Evolutionary genetics of malaria

Abstract

Many standard-textbook population-genetic results apply to a wide range of species. Sometimes, however, population-genetic models and principles need to be tailored to a particular species. This is particularly true for malaria, which next to tuberculosis and HIV/AIDS ranks among the economically most relevant infectious diseases. Importantly, malaria is not one disease—five human-pathogenic species of Plasmodium exist. P. falciparum is not only the most severe form of human malaria, but it also causes the majority of infections. The second most relevant species, P. vivax, is already considered a neglected disease in several endemic areas. All human-pathogenic species have distinct characteristics that are not only crucial for control and eradication efforts, but also for the population-genetics of the disease. This is particularly true in the context of selection. Namely, fitness is determined by so-called fitness components, which are determined by the parasites live-history, which differs between malaria species. The presence of hypnozoites, i.e., dormant liver-stage parasites, which can cause disease relapses, is a distinct feature of P. vivax and P. ovale sp. In P. malariae inactivated blood-stage parasites can cause a recrudescence years after the infection was clinically cured. To properly describe population-genetic processes, such as the spread of anti-malarial drug resistance, these features must be accounted for appropriately. Here, we introduce and extend a population-genetic framework for the evolutionary dynamics of malaria, which applies to all human-pathogenic malaria species. The model focuses on, but is not limited to, the spread of drug resistance. The framework elucidates how the presence of dormant liver stage or inactivated blood stage parasites that act like seed banks delay evolutionary processes. It is shown that, contrary to standard population-genetic theory, the process of selection and recombination cannot be decoupled in malaria. Furthermore, we discuss the connection between haplotype frequencies, haplotype prevalence, transmission dynamics, and relapses or recrudescence in malaria.

Full text

Evolutionary genetics of malaria
Kristan Alexander Schneider
1
* and Carola Janette Salas
2
1
Department of Applied Computer- and Biosciences, University of Applied Sciences Mittweida,
Mittweida, Germany,
2
Department of Parasitology, U.S. Naval Medical Research Unit No 6 (NAMRU-6),
Lima, Peru
Many standard-textbook population-genetic results apply to a wide range of
species. Sometimes, however, population-genetic models and principles need
to be tailored to a particular species. This is particularly true for malaria, which
next to tuberculosis and HIV/AIDS ranks among the economically most relevant
infectious diseases. Importantly, malaria is not one disease—five human-
pathogenic species of Plasmodium exist. P. falciparum is not only the most
severe form of human malaria, but it also causes the majority of infections. The
second most relevant species, P. vivax, is already considered a neglected
disease in several endemic areas. All human-pathogenic species have
distinct characteristics that are not only crucial for control and eradication
efforts, but also for the population-genetics of the disease. This is particularly
true in the context of selection. Namely, fitness is determined by so-called
fitness components, which are determined by the parasites live-history, which
differs between malaria species. The presence of hypnozoites, i.e., dormant
liver-stage parasites, which can cause disease relapses, is a distinct feature of P.
vivax and P. ovale sp. In P. malariae inactivated blood-stage parasites can cause
a recrudescence years after the infection was clinically cured. To properly
describe population-genetic processes, such as the spread of anti-malarial drug
resistance, these features must be accounted for appropriately. Here, we
introduce and extend a population-genetic framework for the evolutionary
dynamics of malaria, which applies to all human-pathogenic malaria species.
The model focuses on, but is not limited to, the spread of drug resistance. The
framework elucidates how the presence of dormant liver stage or inactivated
blood stage parasites that act like seed banks delay evolutionary processes. It is
shown that, contrary to standard population-genetic theory, the process of
selection and recombination cannot be decoupled in malaria. Furthermore, we
discuss the connection between haplotype frequencies, haplotype prevalence,
transmission dynamics, and relapses or recrudescence in malaria.
KEYWORDS
complexity of infection (COI), co-infection, mixed-species infection, recrudescence,
relapse, seed bank, hypnozoites, multiplicity of infection (MOI)
OPEN ACCESS
EDITED BY
Rongling Wu,
The Pennsylvania State University (PSU),
United States
REVIEWED BY
Chenqi Wang,
University of South Florida,
United States
Edith Christiane Bougouma,
Groupe de Recherche Action en Santé
(GRAS), Burkina Faso
*CORRESPONDENCE
Kristan Alexander Schneider ,
kristan.schneider@hs-mittweida.de
SPECIALTY SECTION
This article was submitted to
Evolutionary and Population Genetics,
a section of the journal
Frontiers in Genetics
RECEIVED 29 August 2022
ACCEPTED 26 September 2022
PUBLISHED 03 November 2022
CITATION
Schneider KA and Salas CJ (2022),
Evolutionary genetics of malaria.
Front. Genet. 13:1030463.
doi: 10.3389/fgene.2022.1030463
COPYRIGHT
© 2022 Schneider and Salas. This is an
open-access article distributed under
the terms of the Creative Commons
Attribution License (CC BY). The use,
distribution or reproduction in other
forums is permitted, provided the
original author(s) and the copyright
owner(s) are credited and that the
original publication in this journal is
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not comply with these terms.
Frontiers in Genetics frontiersin.org01
TYPE Original Research
PUBLISHED 03 November 2022
DOI 10.3389/fgene.2022.1030463

