Challenges in forming inferences from limited data: a case study of malaria parasite maturation

Abstract

Inferring biological processes from population dynamics is a common challenge in ecology, particularly when faced with incomplete data. This challenge extends to inferring parasite traits from within-host infection dynamics. We focus on rodent malaria infections (Plasmodium berghei), a system for which previous work inferred an immune-mediated extension in the length of the parasite development cycle within red blood cells. By developing a system of delay-differential equations to describe within-host infection dynamics and simulating data, we demonstrate the potential to obtain biased estimates of parasite (and host) traits when key biological processes are not considered. Despite generating infection dynamics using a fixed parasite developmental cycle length, we find that known sources of measurement bias in parasite stage and abundance data can affect estimates of parasite developmental duration, with stage misclassification driving inferences about extended cycle length. We discuss alternative protocols and statistical methods that can mitigate such misestimation.

1. Background

Malaria weighs heavily on human health, causing roughly half a million deaths per year, with most deaths occurring in young children and the elderly [1]. Malaria results from infection with protozoan parasites of the genus Plasmodium, which follow a complex life cycle involving a vertebrate host and mosquito vector. Within the vertebrate host, the parasite undergoes several rounds of invasion into and replication within red blood cells (RBCs). The majority of parasites produce new asexual forms that burst from an infected RBC and can infect new RBCs, while a smaller fraction differentiate into gametocytes (i.e. sexual forms transmissible to vectors). Asexual malaria parasites develop in RBCs following characteristic cycle lengths, which are typically multiples of 24 h [2]. The two most common human malaria parasites, Plasmodium falciparum and P. vivax, develop over 48 h, while the rodent malaria parasite P. berghei develops over a 22- to 24-h cycle [3,4]. Some malaria species exhibit synchronous development, such that all infected RBCs within a host burst simultaneously [5].

The regularity and coordination of parasite developmental cycles can produce periodic spikes in fever, a known hallmark of malaria infections since Hippocrates [6]. Yet, several questions about this coordination remain unanswered: to what extent are cycle lengths and synchrony under the control of the host or the parasite? Are these traits adaptive for either player [7–9]? That different parasite species maintain different developmental cycle lengths within the same host species (e.g. 48 versus 72 h for P. falciparum and P. malariae in humans, respectively) suggests some degree of parasite control. Evidence also shows variation in cycle length within parasite species in vitro (e.g. [10,11]) and in vivo due to host factors, such as nutrient availability [12,13] and antimalarial drugs [14–18], suggesting either response by the parasite to host conditions or host control. Understanding the drivers of, and constraints on, parasite developmental schedules may bring us closer to controlling parasite growth and thus infectiousness and disease severity.

Recent experimental work, using the rodent malaria parasite P. berghei, suggests that host immunity may modulate parasite developmental schedules [19]. By tracking parasites and RBCs transfused into naive and acutely infected mice, Khoury et al. [19] inferred substantially slowed development of parasites in mice already mounting immune responses to malaria. Since this parasite species follows a (roughly) 24-h cycle, the authors expected none of the transfused labelled and infected cells to remain after 24 h. A higher proportion of infected RBCs remaining at 24 h in acutely infected mice (32% of the initial value verses 2.7% remaining in naive mice) suggested an extended cycle length. The authors fit their time series data—separated into early (i.e. ring) and late (i.e. trophozoite and schizont) parasite stages—using a mathematical model of parasite maturation and estimated that the asexual developmental cycle increased in acutely infected mice from 24 to ∼37 h [19]. Parasite development in acutely infected immune-deficient mice (rag1^−/−) resembled the expected 24 h schedule, leading to the conclusion that the host immune system mediated the inferred delayed maturation.

This fascinating result raises the question of whether delayed maturation results from strong host control or if slowed growth helps parasites evade host immunity, akin to dormancy in the face of drug treatment [15,17]. The result is also surprising because any experiment following infections for at least 5 days—of which there are many (e.g. P. chabaudi [13,20–22], P. yoelii [20,22], P. berghei [23,24])—should see a similar effect as immune responses strengthened in those infections. Assigning process to pattern in malaria infections can be difficult, and previous work showed that synchrony and replication rates in malaria infections are likely to be confounded [25]. We thus test the plausibility of alternative explanations for this inferred delayed maturation, focusing on nuances of the biology and sources of measurement error that may bias estimates of parasite stage and abundance (outlined in figure 1), which in turn may bias estimates of cycle length. We specifically consider three forms of measurement error. First, the parasite species used in this study is a reticulocyte specialist, preferentially invading young RBCs [27]. Reticulocytes, like their mature counterparts, lack a nucleus. Unlike mature RBCs, however, reticulocytes contain ribosomal RNA. Reticulocyte invasion can confound stage-structured parasite measurements that are based on nucleic acid content, leading to misclassification of early-stage parasites as late-stage parasites [28] (figure 1a). Second, any measure of circulating parasites does not capture parasite sequestration in host tissues, a feature of P. berghei (figure 1b). Finally, measurement of parasitaemia (i.e. per cent of circulating RBCs that are infected) risks conflating changes in total RBC densities with changes in infected cell densities, both of which may change over short windows, particularly during acute infection [27,29] (figure 1c).

Figure 1.

Overview of potential sources of measurement bias in within-host malaria parasite dynamics. (a) FACS plots can measure stage-structured parasite abundance using RNA- and DNA-fluorescence (heuristic adapted from [26]). However, with a reticulocyte specialist, parasites preferentially invade young RBCs, which contain RNA. This pairing of reticulocyte RNA and parasite DNA may confound distinguishing between ring- and late-stage parasites. (b) Parasite sequestration in host tissues complicates estimates of parasite loss due to immunity. Measuring density of only circulating uninfected red blood cells (RBCs) and parasitized RBCs (pRBCs), samples 2a and 2b both indicate a loss of pRBCs from circulation since initial sample 1. Without later samples, distinguishing the processes underlying these losses requires measuring total RBC and pRBCs densities, including sequestered pRBCs. Even with later samples, omitting sequestered pRBC abundance in sample 2b will complicate estimates of immune clearance and replication rate. (c) Changes in uninfected RBC count confound relative abundance measures of infected RBCs, such as parasitaemia (i.e. the per cent of circulating RBCs infected). Here, pRBC density decreases identically across samples. Yet, parasitaemia is higher in sample 4b than 4a, because densities of uninfected cells are decreasing in 4b. Sample 4c is shown for comparison, but in the experimental scenario we consider, it is not possible for uninfected cell density to increase.

We develop a mechanistic model of within-host malaria infection dynamics and simulate data following a fixed, 24 h cycle length to test how the aforementioned biological realities and measurement limitations influence cycle length estimates. We frequently infer delayed maturation despite a fixed, true 24 h cycle length. This problem is most pronounced when ring-stage-infected reticulocytes are frequently misclassified as late-stage parasites, suggesting that adjudicating between alternative explanations of the experimental observations requires direct data on RBC age structure. Critically, whether stage misclassification results in estimates of delayed maturation depends on parasite invasion of labelled and tracked donor reticulocytes. There is no reason to expect that P. berghei should inherently prefer donor or host RBCs, however it does prefer to invade young RBCs. Reticulocytes are more prevalent among RBCs in the donor verses the acutely infected host, meaning parasites will frequently invade donor reticulcoytes. This reticulocyte invasion introduces opportunities for stage misclassification, prolonging cycle length estimates. The greater reticulocyte abundance in the naive host relative to the donor limits the invasion of donor reticulocytes, thus limiting stage misclassification and avoiding a bias in cycle length estimates.

