A Statistical Method to Estimate the Population-Level Frequencies of Plasmodium falciparum Haplotypes with Pfhrp2/3 Deletions in the Presence of Mixed-Clone Infections

This paper introduces and validates a novel statistical model using the expectation-maximization algorithm to accurately estimate population-level frequencies of *Plasmodium falciparum* haplotypes with *Pfhrp2/3* deletions, overcoming the limitations of standard molecular assays in detecting these deletions within mixed-clone infections.

The Gist

The Big Problem: The "Invisible" Malaria Bug

Imagine you are a detective trying to catch a criminal (the malaria parasite). You have a special high-tech metal detector (the Rapid Diagnostic Test, or RDT) that beeps when it finds a specific piece of metal (a protein called HRP2) that the criminal always carries.

For years, this detector worked perfectly. But recently, some criminals started wearing invisible cloaks. They deleted the genes that make the metal, so they have no HRP2 protein. When the detector scans them, it stays silent. The detective thinks, "No criminal here!" and walks away. But the criminal is actually right there, hiding in plain sight.

This is the real-world crisis: The malaria parasite Plasmodium falciparum is evolving to delete the very thing our tests look for. If too many of these "invisible" parasites exist, our tests will fail, people won't get treated, and the disease will spread.

The Complication: The "Crowded Room"

Now, imagine the detective doesn't just scan one person; they scan a crowded room where many people are standing close together. In malaria-endemic areas, a single person can be infected by multiple different strains of the parasite at the same time. This is called a "Mixed-Clone Infection."

Here is the tricky part:

• Strain A is wearing the invisible cloak (no HRP2).
• Strain B is wearing the normal metal jacket (has HRP2).

When the detector scans the room, it hears the loud "beep" from Strain B. The detective assumes, "Great, everyone is wearing metal!" and completely misses Strain A hiding in the crowd.

The Problem: Standard lab tests can't tell the difference between "no one has the metal" and "someone has the metal, but it's being drowned out by someone else." This leads to a massive underestimation of how many invisible parasites are actually out there.

The Solution: A New Statistical "Sherlock Holmes"

The authors of this paper built a new mathematical tool (a statistical model) to solve this mystery. Instead of just looking at the metal detector, they decided to look at the shoes everyone is wearing.

1. The Metal (HRP2/3): This is the part that might be missing (deleted).
2. The Shoes (Neutral Markers): These are parts of the parasite's DNA that never get deleted. They are like unique shoe prints or distinct patterns on a sneaker.

The Analogy:
Imagine you walk into a room and see a pile of shoes.

• If you see a pile of red sneakers, you know red sneakers are there.
• If you see a pile of only blue sneakers, you know blue sneakers are there.
• But what if you see a mix of red and blue sneakers, but you know that some people in the room are wearing invisible shoes?

The new method uses the shoe patterns (the neutral markers) to figure out how many different people are in the room and how they are mixed up. By understanding the "shoe mix," the math can work backward to guess: "Even though I see a loud beep from the metal, the shoe patterns tell me there must be a hidden group of invisible-cloak-wearers in this pile."

How the Math Works (The "Guess and Check" Machine)

The authors used a computer algorithm called the Expectation-Maximization (EM) algorithm. Think of this as a very smart, iterative guessing game:

1. The Guess: The computer starts by guessing, "Okay, maybe 10% of the parasites are invisible."
2. The Check: It looks at the real data (the shoe patterns and the metal beeps) and asks, "Does this guess fit the data?"
3. The Correction: If the guess doesn't fit, the computer adjusts the numbers. "Okay, maybe it's 15%."
4. Repeat: It keeps doing this thousands of times, getting closer and closer to the truth, until the numbers stop changing.

This allows them to calculate the true frequency of the invisible parasites, even when they are hiding inside a mixed infection.

What They Found (The Real-World Test)

The team tested this new method on real data from a tribal community in India.

• Old Way: If they just looked at the raw test results, they would have missed a lot of the invisible parasites.
• New Way: Using their new math, they found that the "invisible" parasites were actually much more common than the raw tests suggested.
  • In the hospital group, about 1.4% of infections were fully invisible.
  • In the community group, the number jumped to 5.9%.

This is a huge deal because the World Health Organization (WHO) has a rule: If more than 5% of the parasites are invisible, we must stop using the old metal detectors and switch to a different kind of test.

Why This Matters

This paper gives health officials a magnifying glass for the invisible.

• Without this tool: They might think, "Oh, only 2% are invisible, we are safe," and keep using the failing tests.
• With this tool: They can see, "Wait, it's actually 6%! We are in danger!" and switch to better tests before people start dying.

It turns a blurry, confusing picture into a clear one, ensuring that the right tools are used to fight malaria, even when the enemy tries to hide.

Technical Summary

1. Problem Statement

The World Health Organization (WHO) relies on Rapid Diagnostic Tests (RDTs) targeting the Plasmodium falciparum histidine-rich proteins 2 and 3 (HRP2/3) for malaria diagnosis. However, parasite populations are increasingly evolving deletions in the Pfhrp2 and Pfhrp3 genes, leading to false-negative RDT results.

The core challenge addressed in this paper is the "masking effect" in high-transmission settings:

• Mixed-Clone Infections: In endemic areas, patients often harbor multiple genetically distinct parasite clones simultaneously (Multiplicity of Infection, or MOI).
• Unobservability: If a patient is infected with a mix of wild-type (functional HRP2/3) and deletion-carrying parasites, standard molecular assays detect the functional alleles from the wild-type, rendering the deletion-carrying clones unobservable.
• Current Limitations: Existing statistical methods treat missing allele data as laboratory errors or missing data, rather than a biological indicator of gene deletion. Consequently, current surveillance significantly underestimates the true prevalence of Pfhrp2/3 deletions in polyclonal infections.