1 Introduction
After a decade of declining incidence the number of annual
malaria infections rises since 2018, challenging the WHO goal
to reduce malaria incidence by at least 90% by 2030 (WHO,
2021a). This is partly attributed to the rapid emergence and
spread of anti-malarial drug resistance, an evolutionary-genetic
process whose understanding is a global health priority (WHO,
2021b).
Malaria is caused in humans and animals by Plasmodium
parasites. These unicellular, haploid eukaryotes are transmitted
by numerous species of female Anopheles mosquitoes. Both the
parasite and vector species are adapted to specific human or
animal hosts. Five species of Plasmodium are pathogenic to
humans, which can be transmitted by over 100 Anopheles
species (Nicoletti, 2020). Over 95% of the 240 million annual
infections and 620,000 deaths worldwide are attributed to P.
falciparum. Although, the WHO recommended the use of RTS,S,
the first approved malaria vaccine, in children to prevent P.
falciparum infections in areas of moderate to high transmission,
the vaccine’sefficacy is low and malaria control depends strongly
on reliable diagnostics and drug treatments to cure acute
infections (Greenwood et al., 2021). While the second most
relevant species, Plasmodium vivax, receives considerable
attention, the other species P. ovale sp., P. malariae, and P.
knowlesi are somewhat neglected, due to an outdated distinction
between harmful and harmless malaria species (Lover et al.,
2018).
The spread of deletions in the histidine-rich protein 2 and 3
(HRP2/3) genes of P. falciparum, which encode for the antigens
targeted by rapid diagnostic tests (RDTs) as well as drug-resistant
P. falciparum and P. vivax haplotypes substantially challenge
successful malaria control. These evolutionary genetic processes
are tightly linked to the pathogen’s complex transmission cycle,
which besides some species-specific differences, is commonly
shared among all Plasmodia (Su et al., 2019;Beshir et al., 2022).
The transmission cycle starts with an infected mosquito
taking her blood meal. She inoculates parasites in the form of
sporozoites from her salivary glands into the human body. This
is followed by the exo-erythrocytic cycle, during which
sporozoites reach the liver to infect hepatocytes. In the
infected liver cells parasites mature into schizonts. The
erythrocytic cycle is initiated when the schizonts rupture and
merozoites are released into the bloodstream. Erythrocytes are
invaded by merozoites, which form ring stage trophozoites and
then mature into schizonts. Once they rupture, new merozoites
are released into the bloodstream. During this step of asexual
reproduction, some parasites differentiate into male or female
gametocytes, which do not reproduce in the human host. Once
a mosquito ingests male and female gametocytes, the
sporogonic cycle is initiated. Gametes released by male and
female gametocytes fertilize and form zygotes. Following a step
of meiosis, and hence recombination, the zygote becomes
tetraploid and develops into ookinetes, which migrate
through the midgut wall and transform into oocysts. In the
oocyst sporozoite budding occurs in the haploid state. Division
of each oocyst produces thousands of sporozoites that move
into the mosquito salivary glands, completing the transmission
cycle. Because gametocytes immediately release gametes, only
parasites exiting the same host recombine, potentially leading
to a high degree of inbreeding during the sexual reproduction of
the parasite (Ngwa et al., 2016).
Species-specific differences occur in the number of parasites
within an infection (parasitemia and gametocytemia counts), and
the duration of the various phases in the transmission cycle. The
replication of merozoites in 72-hour- rather than 48-hour-cycles
distinguishes P. ovale sp. from other species. The onset of
gametocytogenesis and the longevity of gametocytes were
argued to accelerate drug-resistance evolution in P. falciparum
compared to P. vivax (Schneider and Escalante, 2013). Dormant
liver-stage parasites (hypnozoites), can result in disease relapses
weeks, months, or even years after the clearance of blood stage
parasites and occur only in P. vivax and P. ovale sp. Currently
primaquine (PQ) and tafenoquine (TQ) are the only approved
drugs to clear hypnozoites (Watson et al., 2021). Unfortunately,
patients with (glucose-6 phosphate dehydrogenase) G6PD
deficiency, which is widespread in many malaria-endemic
areas, cannot be treated with these drugs (Baird et al., 2018;
Dean et al., 2020). Extremely prolonged carriage of blood-stage
parasites causing recrudescences occur in P. malariae (Collins
and Jeffery, 2007). It is commonly accepted, although not
completely ruled out, that the rebounce of parasitaemia in P.
malariae is not caused by quiescent pre-erythrocytic stages such
as hypnozoites. Because of relapses occurring in P. vivax,P. ovale
sp., and prolonged blood stage parasite carriage in P. malariae,
these species are resilient in areas in which P. falciparum
transmission cannot be sustained. While all other human
malaria species can—at least in theory—be eradicated by
concentrating on the human host, this is not possible for P.
knowlesi, which is characterized by zoonotic transmission. It
became the predominant species in several endemic countries in
Southeast Asia, which shifted from malaria control toward
elimination (Sutherland, 2016).
The characteristics of the transmission cycle render the
application of standard textbook population-genetic results
incorrect. Particularly it was shown that the process of