Because estimating survival and recruitment is fundamental to both basic and applied ecology [30–32], we also explore how sources of measurement error affect estimates of parasite clearance and replication. We find that replication (but not clearance) rate estimates are sensitive to both stage misclassification and parasite sequestration. Finally, we detail methodology used to study malaria, specifically, and population dynamics in ecology more broadly that can mitigate similar measurement biases and their consequences for making biological inferences.

2. Methods

We begin with a brief description of the experiment that generated the empirical data we explore. Khoury et al. [19] transfused blood samples containing parasitized and uninfected RBCs from donor mice infected with P. berghei into recipient mice. Recipient mice were either naive (i.e. not previously infected) or acutely infected, in which RBCs would be limited and immune responses would be upregulated. Using fluorescence-activated cell sorting (FACS), a form of flow cytometry, Khoury et al. obtained time series of parasitaemia (i.e. per cent of circulating RBCs parasitized) and further delineated these into early and late stages using the differential nucleic acid content between parasite stages. Not included in these data are estimates of sequestered parasites (which are difficult to obtain directly) and density (e.g. RBCs per microlitre).

To mimic the experimental scenario in [19], we simulated dynamics of P. berghei-like infection in a cohort of RBCs, representing the transfused (or ‘donor’) population in [19]. We developed a delay-differential model to capture complexities of RBC age structure, parasite stage structure and sequestration of infected cells and use this model to simulate our data. In all simulations, we assume the duration of parasite development is 24 h. From simulated data, we sampled counts of ring-stage and late-stage parasites and adjusted these data to reflect plausible sources of bias. Below, we describe the delay-differential model used to generate the data, the aforementioned data adjustments and the parasite maturation model used to fit the simulated data.

2.1. Delay-differential equation model of within-host malaria infection dynamics

Our model tracks the dynamics of a ‘donor’ RBC population, comprised of both uninfected and infected RBCs, over 1 day. We separately track the fate of the initial cohort of infected RBCs—divided into ring-stage parasites (P_R), trophozoite-stage parasites (P_T), schizont-stage parasites that are circulating in the blood (P_S) and schizont-stage parasites that are sequestered in the microvasculature (P_Q)—and subsequently infected RBCs from this donor population. The dynamics of that initial cohort are governed by:

$$\frac{\mathrm{d}{P}_R(t)}{\mathrm{d}t}=\{\begin{array}{ll}-P_0\mathrm{Beta}(0.5-t;\sigma ),& t\le 0.5\\ 0,& \mathrm{else}\end{array}$$ 2.1

$$\begin{array}{ll}& \frac{\mathrm{d}{P}_T(t)}{\mathrm{d}t}\\ & =\{\begin{array}{ll}{P}_0\mathrm{Beta}(0.5\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}t;\hspace{0.17em}\sigma )\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}\gamma P_T(t)\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{P}_0\mathrm{Beta}(0.75\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}t;\hspace{0.17em}\sigma )S_T,& t\le 0.5\\ -\gamma \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{P}_T(t)\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{P}_0\mathrm{Beta}(0.75\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}t;\hspace{0.17em}\sigma )S_T,& 0.5<t\le 0.75\\ 0,& \mathrm{else}\end{array}\end{array}$$ 2.2

$$\begin{array}{ll}& \frac{\mathrm{d}{P}_S(t)}{\mathrm{d}t}\\ & =\{\begin{array}{ll}{P}_0\mathrm{Beta}(0.75\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}t;\sigma )S_T\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}(\gamma +\varphi )P_S(t)\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{P}_0\mathrm{Beta}(1\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}t;\sigma )S_S,& t\le 0.75\\ -(\gamma +\varphi )P_S(t)-P_0\mathrm{Beta}(1-t;\sigma )S_S& \mathrm{else}\end{array}\end{array}$$ 2.3

$$\mathrm{and}\frac{\mathrm{d}{P}_Q(t)}{\mathrm{d}t}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}\varphi P_S(t)\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}{P}_0\mathrm{Beta}(1\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}\hspace{0.17em}t;\hspace{0.17em}\hspace{0.17em}\sigma )S_Q,$$ 2.4

where γ is the rate of clearance by immunity, assumed only to act on accessible late-stage parasites (i.e. trophozoites and circulating schizonts; [33]), ϕ is the rate at which circulating schizonts sequester thus rendering them inaccessible to immunity [34], P₀ is the initial abundance of donor infected cells and Beta(x; σ) is a beta distribution, with shape parameters, σ = {σ₁, σ₂}, describing the parasite stage distribution in the initial parasite cohort at the time of transfusion, evaluated at developmental index x. Developmental indices range from 0 to 1, where x = 0 represents a ring-stage parasite for which RBC invasion has just occurred, and fully developed schizonts will burst to release progeny parasites at x = 1. Assuming a fixed duration of development of 1 day, and noting that we track time (and development) in units of days, we assume that ring-stage parasites become trophozoites at x = 0.5 (i.e. parasites are ring stages for 12 h) and that trophozoites mature into schizonts at x = 0.75. The survival terms S_T, S_S and S_Q define the probability of a parasite in the initial cohort surviving to the end of the trophozoite, schizont or sequestered schizont stages, respectively, and are defined below.

To understand the indexing on the Beta terms in equations (2.1)–(2.4), consider that at the time of transfusion, t = 0, there will be some number of ring-stage parasites immediately maturing into the next developmental stage, i.e. parasites at stage x = 0.5. Thus, at t = 0, exactly P₀Beta(0.5; σ) parasites will mature from ring-stage class, P_R, into the trophozoite class, P_T, assuming no loss since immunity is restricted to acting on later stages (figure 2a). More generally, for any time up to t = 0.5, the exact abundance of parasites developing out of the ring-stage class is given by the number of parasites that were at a developmental stage of exactly 0.5 − t at the time of transfusion; in other words, P₀Beta(0.5 − t; σ) (figure 2a–c). A similar logic explains the maturation terms for the other stages: parasites developing out of the trophozoite (P_T) or schizont (either P_S or P_Q for circulating or sequestered) classes were at exactly stage 0.75 − t or 1 − t, respectively, at the time of transfusion. In these cases, however, we have to account for any losses from the initial cohort due to immunity through the survival terms. Finally, note that since equations (2.1)–(2.4) track only the initial cohort of infected cells, there is no input into the ring-stage class, movement from P_R to P_T can only occur for t ≤ 0.5 (after which time no parasite in the initial cohort can still be a ring-stage, given our assumptions about fixed development) and movement from P_T to P_S can only occur for t ≤ 0.75 (after which time no parasite in the initial cohort can still be a trophozoite).

Figure 2.

Visualization of stage distribution and maturation of the transfused parasite cohort. Colours reflect parasite stages at the time of transfusion, t = 0. (a) The initial stage distribution f (x) of the transfused parasite cohort contains 50% ring stages (developmental index x < 0.5) and 50% older stages, including trophozoites and schizonts. (b) To obtain the density of transfused parasites at a particular stage x at time t = 0.25, we transform the initial stage distribution such that f (x) becomes f (x − 0.25). Thus, the density of parasites maturing from rings to trophozoites at t = 0.25 is f (0.5–0.25). Similarly, the density of parasites maturing from trophozoites to schizonts is f (0.75–0.25) and the density of schizonts completing development is f (1–0.25). (c) At time t = 0.5, none of the remaining transfused parasites are ring stages. All of the initial rings have matured into trophozoites and schizonts. The density of trophozoites maturing into schizonts at t = 0.5 is given by f (0.75–0.5) and the density of schizonts completing development is f (1–0.5). Our delay-differential model describes the transfused parasite cohort using a beta distribution, f (x) = Beta(x, σ).

