laplace transform to derive correction factors, resulting in reproduction numbers that are 20–30% higher than unconstrained estimates.
• This approach is critical for accurate epidemiological modeling and effective assessment of interventions in diseases with presymptomatic transmission.

The Lotka–Euler equation provides a fundamental relationship between the exponential growth or decay rate of an epidemic and the statistical properties of the time intervals between primary and secondary infections, known as generation times. When the underlying distribution of these intervals, or the observable serial interval, can take negative values—as occurs with presymptomatic transmission in diseases such as COVID-19—a Gaussian model with explicit lower truncation is required. The application of the Lotka–Euler equation to truncated Gaussians, rather than unconstrained Gaussians, yields systematically higher estimates for reproduction numbers ($R_0$, $R_t$), directly impacting epidemiological modeling and intervention assessment.

1. Truncated Gaussian Distribution of Generation and Serial Intervals

In renewal-equation epidemiological models, the probability density function (PDF) of the generation time or serial interval is often modeled as a Gaussian distribution parameterized by mean $\mu$ and standard deviation $\sigma$. However, negative intervals—possible due to presymptomatic transmission—necessitate a lower cut-off ($\tau_0$), commonly determined by the minimum possible incubation period. The truncated Gaussian is given by:

$$w(\tau) = \begin{cases} \displaystyle \frac{1}{\sigma\sqrt{2\pi}Z}\exp\left(-\frac{(\tau-\mu)^2}{2\sigma^2}\right), & \tau \geq \tau_0 \ 0, & \tau < \tau_0 \end{cases}$$

where $Z = 1 - \Phi(\alpha)$ is the normalization constant for a lower truncation at $\tau_0$, $\alpha = (\tau_0 - \mu)/\sigma$, and $\Phi(x)$ is the standard normal cumulative distribution function.

Explicit modeling of such truncation ensures that intervals cannot be physically shorter than $\tau_0$, reflecting biological constraints such as the incubation period.

2. Lotka–Euler Self-Consistency Equation with Truncation

During periods of exponential growth ($C(t) \propto e^{rt}$) or decay, the Lotka–Euler equation links the observed rate $r$ to the reproduction number $R$ by integrating over possible intervals:

$$1 = R \int_{\tau_0}^{\infty} w(\tau) e^{-r\tau} d\tau$$

Solving for $R$ yields

$$R = \left[\int_{\tau_0}^{\infty} w(\tau) e^{-r\tau} d\tau\right]^{-1}$$

This integral, with the appropriate limits and normalized $w(\tau)$, forms the foundation for calculating $R_0$ and $R_t$ with truncated Gaussian distributions.

3. Closed-Form Solution and Correction Factor

The integration of the truncated Gaussian can be performed analytically by change of variable, resulting in a Laplace transform: $$L(r) = \int_{\tau_0}^{\infty} w(\tau) e^{-r\tau} d\tau = \frac{e^{-r\mu + \frac{1}{2}\sigma^2 r^2}}{Z} [1 - \Phi(\alpha + \sigma r)]$$ The reproduction number estimate becomes: $$R(r; \mu, \sigma, \tau_0) = \exp(r\mu - \frac{1}{2}\sigma^2 r^2) \frac{1-\Phi(\alpha)}{1-\Phi(\alpha+\sigma r)}$$ or, equivalently, using the complementary error function ($\mathrm{erfc}$):

$$R(r) = \exp(r\mu - \frac{1}{2}\sigma^2 r^2) \frac{\mathrm{erfc}\left(\frac{\alpha}{\sqrt{2}}\right)}{\mathrm{erfc}\left(\frac{\alpha + \sigma r}{\sqrt{2}}\right)}$$

This formulation introduces a systematic, closed-form correction to the classical Lotka–Euler calculation, encapsulated by the factor:

$$\frac{1-\Phi((\tau_0-\mu)/\sigma)}{1-\Phi((\tau_0-\mu)/\sigma+\sigma r)} > 1$$

which multiplies the usual full-Gaussian result, always yielding a higher reproduction number estimate compared to ignoring the truncation.

4. Comparison with Unconstrained Gaussian and Epidemiological Significance

If the lower cut-off $\tau_0$ is allowed to approach $-\infty$, $\alpha \rightarrow -\infty$, $Z \rightarrow 1$, and $1 - \Phi(\alpha + \sigma r) \rightarrow 1$, reducing the truncated formula to the classical unconstrained Gaussian result:

$R_\infty(r) = \exp(r\mu - \frac{1}{2}\sigma^2 r^2)$

Whenever a physical cut-off is present ($\tau_0 > -\infty$), the truncated-Gaussian estimate $R(r)$ always exceeds $R_\infty(r)$. Empirical studies with COVID-19 parameters (e.g., $\mu \approx 4$ days, $\sigma \approx 4$ days, $\tau_0 \approx -5$ days) show that the truncated-Gaussian $R_0$ can be 20–30% higher than the unconstrained estimate. Neglecting the lower cut-off biases $R_0$ and $R_t$ downward, impacting risk assessment and public health decision-making.

5. Implementation Considerations in COVID-19 Modeling

COVID-19 serial intervals can be negative due to presymptomatic transmission, with the smallest observable value constrained by the minimum incubation period ($\tau_0 \approx$ –5 to –6 days). Accurate application of the truncated-Gaussian Lotka–Euler equation requires:

• Specification of the empirical lower limit $\tau_0$ reflecting biological reality.
• Computation of the normalization $Z = 1 - \Phi(\alpha)$.
• Use of the closed-form correction in the reproduction number estimate rather than the unconstrained formula.

Any renewal-equation approach or discrete calculation of $w(\tau)$ must explicitly enforce the lower limit, ensuring consistency between instantaneous and basic reproduction number estimates and between continuous and discretized models.

6. Broader Methodological Implications and Required Care

The presence of pre-symptomatic transmission and negative serial intervals necessitates caution when using serial interval as a proxy for generation time. Both involve potential negative values, but true generation times cannot be negative. Methodological rigor demands that all such models, and by extension their software implementations, impose a lower cut-off when applying Lotka–Euler-based or renewal equation-derived estimators for epidemic growth.

A plausible implication is that models which fail to incorporate the explicit truncation may systematically underestimate critical epidemic control thresholds, affecting optimization of interventions such as vaccination coverage.

7. Summary Table: Key Formulae and Correction Factors

Description	Formula	Notes
Truncated Gaussian $w(\tau)$	$$\frac{1}{\sigma\sqrt{2\pi}Z}\exp\left(-\frac{(\tau-\mu)^2}{2\sigma^2}\right),\ \tau \geq \tau_0$$	$Z = 1-\Phi(\alpha)$
Lotka–Euler with truncation	$$R(r) = \exp(r\mu-\frac{1}{2}\sigma^2 r^2) \cdot \frac{1-\Phi(\alpha)}{1-\Phi(\alpha + \sigma r)}$$	Correction always $>1$
Unconstrained Gaussian	$R_\infty(r) = \exp(r\mu-\frac{1}{2}\sigma^2 r^2)$	Recovery as $\tau_0\to -\infty$

Accurate estimation of reproduction numbers in the presence of negative or truncated serial intervals necessitates use of the truncated-Gaussian Lotka–Euler equation. The correction factor is essential for precise epidemiological inference and for avoiding systematic downward bias in epidemic growth metrics (Marsh, 6 Nov 2025).