selection acting on parasites in the human hosts (including
selection for drug resistance) and recombination cannot be
separated (Schneider and Kim, 2010). Hence, population-
genetic theory and models have to be tailored to the malaria
transmission cycle. This has been done mainly for P. falciparum.
Because a clear path to eradication has been chartered only for P.
falciparum, the other malaria species gain more importance due
to their resilient nature (Lover et al., 2018). This requires to
further adapt population-genetic theory to the characteristics of
other human-pathogenic malaria species.
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Here, we extend a population-genetic framework, originally
developed for P. falciparum, to be applicable to all other malaria
species.
We exemplify the importance of species-specific differences
by clarifying the role of hypnozoites in the evolution of drug
resistance in P. vivax vs. P. falciparum. We also clarify, how
haplotype frequencies (i.e., their relative abundance in the
parasite population) and prevalence (i.e., the likelihood that a
given haplotype occurs in an infection) are affected by relapses/
recrudescence in other malaria species. Based on this framework,
we discuss past and current developments with relevance for the
evolutionary genetics of malaria.
2 Methods
We extend the population-genetic framework of (Schneider
and Kim, 2010;Schneider and Kim, 2011;Schneider, 2021) that
describes the temporal change in the distribution of parasite
haplotypes due to recombination and selection in generations of
FIGURE 1
Transmission cycle of human malaria. All species have the same cycl e, but parasites life-stages have different morphology (illustrated here for P.
falciparum). In P. vivax and P. ovale sp. dormant hypnozoites remain in the liver. In P. malariae recrudescence form prolonged blood stage parasites
occur. In P. knowlesi humans and non-human primates can be infected.
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transmission cycles. While the original framework was tailored to
P. falciparum, the extension captures the characteristics of all
human-pathogenic malaria species.
The model is based on an idealization of the complex malaria
transmission cycle (cf. Figure 1), which is illustrated in Figure 2.
Although, pathogen, mosquito vector, human hosts (and, in the
case of P. knowlesi the animal host) are involved in transmission,
the framework does not require to model transmission dynamics
(i.e., the interaction of mosquito vectors and human or animal
hosts) explicitly. This conceptional advantages arise, because
haplotype frequencies are considered at the end of the
sporogenic cycle (cf. Figure 2). Thus, the frequency
distribution of parasite haplotypes in the mosquitoes’salivary
glands, which are ready for vector-host transmission, is followed.
Host and vector populations are assumed to be sufficiently
large and malaria infections sufficiently frequent to justify a
deterministic description of the evolutionary dynamics. Steps
of full transmission cycles correspond to steps of sexual
reproduction, because only one step of sexual reproduction
occurs during one full transmission cycle, namely inside the
mosquito vector. Many steps of asexual reproduction occur
inside the vectors and hosts.
2.1 Genetic architecture of haplotypes
The genetic architecture of haplotypes is determined by their
allelic configuration at one or several loci. We denote the number
FIGURE 2
Illustrated is the idealization of the malaria transmission cycle underlying the population-genetic framework. The illustrated genetic
architecture of malaria haplotypes assumes two biallelic loci, leading to four possible haplotypes. Furthermore, two groups of hosts are illustrated.
Each host is infected by randomly drawing haplotypes from generation t, or a relapse/recrudescence from a previous generation occurs, which
corresponds to randomly draw parasites from a previous generation (haplotype reservoir). With probability G(t)
ga host belongs to group gin
generation t. The selective environment is different in the two groups. Recombination occurs exclusively between haplotypes exiting the same host.
After recombination, haplotypes in the mosquitoes are pooled together to derive their distribution in generation t+1.
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of all possible haplotypes by H. E.g., Lbiallelic loci lead to H=2
L
haplotypes. In general, if haplotypes are determined by Lloci, and
n
l
alleles are segregating at locus l,H=n
1
·n
2
·...·n
L
. The
frequency of haplotype hin generation tis denoted by P(t)
h.
Collectively, the vector of haplotype frequencies is Pt
(P(t)
1,...,P
(t)
H).
2.2 Idealizing the transmission cycle
The idealized transmission cycles allows to describe the
evolutionary genetics of malaria in generations of full
transmission cycles (Figure 2). In generation t, it is assumed
that all hosts are infected (or have a relapse or recrudescence) at
the same time. Moreover, host-vector transmission is also
synchronized. Inside the mosquito, parasites, which were
ingested by the mosquitoes, can recombine during one step of
sexual reproduction. This determines the distribution of
haplotypes in the mosquitoes’salivary glands of the parasite
(sporozoite) population in generation t+1.
2.2.1 Heterogeneity
Disease exposure and transmission intensities are
heterogeneous in endemic areas and change over time (e.g. in
the context of seasonal transmission) (Bousema et al., 2011;
Selvaraj et al., 2018). Moreover, hosts are heterogeneous