Survival through the trophozoite, S_T, circulating schizont, S_S, and sequestered schizont, S_Q, stages are given by:

$$\begin{array}{l}{S}_T=\{\begin{array}{cc}{\text{e}}^{-\gamma t},& t\le 0.25\\ {\text{e}}^{-\gamma 0.25},& \text{else}\end{array}\\ S_S=\{\begin{array}{ll}{\text{e}}^{-(\gamma +\varphi )t},& t\le 0.25\\ {\text{e}}^{-(\gamma +\varphi )0.25}{\text{e}}^{-\gamma (t-0.25)},& 0.25\text{<}t\le 0.5\\ {\text{e}}^{-(\gamma 0.5+\varphi 0.25)},& \text{else}\end{array}\\ \text{and}{S}_Q=\frac{\varphi }{\gamma +\varphi }(1-S_S).\end{array}$$

Given our assumption of fixed developmental duration, all parasites spend maximally 0.25 time units each as trophozoites and as schizonts, and so are maximally exposed to immunity (or sequestration, for schizonts) for this length of time. Of course, parasites may spend less time in these stages, depending on their stage at the time of transfusion. The survival of trophozoites, S_T, is relatively straightforward: when t ≤ 0.25 any parasite maturing out of the trophozoite stage was in this stage at the time of transfusion, and so was subjected to a constant rate of immune clearance, γ, for duration t. Thus, a proportion e^−γt of these parasites will have survived. When t > 0.25, then trophozoites maturing out of that stage were ring stages at the time of transfusion, and so were exposed to a constant clearance rate for the entire duration of the trophozoite stage; a proportion e^−γ0.25 will have survived.

The survival of circulating schizonts, S_S, is a bit more complex for two reasons. First, these stages are subject to a constant rate of sequestration, ϕ, so ‘survival’ in this stage represents evading both immunity and sequestration. Second, all transitions between parasite stages in equations (2.1)–(2.4) require considering the number of parasites that were in the initial transfused cohort (i.e. the P₀Beta(x; σ) terms). Thus, while our model predicts the instantaneous abundance of parasites in different stages, when transitions occur, our survival terms have to account for losses accumulated in previous stages. (This was true for S_T too, except we assumed no loss from the ring-stage class.) When t ≤ 0.25, parasites maturing out of the circulating schizont stage were in that stage at the time of transfusion and so were subjected to a constant rate of immune clearance, γ, and sequestration, ϕ, for duration t; a proportion e^−(γ+ϕ)t of these parasites will have ‘survived’. When 0.25 < t ≤ 0.5, parasites maturing out of the circulating schizont stage were trophozoites at the time of transfusion, and thus were subjected to clearance and sequestration for the full duration of the schizont stage (0.25 time units) and subject to clearance for the length of time they remained trophozoites (exactly t − 0.25). When t > 0.5, parasites maturing out of the circulating schizont class were ring stages at the time of transfusion, and thus were subjected to clearance for the full duration of the trophozoite and schizont stages and sequestration for the full duration of the schizont stage. Finally, to arrive at estimates for survival in the sequestered schizont stage, S_Q, note that for every cohort of schizonts that are maturing, a proportion 1 − S_S of them will no longer be in the circulating class, and a fraction ϕ/(γ + ϕ) of those will have been sequestered rather than cleared by immunity.

Uninfected RBCs in the initial transfused population are separated into old (normocyte), N_U, and young (reticulocyte), R_U, age classes. Since there is no further input of uninfected donor cells (i.e. there is a single transfusion event in [19]), and we assume for simplicity that there is no immune clearance of uninfected cells, the abundance of these cells can only decline due to parasite invasion. We do not explicitly track the maturation of reticulocytes to normocytes over the simulated 24 h, given the maturation time of approximately 3 days [35,36], and the expectation that most reticulocytes in the donor cells will be young, due to ongoing infection. These dynamics are given by

$$\frac{\mathrm{d}{N}_U(t)}{\mathrm{d}t}=-\kappa P_0\mathrm{Beta}(1-t;\sigma )(S_Q+S_S)\frac{N_U(t)}{N_U(t)+\rho R_U(t)}$$ 2.5

and

$$\frac{\mathrm{d}{R}_U(t)}{\mathrm{d}t}=-\kappa P_0\mathrm{Beta}(1-t;\sigma )(S_Q+S_S)\frac{\rho R_U(t)}{N_U(t)+\rho R_U(t)},$$ 2.6

where κ is the number of new infected RBCs resulting from one bursting (and surviving) schizont, which are distributed across the RBC age classes according to their relative density and the strength of any parasite ‘preference’ for reticulocytes, ρ. Note that κ is not the probability of invasion given contact between an uninfected RBC and merozoite (i.e. proliferative form of the malaria parasite); rather κ specifies the average number of uninfected RBCs that will become infected following the rupture of a single infected RBC. Dynamics of ring-infected normocytes, N_R, and reticulocytes, R_R, invaded post-transfusion are therefore given by:

$$\frac{\mathrm{d}{N}_R(t)}{\mathrm{d}t}=\kappa P_0\mathrm{Beta}(1-t;\sigma )(S_Q+S_S)\frac{N_U(t)}{N_U(t)+\rho R_U(t)}-\mathrm{A}$$ 2.7

and

$$\frac{\mathrm{d}{R}_R(t)}{\mathrm{d}t}=\kappa P_0\mathrm{Beta}(1-t;\sigma )(S_Q+S_S)\frac{\rho R_U(t)}{N_U(t)+\rho R_U(t)}-\mathrm{B},$$ 2.8

where

$$\mathrm{A}=\{\begin{array}{ll}0& t\le 0.5\\ \kappa P_0\mathrm{Beta}(1.5-t;\sigma )(1-\frac{\gamma }{\gamma +\varphi }(1-{\mathrm{e}}^{-(\gamma +\varphi )(t-0.5)}))\frac{N_U(t-0.5)}{N_U(t-0.5)+\rho R_U(t-0.5)}& 0.5<t\le 0.75\\ \kappa P_0\mathrm{Beta}(1.5-t;\sigma )(1-\frac{\gamma }{\gamma +\varphi }(1-{\mathrm{e}}^{-(\gamma +\varphi )0.25})){\mathrm{e}}^{-\gamma (t-0.75)}\frac{N_U(t-0.5)}{N_U(t-0.5)+\rho R_U(t-0.5)}& \mathrm{else}\end{array}$$

and

$$\mathrm{B}=\{\begin{array}{ll}0& t\le 0.5\\ \kappa P_0\mathrm{Beta}(1.5-t;\sigma )(1-\frac{\gamma }{\gamma +\varphi }(1-{\mathrm{e}}^{-(\gamma +\varphi )(t-0.5)}))\frac{\rho R_U(t-0.5)}{N_U(t-0.5)+\rho R_U(t-0.5)}& 0.5<t\le 0.75\\ \kappa P_0\mathrm{Beta}(1.5-t;\sigma )(1-\frac{\gamma }{\gamma +\varphi }(1-{\mathrm{e}}^{-(\gamma +\varphi )0.25})){\mathrm{e}}^{-\gamma (t-0.75)}\frac{\rho R_U(t-0.5)}{N_U(t-0.5)+\rho R_U(t-0.5)}& \mathrm{else}\end{array}$$

The A and B terms capture the maturation out of the ring stage by parasites that infected donor RBCs after transfusion. This can only happen when t > 0.5. Note that the bracketed terms containing the exponentials are analogous to our survival terms defined above, but with different exponents. Also, note that S_Q + S_S is equivalent to 1 − (γ/(γ + ϕ))(1 − S_S), or one minus the fraction of circulating schizonts that did not ‘survive’ due to clearance rather than sequestration. We use this relationship here for convenience. When 0.5 < t ≤ 0.75, parasites maturing out of the ring stage (in a newly infected cell) were schizonts at the time of transfusion and were subject to clearance and sequestration for t − 0.5 time units. When t > 0.75, these maturing parasites were trophozoites at the time of transfusion and so were subject to these two forces for exactly 0.25 time units as schizonts and an additional t − 0.75 time units of clearance while trophozoites.