2. Methodology

The authors propose a novel statistical framework combining genetic/molecular data from Pfhrp2/3 loci with neutral genetic markers to infer hidden deletion frequencies.

A. Statistical Model

• Genetic Architecture: The model considers $L$ multi-allelic markers. The first $D$ markers correspond to Pfhrp2/3 regions (where allele '1' represents a deletion), and the remaining markers are neutral (e.g., Pfmsp1, Pfmsp2).
• Haplotype Definition: A haplotype $h$ is a vector of alleles across all loci. The model defines "complete" and "partial" deletions based on the combination of alleles at the first $D$ loci.
• Multiplicity of Infection (MOI): The number of independent infection events per host ($m$) is modeled using a Conditional Poisson Distribution with parameter $\lambda$. The mean MOI is denoted as $\psi$.
• Observation Space: Molecular assays yield unphased data (a set of observed alleles at each locus). If a deletion occurs, the assay yields a "null" signal (allele 1 is assumed present in the observation set by convention to account for the possibility of deletion). The model mathematically defines the probability of observing a specific allele set $x$ given the underlying haplotype frequencies and MOI distribution.

B. Estimation Algorithm (EM Algorithm)

Since the likelihood function does not permit a closed-form solution, the authors utilize the Expectation-Maximization (EM) algorithm:

1. E-Step: Computes the conditional expectation of the log-likelihood given the observed data and current parameter estimates. This involves calculating the probability that specific unobserved haplotype configurations generated the observed allele sets, utilizing the inclusion-exclusion principle to handle the complexity of mixed infections.
2. M-Step: Updates the parameters (haplotype frequencies $p$ and MOI parameter $\lambda$) to maximize the expected log-likelihood.
   • Haplotype frequencies are updated using a weighted sum of expected counts.
   • The MOI parameter $\lambda$ is updated via a fixed-point iteration (Newton-Raphson method).
3. Convergence: The algorithm iterates until the change in parameters falls below a threshold, yielding Maximum Likelihood Estimates (MLEs).

C. Theoretical Validation

• Fisher Information: The authors derived the Fisher Information matrix to calculate the Cramér-Rao Lower Bound (CRLB), establishing the theoretical minimum variance for unbiased estimators.
• Prevalence Formulas: The paper derives formulas to convert haplotype frequencies (abundance in the population) into prevalence (probability of a deletion being present in a random infection), accounting for the MOI.

3. Key Contributions

1. Novel Statistical Framework: The first method explicitly designed to estimate Pfhrp2/3 deletion frequencies in the presence of mixed-clone infections, correcting for the "masking effect."
2. Integration of Neutral Markers: The method leverages neutral markers to resolve the ambiguity of mixed infections, allowing the model to distinguish between a true absence of data (deletion) and the presence of multiple strains.
3. Rigorous Theoretical Properties: The study provides a formal derivation of the likelihood function, the EM algorithm, and the asymptotic variance (CRLB), proving the estimator's efficiency.
4. Open-Source Implementation: A stable R implementation is provided, making the method accessible for public health surveillance.

4. Results

A. Simulation Studies

The method was tested via Monte Carlo simulations across various sample sizes ($N=50$ to $300$), MOI levels, and allele frequency distributions.

• Bias: The estimator showed low bias (typically within $\pm 0.005$) for haplotype frequencies. Bias decreased as sample size increased.
• Precision: The standard deviation of estimates decreased with larger sample sizes. The empirical variance closely matched the theoretical Cramér-Rao Lower Bound, indicating the estimator is statistically efficient.
• Robustness: The method performed well even with unbalanced allele frequencies at neutral markers and high transmission intensities (high MOI).

B. Empirical Application (India)

The method was applied to data from two studies in Jagdalpur, India:

1. Hospital Study ($N=247$):
   • Estimated complete deletion frequency: 1.27%.
   • Estimated prevalence of infections with any deletion: 7.82%.
   • Mean MOI ($\psi$): 1.12 (low transmission intensity).
2. Community Study ($N=104$):
   • Estimated complete deletion frequency: 5.33%.
   • Estimated prevalence of infections with any deletion: 30%.
   • Mean MOI ($\psi$): 1.11.

Key Finding: The community study revealed a significantly higher prevalence of deletions (30%) compared to the hospital study, highlighting how standard surveillance might miss high-risk clusters in community settings due to mixed infections. The method successfully identified "unobservable" deletions that would have been missed by standard analysis.

5. Significance and Conclusion

• Public Health Impact: This method addresses a critical gap in malaria surveillance. By accurately estimating deletion frequencies in high-transmission settings, it enables health authorities to make data-driven decisions on whether to switch from HRP2-based RDTs to alternative diagnostics (e.g., pLDH-based) before the 5% WHO threshold is breached.
• Transmission Monitoring: The simultaneous estimation of MOI provides a robust metric for monitoring transmission intensity, independent of healthcare-seeking behavior.
• Scalability: While computationally intensive for highly polymorphic markers, the method is scalable and provides a necessary tool for the "test-and-treat" strategy in the era of evolving parasite resistance.

In summary, this paper presents a mathematically rigorous and practically validated solution to the problem of underestimating Pfhrp2/3 deletions in complex, mixed-infection scenarios, offering a vital tool for global malaria control programs.