regarding their level of genetic and naturally acquired
immunity, number of co-morbidities, or the drug treatment
they receive to cure the infection (in case they receive any),
etc. (Hedrick, 2011;Gonzales et al., 2020). All of these factors can
be addressed by modeling hosts in different groups (strata). Let
G(t)
gbe the probability that a host, in which an infection occurs in
generation t, belongs to group g. Hence, G(t)
1+/+G(t)
S1 for
every generation t.
The number of groups, S, has to be chosen to capture the
features important to the specific application of the framework.
For instance, when considering drug resistance evolution, a
simple distinction would be between treated and untreated
infections, i.e., S= 2. In the case of P. knowlesi different
groups can model human and animal hosts. In the simplest
case one would have just two groups (S= 2), namely humans and
animals.
2.2.2 Relapses and recrudescence
Hosts are not modelled explicitly. This becomes relevant
when considering relapses (in P. vivax and P. ovale sp.) and
recrudescence in P. malariae. In the following we use relapse and
recrudescence synonymously, unless a distinction is necessary.
In the idealized transmission cycle, a relapse in generation t,
which occurs after a delay of dgenerations, is equivalent to a new
infection from the sporozoite population from dgenerations in
the past, i.e., from generation t−d. Let R(t)
dbe the probability that
an infection in generation tis a relapse, with a delay of d
generations, where R(t)
0is the probability of a new infection at
time t. Assuming the maximum possible delay is D, the relation
D
d0R(t)
d1 for all t, and 1 −R(t)
0is the probability that a relapse
occurs at time t.
The framework models the haplotype distribution in
generations of transmission cycles not in real-time. The higher
the transmission intensities, the more transmission cycles occur
per year. The choice of the distribution of relapses has to take this
into account (see Results section The effect of recrudescences and
relapses). Moreover, the timing of relapses depends on the
Plasmodium species (White, 2011).
Importantly, a host might have been exposed differently to
the disease in the past, i.e., the host might belong to different
groups in generations t−dand t. Let G(t−d,t)
g′,g be the probability
that a host, who belonged to group g′in generation t−d, belongs
to group gin generation t(d≥0). Marginalisation yields
G(t)
g
S
g′1
G(t−d,t)
g′,g (1)
for all t,d,g. Hence, the probability that a relapse occurs in
generation tin a host in group gafter a delay of dgenerations,
when he belonged to group g′, is given by
R(t)
dG(t−d,t)
g′,g .
2.3 Vector-host transmission and
multiplicity of infection
The presence of multiple genetically distinct parasite
haplotypes within an infection is frequently referred to as
multiplicity of infection (MOI) or complexity of infections
(COI) and considered important in malaria. The terms MOI
and COI are ambiguously defined in the literature (see
(Schneider et al., 2022) for a comprehensive review).
Although, it is unclear whether MOI is affecting the clinical
pathogenesis of malaria, or whether different parasite haplotypes
are competing within infections (intra-host competition), MOI
mediates the amount of meiotic recombination and scales with
transmission intensities (Pacheco et al., 2020) (see Figure 3).
Different parasite haplotypes can occur within an infection,
because they are 1) sequentially transmitted (during the course of
one disease episode) by different mosquitoes (super-infection); 2)
co-transmitted by one mosquito (co-infection); 3) mixed up with
parasites from previous infections by relapses or recrudescence.
Concerning models of MOI, the focus was mainly on super-
infections. More recently, the importance of co-infections is
being emphasized. Namely, more parasite genomics data is
being generated, which has enough resolution to study genetic
relatedness of parasites. Such data is appropriate for molecular
surveillance of transmission routes (Ndiaye et al., 2021). Formal
population-genetic frameworks to describe the evolutionary
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genetics of malaria that consider relapses do not exist.
Mathematical models describing relapses in P. vivax and P.
ovale sp. are limited to epidemiological models, e.g., the
compartmental model of (Chamchod and Beier, 2013), which
neglects parasite genetics. A population-genetic framework
applicable to all human-pathogenic malaria species has to be
flexible enough to accommodate super-infections, co-infections,
relapses, and recrudescence.
To set up the framework an infection is identified by a vector
m=(m
1
,...,m
H
), where m
h
is the number of times haplotype h
is infecting. Hence, m
h
=0orm
h
>0 if haplotype his absent or
present in the infection, respectively. The number m
h
accounts
for super-infections with the same haplotype. Moreover, it can be
interpreted as the “concentration”of haplotype hif several
haplotypes are co-infecting, etc.
Let Pr [m|t] be the probability of an infection with
configuration mgiven generation t. The infection might be a
new infection or a relapse. The probability of infection m, given it
occurs in generation t, when the host belongs to group g, and
given it is a relapse with a delay of dgenerations, when the host
belonged to group g′, is denoted by Pr [m|t−d,g′;t,g]. Hence,
the probability of infection moccurring in a host in group gin
generation t, which is a relapse from generation t−d, when the
host belonged to group g′,is
Pr m;t−d, g′;t, g