Finally, the dynamics of trophozoite-infected normocytes and reticulocytes, N_T and R_T, circulating schizont-infected normocytes and reticulocytes, N_S and R_S, and sequestered schizont-infected normocytes and reticulocytes, N_Q and R_Q, are described by:

$$\frac{\mathrm{d}{N}_T(t)}{\mathrm{d}t}=\mathrm{A}-\mathrm{C}-\gamma N_T(t)$$ 2.9

$$\frac{\mathrm{d}{R}_T(t)}{\mathrm{d}t}=\mathrm{B}-\mathrm{D}-\gamma R_T(t)$$ 2.10

$$\frac{\mathrm{d}{N}_S(t)}{\mathrm{d}t}=\mathrm{C}-(\gamma +\varphi )N_S(t)$$ 2.11

$$\frac{\mathrm{d}{R}_S(t)}{\mathrm{d}t}=\mathrm{D}-(\gamma +\varphi )R_S(t)$$ 2.12

$$\frac{\mathrm{d}{N}_Q(t)}{\mathrm{d}t}=\varphi N_S(t)$$ 2.13

$$\mathrm{and}\frac{\mathrm{d}{R}_Q(t)}{\mathrm{d}t}=\varphi R_S(t),$$ 2.14

where

$$\mathrm{C}=\{\begin{array}{ll}0& t\le 0.75\\ \kappa P_0\mathrm{Beta}(1.75-t;\sigma )(1-\frac{\gamma }{\gamma +\varphi }(1-{\mathrm{e}}^{(t-0.75)(\gamma +\varphi )})){\mathrm{e}}^{-0.25\gamma }\frac{N_U(t-0.75)}{N_U(t-0.75)+\rho R_U(t-0.75)}& \mathrm{else}\end{array}$$

and

$$\mathrm{D}=\{\begin{array}{ll}0& t\le 0.75\\ \kappa P_0\mathrm{Beta}(1.75-t;\sigma )(1-\frac{\gamma }{\gamma +\varphi }(1-{\mathrm{e}}^{(t-0.75)(\gamma +\varphi )})){\mathrm{e}}^{-0.25\gamma }\frac{\rho R_U(t-0.75)}{N_U(t-0.75)+\rho R_U(t-0.75)}& \mathrm{else}.\end{array}$$

C and D terms capture the maturation out of the trophozoite stage by parasites that infected donor RBCs after transfusion. This can only happen when t > 0.75, by parasites that were schizonts at the time of transfusion. Thus, these parasites were subject to immune clearance and sequestration for t − 0.75 time units prior to bursting and invading new RBCs, and were subject to clearance for the full duration of the trophozoite stage. Since we track the transfused population of cells for only 1 day, no parasite that is infecting an RBC at the time of transfusion can complete development in a subsequently infected RBC, assuming a fixed duration of development of 24 h. Thus, we need not account for any loss due to maturation in the schizont classes (equations (2.11)–(2.14)).

2.2. Simulating infection dynamics

Using the above (hereafter, ‘data-generating’) model, we simulate datasets capturing the fate of the transfused cell population in ‘naive’ and ‘acutely infected’ recipient hosts. We vary two parameters across recipient hosts types. First, the clearance rate, γ, should be higher in a host with an active immune response, so clearance in an acutely infected host, γ_A, is greater than or equal to that in a naive host, γ_N. Second, κ, the number of new infected RBCs given one bursting schizont, may differ across host types (κ_N, κ_A) due to differences in the relative abundance and age structure of host RBCs (additional details below). Default parameter values (i.e. best available estimates), with which ‘focal’ datasets are simulated, are given in table 1. For every parameter combination, we simulate a single time series each for a naive and an acutely infected host. We then adjust these datasets to account for three ways that experimental measures are expected to differ from predicted dynamics.

Table 1.

Parameters used for simulating the focal datasets with the delay-differential equation model of within-host parasite dynamics.

parameter		value	refs.
κA	number new donor pRBCs* per burst pRBC in acutely infected host	0.47	[19,37]
κN	number new donor pRBCs per burst pRBC in naive host	0.13
γA	clearance rate of late-stage circulating parasites in acutely infected host	0.90 day−1	[23]
γN	clearance rate of late-stage circulating parasites in naive host	0.45 day−1	[23]
ϕ	sequestration rate of circulating schizont-stage parasites	0–3 day−1
σ1	parameter 1 for beta distribution of donor parasite stage-distribution	1
σ2	parameter 2 for donor parasite stage-distribution	1.1
P0	initial per cent donor RBCs parasitized	5%
ρ	parasite preference to invade reticulocytes (relative to normocytes)	150-fold	[27]
ε	fraction ring-stage parasites in young RBCs measured as late stages	0–1.0

2.3. Calculating κ_A and κ_N for the data-generating model

The parameters κ_A and κ_N track new infections of ‘donor’ RBCs only. Since P. berghei prefers infecting reticulocytes (ρ), estimating these parameters require estimates of the proportion of all RBCs that are from the donor (p_d) and the fraction reticulocytes among donor d_R, and host h_R, RBCs. Assuming ω is the total number of new donor and host RBCs infected by a bursting schizont,

$${\kappa }_i=\omega \frac{\hspace{0.17em}{p}_d((1-d_R)+\rho d_R)}{p_d((1-d_R)+\rho d_R)+(1-p_d)((1-h_R)+\rho h_R)},$$ 2.15

where i = A or N. When there is no preference for reticulcoytes ( ρ = 1), equation (2.16) reduces to ωp_D: the number of infected donor RBCs is the total number of RBCs infected weighted by the relative abundance of donor RBCs. As ρ increases, new infections disproportionately occur in the RBC population—recipient or donor—with a greater relative abundance of reticulocytes (decreasing or increasing estimates of κ_i, respectively). For either host type, we assume ω = 9 [38–40], ρ = 150 [27] and reticulocytes make up 1.0% of donor RBCs (d_R = 0.01, assuming the donor is 4-days infected at transfusion [37]). For an acutely infected recipient, donor cells make up 3.6% of the initial total RBC population (p_d = 0.036; [19]) and reticulocytes make up 0.045% of host RBCs (h_R = 0.0045, assuming the host is 5-days infected at transfusion [37]). For a naive recipient, p_d = 0.031 [19] and h_R = 0.031 [41]. From these estimates, we arrive at the κ_N and κ_A values in table 1, 0.12 and 0.47, respectively, which are consistent with corresponding estimates in [19] (0.05 and 0.44, respectively). Overall, these values suggest greater invasion of donor RBCs in the acutely infected host relative to the naive host.