Pr m|t−d, g′;t, g

R(t)
dG(t−d,t)
g′,g .(2)
The conditional probability Pr [m|t−d,g′;t,g]reflects the model
of super- and co-infections. There are many possible models.
Super- and co-infections are both notoriously difficult to address.
Namely, knowledge about the vector dynamics and the
distribution of haplotype combinations in the mosquito
population must be known. This is a difficult task and
research on the topic is currently expanding, (cf. Nkhoma
et al., 2012;Wong et al., 2018;Zhu et al., 2019;Nkhoma
et al., 2020;Dia and Cheeseman, 2021;Neafsey et al., 2021).
FIGURE 3
Illustration of the relationship between inbreeding and MOI. Top: An infection with MOI = 1 (single-clone infection) leads only to recombination
between clones, i.e., effectively to no recombination. Bottom: Shown is a super-infection with four infective events (MOI = 4) and three different
haplotypes being transmitted (one haplotype is transmitted independently by two mosquitoes). Recombination between the illustrated haplotypes
leads to the creation of new haplotypes.
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2.3.1 A model for super-infections
Many approaches to estimate MOI or COI by Bayesian or
maximum-likelihood methods (e.g. (Hill and Babiker, 1995;
Stephens et al., 2001;Rastas et al., 2005;Li et al., 2007;Hastings
and Smith, 2008;Druet and Georges, 2010;Ross et al., 2012;Wigger
et al., 2013;Taylor et al., 2014;Galinsky et al., 2015;Ken-Dror and
Hastings, 2016;Schneider, 2018;Hashemi and Schneider, 2021)) are
based on a model, which assumes only super-infections, but no co-
infections. The number of super-infections mis referred to as
multiplicity of infection (MOI; see Figure 3).
Let M(t,g)
mbe the probability that a host belonging to group g
is super-infected exactly mtimes in generation t. This is a
probability distribution, hence

∞
m1
M(t,g)
m1(3)
for all tand g. At each infectious event, exactly one haplotype is
randomly drawn from the mosquito population, i.e., the
haplotype distribution P
t
. Hence, given MOI min generation
t, the infection m=(m
1
,... ,m
H
), which indicates how many
times haplotype hwas transmitted, follows a multinomial
distribution with parameters mand P
t
, i.e.,
Pr m|m;t
[]
m
m

Pm
t,(4)
where m
m
≔m!
H
h1mh!is a multinomial coefficient, and
Pm
t≔H
h1P(t)
hmh. Clearly, the constraint |m|≔H
h1mhm
must hold. If an infection is a relapse with a delay of d
generations, the haplotypes have to be drawn according to the
distribution P
t−d
.
Therefore, the probability of infection mgiven it has MOI
m=|m| and occurs in generation t, when the host belongs to
group g, from a relapse with a delay of dgenerations, when the
host belonged to group g′, is given by
Pr m,m|t−d, g′;t, g

M(t−d,g′)
m
m
m

Pm
t−d,(5)
where M(t−d,g′)
mis the probability of MOI min generation t−dof
a host in group g′. This model makes the expression (WHO,
2021b) much more explicit.
2.3.2 Choices for the distribution of super-
infections
The model (WHO, 2021b) becomes even more explicit for
specific choices of the distribution of MOI. A popular choice
emerges from the assumption of rare and independent infections,
namely that MOI is conditionally Poisson distributed (cf.
Schneider, 2021), i.e.,
M(t,g)
m1
exp λt,g

−1
λm
t,g
m!,(6)
where λ
t,g
>0 is the Poisson parameter of group gin generation t
and m=1,2,....
Another popular choice is the conditional negative-binomial
distribution. It is similar to the Poisson distribution but over-
dispersed (cf. 17).
2.4 The exo-erythrocytic and erythrocytic
cycles
Assume an infection subsumed by the vector mhaving MOI
m=|m|. Since all steps of reproduction are clonal inside the host,
it is not necessary to model the different parasite stages explicitly.
Rather, it suffices to model the change in haplotype frequencies
inside the host as a single step.
If the host belongs to group g, the ‘absolute’frequency of
haplotype his mh
mW(t,g)
m,h . Here, W(t,g)
m,h is the fitness in generation t
of haplotype hin infection mof a host belonging to group g.Itis
interpreted as the expected number of gametocyte descendants of
a single copy of haplotype hinfecting the host at the time a
mosquito takes her blood meal.
2.4.1 Host-vector transmission
Concerning host-vector transmission, a mosquito ingests a
fraction fof male and female gametocytes at her blood meal. The
gametocyte haplotypes ingested are assumed to be proportional
to the haplotype frequencies within the host. More precisely,
fmh
mW(t,g)
m,h male and female haplotype hare ingested from
infection min group g. (Note different fractions fcan also be
assumed for male and female gametocytes, reflecting an unequal
sex ratio.)
2.4.2 Sporogonic cycle
Recombination occurs immediately after the blood meal (see
Figure 1), and only parasites descending from the same host can
recombine (see Figure 3). Assuming the mosquito bite a host
from group gwith infection m, the probability that a male gamete
of haplotype hfertilizes a female i-gamete is the product of their
relative frequencies in the mosquito’s gut, i.e.,
fmh
mWt,g
()
m,h
fW t,g
()
m
·fmi
mWt,g
()
m,i
fW t,g
()
m
mhWt,g
()
m,h miWt,g
()
m,i
m2Wt,g
()
m2
,(7)
where
fW t,g
()
m≔f
H
j1
mj
mWt,g
()
m,j (8)
is the total amount of parasites in the mosquito’s gut. Therefore,
the absolute number of such matings is obtained by multiplying
the probability of the mating by the total amount of parasites, i.e.,
fA t,g
()
h,i (9)
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where
A(t,g)
m,h,i ≔mhWt,g
()
m,h miWt,g
()
m,i
m2Wt,g
()
m
. (10)
The absolute frequency of haplotype hin the population of
mosquitoes, which descends from infections with configuration
m, given 1) MOI m=|m|, 2) the infections occur in generation t,
3) in hosts in group g, which 4) are either novel infections (delay
d= 0) or relapses with a delay of dgenerations, is
Pr m|m;t−d;t, g


H
j,l1
fA(t,g)
m,j,lrjl→h

,(11)
where r(jl →h) is the probability that a mating between
gametes with haplotypes jand llead to offspring of
haplotype h.
The absolute number of haplotype hin the mosquito
population, which descend from hosts in group gwith MOI
m, is calculated from the theorem of total probability, i.e., by
‘averaging’over all possible infections mwith MOI m.
Incorporating all relapses it is given by
Ppg,m
()
ht
()
D
d0
R(t)
d
m:|m|m
Pr m|m;t−d;t, g

×
H
j,l1
fA(t,g)
m,j,lrjl→h

. (12)
If an infection in generation tis a relapse from generation t−d
the host might have belonged to a different group g′then. Noting,
that
Pr m|t−d;t, g


S
g′1
G(t−d,t)
g′,g Pr m|m;t−d, g′;t, g

(13)
equation (Su et al., 2019) can be rewritten as
Ppg,m
()
ht
()
D
d0
R(t)
d
S
g′1
G(t−d,t)
g′,g 
m:|m|m
Pr m|m;t−d, g′;t, g

×
H
j,l1
fA(t,g)
m,j,lrjl→h

.
(14)
2.5 Evolutionary dynamics
To determine the number of haplotypes hin generation t+
1, equation (Ngwa et al., 2016) has to be averaged over all
possible groups and values of MOI. Hence, the absolute
frequency of haplotype hin the next generation’s
sporozoite population is
Pp
ht+1
()
f
D
d0
R(t)
d
S
g,g′1
G(t−d,t)
g′,g 
∞
m1
×
m:|m|m
Pr m,m|t−d, g′;t, g

×
H
j,l1
A(t,g)
m,j,lrjl→h

. (15)
The relative frequency of haplotype hin the sporozoite
population in generation t+ 1 is hence
Pht+1
()
Pp
ht
()

H
i1
Pp
it
()
. (16)
The dynamics (Watson et al., 2021) are extremely flexible.
They allow to model, e.g., temporal changes in selection pressures
(for instance changing treatment policies in the context of drug-
resistance evolution, temporally varying transmission intensities,
intra-host competition of parasites, super- and co-infections,
relapses, recrudescences etc.). This however requires to specify
the model more explicitly.
Next, we show how this is done if only super-infections but
no co-infections are considered.
2.6 Evolutionary dynamics with super-
infections
We introduce a couple of simplifying assumptions, which
make the model more explicit. First, only super- but no co-
infections are assumed. I.e., the super-infection model (Lover
et al., 2018) applies and is substituted into (Schneider and
Escalante, 2013). Thus, (Schneider and Escalante, 2013), becomes
Pp
ht+1
()
f
D
d0
R(t)
d
S
g,g′1
G(t−d,t)
g′,g 
∞
m1
M(t−d,g′)
m
m:|m|mm
mPm
t−d
×
H
j,l1
A(t,g)
m,j,l rjl→h