2.4. Measurement biases

2.4.1. Reticulocyte preference and stage misclassification

Estimation of parasite stage by FACS relies on differential fluorescence of parasite stages for DNA- and RNA-stains [26] (figure 1a): early stages (i.e. rings), contain mostly DNA, while trophozoites increase RNA content and schizonts undergo DNA replication [26]. For a parasite that prefers reticulocytes, several complications emerge. RNA content of uninfected reticulocytes can confound parasite abundance estimates based on RNA fluorescence, inflating estimates upwards of 10% relative to microscopy [42]. When quantifying parasite stages based on DNA and RNA fluorescence, reticulocyte RNA, coupled with the DNA content of a ring-stage parasite, may appear in fluorescence space as an older-stage parasite (figure 1a). Since reticulocytes are a limited resource (e.g. ∼5% of RBCs in naive C57BL/6 mice [29]), multiple parasites can infect a single RBC, increasing nucleic acid content further. Consequently, distinguishing between early and late-stage P. berghei parasites has proved challenging [28], with the abundance of late (early) stage parasites being overestimated (underestimated) (figure 3a).

Figure 3.

Consequences of different measurement biases on stage-structured parasite abundance estimates over 24 h. (a) Misclassification of ring-stage parasites in reticulocytes as older-stage parasites decreases estimates of ring stages and inflates estimates of late stages. (b) When parasites sequester in tissues, the measured abundance of late stages is lower than the true abundance. (c) When parasite abundance is measured relative to the total abundance of RBCs (i.e. parasitaemia), then this measure is sensitive to changes in RBC abundance unrelated to parasites. With no input of new RBCs (as in the experimental data we consider [19]), the total RBC abundance can only decrease, hence decreasing the denominator of this metric and leading to overestimation of abundance through parasitaemia.

To capture misclassification of ring-stage-infected reticulocytes, R_R(t), as late stages, we shift a fraction, ε, of them to the late-stage-infected reticulocytes class, R_L(t), giving adjusted estimates of abundance at time t:

$$R_{R,\mathrm{adj}}(t)=(1-ϵ)R_R(t)$$ 2.16a

and

$$R_{L,\mathrm{adj}}(t)=R_T(t)+R_S(t)+R_Q(t)+ϵR_R(t).$$ 2.16b

Note that for cases where we assume sequestered parasites are not measured, R_Q(t) is omitted from equation (2.16b). Although the exact value of ε is unknown, we calculate a conservative estimate in electronic supplementary material, appendix A, which suggests ε ≥ 0.55. To avoid dependence on our specific estimate, we explore the full range possible, 0 ≤ ε ≤ 1.

2.4.2. Sequestration

Late-stage P. berghei parasites sequester into tissues [34,43], preventing those pRBCs from being measured via FACS or microscopy [37,44], and leading to underestimated parasite burdens [43,44] (figures 1b and 3b). As our model tracks sequestered pRBCs, we can calculate parasite abundance using only circulating pRBCs or all pRBCs to determine how sequestration affects estimates of parasite and host traits. The appropriate sequestration rate ϕ is unclear in the literature, thus we explore rates between 0 and 3 (table 1). At the upper end of this range, over 41% of schizonts sequestered at a given time, which equates to roughly 22% of all late stages being sequestered schizonts. This may be conservative, as work on another rodent malaria species P. chabaudi found upwards of 70% of parasites in the microvasculature versus peripheral blood [45].

2.4.3. Parasitaemia

While it is possible to obtain RBC counts from flow cytometry, FACS frequently estimates parasitaemia—the proportion of RBCs infected. Mathematically, this can be simply expressed as

$$\mathrm{parasitaemia}=100\text{\%}\times \frac{\hspace{0.17em}pRBC}{uRBC+pRBC},$$ 2.18

where pRBC is the density (or total number) of infected RBCs and uRBC is the density of uninfected RBCs. Changes in parasitaemia can be caused by changes in pRBC, uRBC or both (figure 1c).

The model used to infer cycle length ([19] and below) tracks densities of parasitized RBCs but was fit in [19] to estimates of the proportion of labelled, transfused donor RBCs that are infected with labelled parasites. While changes in this estimate may reflect changes in the abundance of infected donor cells (pRBC), the abundance of uninfected donor cells (uRBC) is equally likely to change due to parasite invasion and immune clearance [46–48]. We therefore adjust our abundance time series to reflect parasitaemia by dividing the number of RBCs infected by the total number of RBCs at a given time, which will influence dynamics (figure 3c).

2.4.4. Summary of simulated data

For every parameter set, we generate five simulated parasite time series: the ‘true’ dynamics (the raw outputs from the data-generating model), dynamics including each source of bias independently (stage misclassification, parasitaemia measurement and omission of sequestered parasites) and dynamics including all sources of bias. Except when the sequestration bias is applied, we assume sequestered schizonts are measured. This means our definition of parasitaemia in this case (i.e. relative abundance of all pRBCs) differs from typical parasitaemia estimates (i.e. relative abundance of circulating pRBCs).

2.5. Fitted parasite maturation model

Khoury et al. [19,23] developed a stage-structured, partial differential equation model of asexual malaria parasite maturation within a host that estimates the parasite developmental time. We fit this model to our simulated data, generated assuming a fixed 24 h developmental period. Details of the model can be found in electronic supplementary material, appendix B and in the original papers [19,23]. We repeated the approach in [19] and assumed most model parameters are identical across host types, expect (as in [19]), the number of new invasions into donor RBCs (κ parameters).

While the parasite maturation model (hereafter the ‘fitted model’) includes a number of parameters, our main focus is λ, whose inverse gives the length of the developmental cycle relative to 24 h. When λ = 1, parasite development takes 24 h; when λ < 1, developmental duration is longer. Khoury et al. investigated the evidence for delayed development by comparing the fit of a ‘reduced’ model, where λ is fixed at 1 to a ‘full’ model, where λ can vary in acutely infected mice only (i.e. λ_N = 1 while λ_A is a fitted parameter).

We refit the original data from [19] with the full and reduced fitted model forms and compare model performance. We compared hierarchical versions of both model forms, which allowed for variation between individuals within a host type. We fit models to data using ‘rjags’ v. 4-6 [49], an R package that allows easy implementation of JAGS in R [50]. JAGS is a program that allows fitting of Bayesian hierarchical models using Markov chain Monte Carlo [51]. We compared fits using DIC scores (‘dic.samples’ function from the ‘rjags’ package in R). We use data cloning (‘dc.fit’ function in the ‘dclone’ R package [52]) to obtain maximum-likelihood estimates for each parameter. For additional details on data cloning, see electronic supplementary material, appendix C.

Turning to our simulated data, we compare full and reduced model fits using minimum sum of squares and an F-test. Because each simulated dataset includes time series for only one naive and one acutely infected host, we did not require hierarchical model fitting. As we simulate our data assuming only late-stage parasites are cleared, we fit our data with the same assumption (i.e. x_c = 0.5 in the fitted model).

When simulating our focal datasets, we initially assume perfect stage classification of transfused parasites, although estimating these parasites should be equally subject to the biases described above. We explore the impact of misclassification of transfused parasites, which requires specifying the per cent reticulocytes among transfused pRBCs. For a parasite with no reticulocyte preference, we expect the fraction of parasites invading reticulocytes rather than normocytes at a given point in time to reflect the relative abundance of reticulocytes (i.e. 3.1% at the time of transfusion). If parasites prefer to invade reticulocytes, we expect reticulocytes to make up a fraction of parasitized RBCs greater than their relative abundance among all RBCs. We use values between 0% (functionally equivalent to our approach for the focal datasets) and 80%, an estimate from [27]. The fraction of misclassified parasites in the initial cohort is given as the product of this per cent reticulocytes, the relative abundance of ring stages among transfused pRBCs and the extent of misclassification ε. For these simulations, we only adjust data using stage misclassification and ignore sequestration (i.e. ϕ = 0).

While our primary interest is how the fitted model estimates cycle length under various measurement biases, we are also interested in how these biases affect estimates of parasite replication and host clearance rates. To test the dependency of these parameter estimates on stage misclassification, sequestration and the per cent reticulocytes among transfused pRBCs, we used multiple linear regression. For datasets adjusted by stage misclassification, we fit a model including both stage misclassification ε and either per cent reticulocytes or sequestration rate ϕ as well as their interaction term. For those not adjusted by misclassification, we include only sequestration rate. For each model term, we obtain beta coefficients and p-values (‘beta’ function, part of the ‘reghelper’ package [53] in R [50]).

3. Results

3.1. Refitting original data

We tested our ability to reproduce the results by Khoury et al. [19]. Here, we present our results from applying the fitted model, with immune clearance acting on all parasite stages (i.e. x_c = 0), to pooled data, but we also provide the results of a hierarchical approach in electronic supplementary material, appendix C in which we fit either developmental rate λ_A or clearance rate c as a random effect and assume all remaining parameters are fixed effects. We found the full model (fitting λ_A⩽1) outperformed the reduced model (fixing λ_A = 1; 27.19 verses 353.1, DIC). While the reduced model performed qualitatively similarly to the full model in describing ring-stage dynamics in both host types and describing late-stage dynamics in naive hosts, it under-predicted late stages in acutely infected hosts towards the end of the 24 h post-transfusion (electronic supplementary material, figure S1). Qualitatively, this result is consistent with [19].

Using the full model, we estimated λ_A = 0.64 ± 0.02, which is equivalent to a developmental cycle length of 37.5 h in the acute host and nearly identical to the 37 h estimate by Khoury et al. [19]. We estimated a parasite stage structure similar to the original estimates (μ_s = 0.35 ± 0.01, σ_s = 0.26 ± 0.015 versus μ_s = 0.32, σ_s = 0.26 reported in [19]); however, our estimates of replication rates in acutely-infected and naive mice were lower than the corresponding estimates (k_A = 0.20 ± 0.06, k_N = 0.01 ± 0.01 versus k_A = 0.44, k_N = 0.05) reported in [19]. Similarly, we estimated a lower clearance rate (c = 0.61 ± 0.05 versus c = 0.89 reported in [19]). While we suspect that replication and clearance rate estimates are not independent, we did not find evidence for this through data cloning. It is worth noting that our fitting approach and chosen software differed from [19]. All parameter estimates are provided in electronic supplementary material, table S1.

3.2. Fitting simulated data

Using simulated datasets, we explored how measurement biases both independently and together affect estimates of parasite development and proliferation. Below, we consider the conditions under which the fitted model with delayed development (i.e. the full model form) produced a significantly better fit to simulated data and summarize results from linear models used to test the dependency of parasite estimates on stage misclassification and sequestration. Detailed results for these statistical analyses of the focal datasets are included in electronic supplementary material, tables S4–S6.

3.2.1. Stage misclassification in the host increases the inferred cycle length

Stage misclassification was the key measurement bias affecting inferences of developmental duration. Using the focal datasets, we found the full model provided the best fit, implying delayed development in the acutely infected host (λ_A < 1), even though cycle length was fixed to 24 h in the data-generating model. We only found this effect with sufficient misclassification of ring stages in reticulocytes as late stages. Without stage misclassification, the reduced form of the fitted model always outperformed the full model (F-test, α = 0.05).

We found stage misclassification (ε) to be a significant negative predictor of inferred developmental duration (electronic supplementary material, table S4). Across sequestration rates (ϕ), we found that allowing delayed maturation significantly improved model fit when ε was at least 0.75, i.e. 75% ring-stage reticulocytes misclassified (F-test, α = 0.05; figure 4a). This result held when we applied stage misclassification bias either alone or together with the sequestration and parasitaemia biases. This effect stems from a ‘build-up’ of measured late-stage parasites: misclassified ring stages move into the measured late-stage cohort early and remain there for the duration of their development, appearing to undergo an extended period of late-stage development. When we applied all measurement biases to the data, we also found a significant interaction between misclassification and sequestration (β = 0.35, p < 0.05). We can consider stage misclassification as shuttling ring stages into the late-stage compartment, while sequestration removes late stages (if only circulating parasites are measured). While misclassification decreases developmental duration estimates, sequestration mitigates this effect through removal of true late stages from circulation.

Figure 4.

Estimated parasite developmental cycle length at different values of stage misclassification ε and either (a) sequestration ϕ or (b) per cent of reticulocytes in the initial cohort of transfused, parasitized red blood cells. (a) We find the fitted model estimates delayed parasite development (λ_A < 1) when the extent of stage misclassification ε is at least 75%, while sequestration rate does not notably affect λ_A estimates. We explored the effects of stage misclassification and sequestration on developmental duration estimates using multivariate linear regression (electronic supplementary material, table S4). (b) In the absence of sequestration (ϕ = 0), we find that the presence of reticulocytes in the initial cohort of parasitized RBCs works in concert with greater stage misclassification to increase estimates of cycle length, up to roughly 49 h (λ_A ≈ 0.49). We considered the effects of stage misclassification and reticulocytes in the initial cohort on developmental duration estimates using multivariate linear regression (electronic supplementary material, table S2). Black boxes indicate the values of ε and per cent reticulocytes for which we considered sensitivity of cycle length estimates to additional data-generating model parameters.

3.2.2. Stage misclassification of transfused parasites further increases cycle length estimates

We next extended the impact of stage misclassification to the transfused parasite cohort, a scenario we consider to more realistic than misclassification of only parasites in subsequently infected cells. We found that increasing representation of reticulocytes in the transfused cohort led to increased estimates of cycle length (electronic supplementary material, table S2). We also found a significant, negative interaction between stage misclassification and the per cent reticulocytes on estimates of developmental rate (figure 4b). With less than 50% misclassification (ε < 0.5), we estimate a cycle length of 24 h. Above 50% misclassification, cycle length estimates increase with greater values of ε and per cent reticulocytes among the transfused cohort. For example, with 50% of the transfused cohort being reticulocytes, and complete misclassification (ε = 1), we estimated a maturation rate λ_A ≈ 0.67, which equates to a 36 h cycle length. At even higher fractions of reticulocytes, cycle length estimates increased, reaching approximately 49 h in the parameter space we explored. We note that at combinations of higher values of ε and per cent reticulocytes, model fit (minimum sum of squares) worsened; however, the difference between reduced and full model fits also increased (electronic supplementary material, figure S3).