.
(17)
3 Results
The framework is appropriate to investigate numerous
evolutionary-genetics aspects in malaria. It would be far too
comprehensive to exemplify the full flexibility. Hence, only
special cases are illustrated here. We assume that only super-
infections but no co-infections occur, i.e., the dynamics (Baird
et al., 2018) are assumed. First, we clarify the difference
between haplotype frequency and prevalence. Then we
focus on a simple model of drug resistance. Although it is
applicable to all malaria species, primarily it shall illustrate the
differences between P. falciparum and P. vivax, because there
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were no reports on drug resistance in any of the other species
(Tseha and Tyagi, 2021).
3.1 Frequency and prevalence
The evolutionary genetics of malaria are described as the
time-change in the frequency distribution of parasite haplotypes.
For instance, monitoring the frequencies of haplotypes, which
confer drug resistance is essential. However, concerning the
clinical pathogenesis, the occurrence of resistance-conferring
haplotypes in infections is more relevant. Due to super- and
co-infections the frequency of a haplotype h, i.e., its relative
abundance among sporozoites in the mosquito population does
not coincide with the probability that haplotype hoccurs in an
infection. The latter is referred to as the haplotype’s prevalence.
If only super-infections are considered, the prevalence of
haplotype hin generation t, denoted by q(t)
his derived in section
Prevalence in the Mathematical Appendix. It is given by
qt
()
h1−
D
d0
R(t)
d
S
g′1
G(t−d)
g′U(t−d)
g′1−P(t−d)
h

,(18)
where U(t−d)
g′(x)is the probability generating function of the
MOI distribution in group g′in generation t−d. This function
characterizes transmission in group g′in generation t−d. From
the above expression it is clear that prevalence depends on (i) the
frequency of haplotype h, (ii) the distributions of MOI in the
various groups, and (iii) the distribution of relapses/
recrudescence. If no relapses or recrudescences occur, as it is
the case for P. falciparum and P. knowlesi, the prevalence
simplifies to
qt
()
h1−
S
g1
G(t)
gU(t)
g1−P(t−d)
h

. (19)
Hence, for P. falciparum and P. knowlesi prevalence is
characterized by the haplotype frequency distribution in t, the
distribution of groups, and the MOI distributions in the groups.
We illustrate the effect of relapses on prevalence in a simple
example below.
3.2 Selection at a single locus without
intra-host competition
Assume drug resistance is determined by a single locus. This
is a reasonable assumption since often drug resistance is
determined mainly by mutations at one locus. For instance, in
P. falciparum resistance to chloroquine is determined by
mutations at the Pfcrt locus, while resistance artemisinin is
determined by mutations in the Kelch-13 propeller region
(Cui et al., 2015). The assumption is even justified in
sulfadoxine-pyrimethamine resistance, determined by the
Pfdhfr and Pfdhps loci, because mutations at the Pfdhfr locus
seem to have a much stronger effect (McCollum et al., 2012).
Assume nalleles A
1
,... ,A
n
are segregating at the selected
locus. The ndifferent alleles confer different levels of drug
resistance. All other alleles are assumed to be neutral. Thus,
the number of possible haplotypes, H, is a multiple of n, i.e., H=
nN. Hence, Nis the number of all possible haplotypes when the
resistance-conferring locus is disregarded. Let us assume that the
haplotypes are ordered such that haplotypes h=(a−1)N+1,...,
aN carry allele A
a
at the resistance-conferring locus. Therefore,
the frequency of allele A
a
at time t+ 1, denoted by p(t+1)
ais
given by
p(t)
a
aN
h(a−1)N+1
P(t)
h. (20)
Cumulatively, we denote the vector of allele frequencies in
generation tby p
t
.
Under the assumption of no intra-host competition of
parasites these dynamics can be made more explicit. In an
infection characterized by mof a host in group g, no intra-
host competition means that the fitness of an infecting haplotype
his independent of what other haplotypes are present in the
infection, i.e., it is independent of m, or formally
W(t,g)
m,h W(t,g)
h. (21)
Furthermore, because fitness is only determined by the
resistance-conferring locus, the fitness of haplotype hdepends
only on its allele at this locus. Let the fitness of haplotypes
carrying allele A
a
at the resistance-conferring locus be denoted by
w(t,g)
a,i.e,
w(t,g)
aW(t,g)
hW(t,g)
m,h for h
a−1
()
N+1,...,aN and for all m. (22)
Moreover, let the average fitness of allele A
a
in generation tbe
w(t)
a
S
g1
w(t,g)
aG(t)
g. (23)
As shown in the Mathematical Appendix the dynamics of the
allele frequencies are given by
p(t+1)
a
w(t)
a
D
d0
R(t)
dpt−d
()
a