3.2.3. Bias from stage misclassification is amplified by within-host dynamics

Because cycle length estimates likely depend not just on the extent of stage misclassification (ε) and the fraction of reticulocytes among transfused, infected RBCs, we considered the sensitivity of those estimates to a number of parameters in the data-generating model. We chose three combinations of ε and per cent reticulocytes (indicated in figure 4b) and estimated cycle length across a range of values for each data-generating model parameter (electronic supplementary material, table S3; figure 5). We initially explored a wide range of values for each parameter (results not shown) but restricted our final analyses to values that increased cycle length estimates (figure 5). With relatively high representation of reticulocytes (80%) among transfused, infected RBCs but modest stage misclassification (ε = 0.25), we found a 24 h cycle length typically outperformed delayed parasite development and the estimated cycle length never exceeded 30 h. A different picture emerged from considering complete stage misclassification (ε = 1) and a moderate per cent reticulocytes among the transfused pRBCs (50%). Both greater replication (κ_A) and reduced clearance (γ_A) in the acutely infected host increased cycle length estimates up to 48 h. With 50% reticulocytes but intermediate stage misclassification (ε = 0.5), we observe similar, albeit dampened, dependency on replication and clearance parameters. Greater replication within donor cells in the acutely infected host (κ_A > κ_N) produces a greater abundance of ring-stage parasites, which can then be misclassified as late-stage parasites, favouring a model with delayed maturation in the acutely infected host. Decreasing γ_A increases the abundance of late-stage parasites remaining in circulation in the acutely infected host. More parasites thus complete development and burst, increasing the abundance of ring-stage parasites. As the fitted model assumes only a single clearance rate (c) for both host types, it compensates for this greater abundance of ring stages by lowering the maturation rate.

Figure 5.

Sensitivity of estimates of parasite developmental cycle length in the acutely infected host to different parameters in the data-generating model. We considered sensitivity at three combinations of misclassification ε and per cent reticulocytes (indicated with black boxes in figure 4b). We chose ranges for model parameters based on a preliminary exploration of parameter space, filtering to those that increased cycle length estimates (data not shown). Parameters related to proliferation, i.e. replication (κ_A) and clearance (γ_A, γ_N) rates, have the greatest effect on cycle length estimates, with higher effective proliferation in the acutely infected host and lower effective proliferation in the naive host favouring estimating delayed development. We also found that with reduced stage misclassification—despite a higher per cent reticulocytes in the transfused parasite cohort—the fitted model rarely infers delayed development, except with sufficiently low values of γ_A and moderate values of γ_N. Replication rate in the naive host (κ_N), sequestration rate (ϕ) and reticulocyte preference ( ρ) do not strongly affect cycle length estimates for the cases considered.

With greater stage misclassification and moderate per cent reticulocytes, we found a non-monotonic relationship between cycle length estimates and the first shape parameter (σ₁) of our transfused parasite stage distribution, a beta distribution. To understand the relationship between σ₁ and cycle length, we considered the relationship between σ₁ and the per cent of ring-stage parasites in the transfused parasite cohort (electronic supplementary material, figure S4). Broadly speaking, a smaller value of σ₁ equates to a greater per cent of ring stages. We found that when ring stages comprised roughly a quarter or higher of the transfused parasites, the fitted model inferred delayed development (peaking at around 40 h when ring stages made up around 40% of transfused parasites).

In summary, we find that estimates of parasite developmental rate are sensitive to the extent of parasite stage misclassification, especially when we allow misclassification of the initial parasite cohort (and there is no reason to expect this bias should not apply to them). In the presence of stage misclassification, developmental rate estimates are further biased by the stage structure of the parasite population. When ring stages within reticulocytes may be misclassified as late stages, there is then a cohort of ring stages that will appear as late stages for their entire development, i.e. over 12 h. It is this seemingly ‘stagnant’ population of parasites that (incorrectly) require a model with delayed development. Greater replication in the acutely infected host relative to the naive host and a larger representation of ring stages in the transfused parasite cohort both increase the size of this seemingly stagnant population, explaining their role in inflating estimates of developmental rate.

3.2.4. Stage misclassification biases estimates of replication and clearance rates

When data are biased only by omission of sequestered parasites, we found greater sequestration rates (ϕ) increased replication rate estimates in both host types (electronic supplementary material, table S4; figure 6). To understand this effect, we can consider how sequestration changes the measured abundance of different parasite stages. Sequestration moves late-stage parasites out of circulation and into host tissues. When sequestered parasites are omitted from measurements, the proportion of ring-stage parasites among the total measured parasite population increases. In order to produce a larger observed abundance of new parasites from an apparent lower abundance of late-stage parasites, the fitted model requires a higher replication rate. As previously discussed, stage misclassification has an opposing effect on measured parasite stage structure compared to sequestration (i.e. increasing measured late stages, decreasing measured ring stages). When we adjusted data with both stage misclassification and sequestration biases, sequestration rate did not significantly predict replication rates. We thus suspect the effect of stage misclassification on replication rate estimates overwhelms the effect of sequestration.

Figure 6.

Estimated values of replication (k_N, k_A) and clearance (c) rates at different values of stage misclassification ε and sequestration ϕ (a–c) or per cent reticulocytes in the transfused parasite cohort (d–f). We simulated data assuming late-stage clearance (x_c = 0.5) and obtained parameter estimates from the fitted model after adjustment with either all measurement biases (a–c) or only stage misclassification (d–f). (a) The estimated clearance rate depends on stage misclassification, but not sequestration. (b,c) Misclassification of ring-stage parasites as older-stage parasites lowers the measured abundance of young parasites, which the fitted model interprets as lower replication rates. We report statistical analyses of these effects in electronic supplementary material, table S4. (d) Clearance rate estimates depend on the interaction between stage misclassification and the per cent reticulocytes among the transfused pRBCs, with greater misclassification and more reticulocytes decreasing estimates. (e) Estimates of replication rate in the naive host decrease with greater values of misclassification. (f) Both misclassification ε and its interaction with per cent reticulocytes decrease estimates of replication rate in the acutely infected host. We report statistical analyses of these effects in electronic supplementary material, table S2.

Indeed, the fraction of ring-stage reticulocytes misclassified (ε) is a significant, negative predictor of clearance rate (c) and replication rates (k_N and k_A) when data are biased by stage misclassification (electronic supplementary material, tables S4–S5), either in isolation (electronic supplementary material, figure S2) or in conjunction with other measurement biases (figure 6a–c). In both cases, simulated ring-stage parasites are not subjected to clearance, thus measuring a ‘late-stage’ parasite population composed of misclassified ring-stage and true late-stage parasites decreases the effective clearance rate per measured ‘late-stage’ parasite. Increasing stage misclassification decreases the abundance of ring-stage parasites measured, including new invasions, as those developing within reticulocytes are misclassified as older parasites. To compensate for this absence of newly developing parasites, the model underestimates replication rates. Under both stage misclassification bias and all measurement biases together, we did not find a significant interaction between sequestration rate (ϕ) and stage misclassification (electronic supplementary material, tables S4–5).

When we considered misclassification of transfused, parasitized reticulocytes (electronic supplementary material, table S2; figure 6d–f), we found a significant interaction between misclassification (ε) and the per cent of reticulcoytes among the transfused pRBCs when estimating clearance rate (c) and replication rate in the acutely infected host (k_A). Increasing the representation of reticulocytes among the transfused pRBCs functionally increases the effect of stage misclassification on clearance and replication rate estimates, as it increases the population of parasites susceptible to misclassification early in infection (when most pRBCs were parasitized in the donor and transfused). Similar to the effect of stage misclassification on replication and clearance estimates explored above, increasing the representation of ring-stage parasites among the measured ‘late-stage’ cohort lowers the effective clearance rate as well as the replication rate in the acutely infected host.

4. Discussion

There are extensive reports of genetic variation in asexual developmental cycle lengths, and thus maturation rates, across malaria parasites [10,11,39,54]. There is evidence to suggest that parasites may plastically alter their developmental schedules in response to environmental changes, including host circadian rhythm [55,56], drug concentration [14] and, as we focus on here, host immune responses [19]. Our analyses reveal a need for caution in inferring parasite schedules from time series of relative abundance data, since various measurement biases directly affect the ability to draw correct inferences regarding parasite development and proliferation.

4.1. Understanding the data we need

After confirming our ability to reproduce the estimate of delayed parasite maturation in [19], we dissected the effects of measurement biases both in isolation and in concert on estimates of malaria parasite development and proliferation. Simulating data conferred a distinct advantage for this purpose, since it allowed us to specify—and thus know—the true biology underlying within-host processes and also precisely quantify how different biases affected estimates of that biology, individually and in concert.

For our simulated focal datasets, the maximum deviation of estimated parasite developmental duration from the expected value of 24 h was just over 10%, with resulting estimates of cycle length reaching upwards of 27 h. When we added the potential to misclassify the original, transfused cohort of parasites, the estimated cycle length reached 36 h (approximately that estimated in [19]) and beyond. We also found developmental duration estimates to be sensitive to disparities in acutely infected and naive host dynamics. For example, when invasion into donor RBCs is greater in the acutely infected host, we inferred prolonged development, as ring-stage parasites in young donor RBCs can be misclassified as late stages. Assuming the probability of invasion into donor RBCs is some function of their relative abundance compared to host RBCs, we then expect greater invasion into donor cells when the host is depleted of its own RBCs, such as during the peak of infection when host immune response is heightened. We also expect stronger clearance of parasites in acutely infected hosts, further biasing estimates of cycle length. In addition to age of infection in the host, infection age within the donor is likely important for estimating developmental duration. We observed a general dependency of developmental duration estimates on the abundance of ring stages in the parasite cohort transfused from the donor. Assuming parasites in the transfused cohort are representative of the parasite population in the donor, we expect shorter cycle length estimates when parasites are transfused from hosts prior to the peak of infection (i.e. from an expanding parasite population). Overall, we expect developmental duration estimates to be sensitive not only to the extent of stage misclassification but also to age of infection in both donor and host. Stage misclassification can be addressed by validating staging from flow cytometry with microscopy, but adequately accounting for differences in donors and hosts requires considering the myriad changes that accrue over the age of infection.

One outstanding question is the role of immunity in the apparent delayed development observed in [19]. Specifically, acutely infected immune-deficient rag1^−/− mice did not display a pattern consistent with delayed development. One possible explanation of this observation involves differences in stage misclassification. During acute infection, mice mount a strong inflammatory response, which includes increasing tumour necrosis factor (TNF-α) and interferon-γ (IFN-γ) [57–59], both of which suppress erythropoiesis in mice [59–64]. We then expect a weaker erythropoietic response in acutely infected wild-type mice compared to their rag1^−/− counterparts, whose TNF-α and IFN-γ production is severely blunted. Should rag1^−/− hosts input more young RBCs into circulation, this would lower the replication rate within donor RBCs (κ) and limit misclassification of ring-stage-infected donor reticulocytes. Measuring the age structure of RBC populations across time during infection of wild-type and rag1^−/− mice should illuminate how the strength of erythropoiesis affects the relative abundance of reticulocytes in the host and thus parasite proclivity for invading donor RBCs.

Though not the focus in [19], we investigated the influence of the different measurement biases on estimates of parasite proliferation (i.e. replication and clearance), again finding the extent of stage misclassification to be critical. With greater misclassification, we increasingly underestimated clearance and replication rates in both acutely infected and naive hosts. Replication rate was also affected by sequestration. If the extent of sequestration is well known, then it is straightforward to translate circulating parasite abundance to total parasite abundance [65]. Measures of total parasite burden (TPB) may also be used to mitigate bias due to sequestration by measuring parasites both in circulation and sequestered in tissues. Briefly, transgenic Plasmodium parasites express bio-markers, which when used with in vivo imaging techniques, provide measures of total biolumenscence (a function of abundance of parasite genomes) in a host [43,66]. Notably, dependence of bio-marker expression on parasite stage will complicate estimating total parasite abundance [67,68], particularly for semi-synchronous or asynchronous parasite species, such as P. berghei. The ability to accurately measure replication and clearance rates is important for understanding diverse infection outcomes and the contributions of parasite and host factors to the severity of infection.

4.2. Extensions of sequestration: unobservable life stages

Although our results show little direct effect of sequestration on inferences of cycle length, sequestration biases estimates of parasite clearance and replication. In the language of ecology, these equate to estimates of survival and recruitment, respectively. Unobservable life stages (e.g. sequestered parasites) are ubiquitous in nature and take numerous forms, including both dormant seedlings and dormant adult plants [69,70], animals with cryptic forms [71,72] and animals with temporary emigration [73–76]. Unfortunately, this ubiquity renders imperfect detection of unobservable life stages a pervasive problem in ecology. Kellner & Swihart [77] found only 23% of 537 ecological articles in a quantitative literature review accounted for imperfect detection of unobservable life stages when estimating species abundance and distribution and that this issue spanned a wide range of taxa, including vertebrates, invertebrates and plants. Indeed, just as failure to measure sequestered parasites drives overestimates of clearance and replication, omission of unobservable life stages can drive underestimates of survival [78] and extinction [79] and overestimates of colonization [79] and population size [80].

Because unobservable life stages are so common, statistical methodology has been developed to obtain unbiased estimates of survival and recruitment. At the broadest level, state equations describing system dynamics are defined and coupled with observation equations, which account for imperfect measurement. Such models can take different forms, including hidden Markov models [81–84] and age-structured population models [85,86]. Modelling that can account for unobservable stages through observation error may be the best option for a species like P. berghei, given the complications of asynchronicity and TPB estimates discussed above.

4.3. Conclusion

Multiple complex processes affecting detection and quantification of different cell populations come together to produce a seemingly simple parasitaemia time series. A summary estimate, such as parasitaemia, represents a loss of information about within-host infection dynamics. Without that information, it may not be possible to resolve the true dynamics underlying the observed data. Indeed, we show that a model of delayed parasite maturation can fit observed infection dynamics, even when these data were produced using a 24 h developmental cycle. Measurement biases that influence such inferences should be quantified, and accounted for, through comparison of empirical stage estimates produced by flow cytometry and microscopy.

Models of murine malaria are important for understanding processes underlying within-host infections (e.g. host immunity [87,88], infection outcomes [89,90]) and developing therapeutics [91,92]. Rodent malaria infections allow exploration of these processes in controlled systems. If reliable inferences are hard to obtain from these infections, this suggests greater difficulties interpreting human infection data. Yet, empirical and statistical paths forward exist for validating infection data and treating missing data appropriately. Such future work may untangle the players and processes controlling parasite development and thus give insight into limiting the spread and severity of infection.

Data accessibility

Scripts for data generation and fitting, subsequent analyses and figures: Github https://github.com/madelineapeters/InferringParasiteTraits.

Authors' contributions

M.A.E.P. conceived of the study, developed the model, carried out data analyses and drafted the manuscript; M.G. helped conceive of the study, assisted with model development and critically revised the manuscript; N.M. conceived of the study, assisted with model development and helped draft the manuscript. All authors gave final approval for publication and agree to be held accountable for the work performed therein.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by Natural Sciences and Engineering Research Council of Canada Discovery grant no. RGPIN-2018-06017.