n
b1
w(t)
b
D
d0
R(t)
dpt−d
()
b
. (24)
As in the case without relapses/recrudescence (cf. 17), these
dynamics are independent of the distribution of MOI. This
holds because no intra-host competition occurs and because
only super-infections are considered. Even without intra-host
competition the dynamics of the allele frequencies at the selected
locus might depend on MOI, depending on the assumed model
for co-infections; a general statement cannot be made.
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Further, the dynamics (Bousema et al., 2011) depend only on the
average fitnesses of the alleles w(t)
a. This implies that the stratification
of the host population into different groups does not need to be
modelled explicitly, when considering selection at a single locus.
Note that the average fitnesses can be scaled by any constant
without affecting the dynamics (Bousema et al., 2011). Hence, it
suffices to consider relative fitnesses, and fitness can be
normalized such that w(t)
11 in every generation.
3.2.1 The effect of recrudescences and relapses
In the dynamics of the allele frequencies (Bousema et al.,
2011) the effect of relapses or recrudescence is clearly visible. In
the case of no relapses or recrudescence, i.e., R(t)
d0 for d≥0 the
dynamics simplify to
p(t+1)
aw(t)
apt
()
a

n
b1
w(t)
bpt
()
b
. (25)
In this situation, the allele frequencies in generation t+ 1 are
solely determined by the fitnesses and the allele frequencies in
generation t. Once relapses or recrudescences are considered, the
allele frequencies in generation t+ 1, depend also on the allele
frequencies in previous generations. This is intuitively clear,
because relapses/recrudescence are equivalent to infections
from the sporozoite population from previous generations (see
Figure 2). Hence, relapses/recrudescence act as “seed banks”.
Intuitively, this will delay the evolutionary dynamics, because the
allele frequencies are averaged over several previous generations.
To further discuss the effect of relapses/recrudescence we
impose some additional assumptions. First, we assume that the
selective environment does not change over time, i.e., w(t)
awa
for all t. This is a reasonable assumption when considering drug
resistance evolution over a time period in which treatment
policies do not change. In this case, the change in allele
frequencies can be solved explicitly only in the absence of
relapses/recrudescence. Namely, the dynamics become
p(t+1)
awt+1
ap0
()
a

n
b1
wt+1
bp0
()
b
,(26)
where p(0)
aare the initial allele frequencies in generation t=0.
From these dynamics it follows that the average fitnesses w
a
can
be estimated from longitudinal data of allele frequencies by fitting
a straight-line regression (see 48, 17).
Once relapses/recrudescence are considered, the dynamics
can no longer be solved explicitly, but need to be calculated
recursively from the frequencies of the last D+ 1 generations,
i.e., they become
p(t+1)
a
wa
D
d0
R(t)
dpt−d
()
a

n
b1
wb
D
d0
R(t)
dpt−d
()
b
. (27)
Importantly, to be able to iterate these dynamics, initial
frequencies need to be known from Dgenerations in the past.
Hence, to calculate the frequencies in generation t= 1, initial
frequencies p(0)
a,p
(−1)
a,...,p
(−D)
aneed to be specified. Moreover,
the distribution R(t)
dneeds to be known. In practice, the
distribution of relapses might change over time. For instance,
changes in control policies impact malaria transmission and
hence the proportion of new infection in comparison to
relapses. If transmission intensities decrease, relapses amount
for a larger fraction of infections. Also the number of
transmission cycles during 1 year decrease. Because the
distribution of the time to relapse measured in years will not
change, the time distribution measured in units of transmission
cycles will change. In the simplest case the distribution of relapses
remains constant over time, i.e., R(t)
dRd, the change of allele
frequencies is given by
p(t+1)
a
wa
D
d0
Rdpt−d
()
a

n
b1
wb
D
d0
Rdpt−d
()
b
. (28)
Unfortunately, even if the distribution of relapses is constant, the
average fitnesses can no longer be estimated by a linear
regression.
The distribution of relapses depends crucially on the specific
parasite strain (White, 2011). Consider the following example of
drug-resistance evolution, with just two alleles: allele A
1
being the
drug sensitive wildtype and A
2
the mutant allele conferring drug
resistance. The mutant allele first occurs in generation t=0at
frequency p(0)
20.001. Let w
2
=1+s, where sis the selective
advantage of the drug resistant allele A
2
. We assume s= 0.1,
i.e., the fitness is increased by 10%, which is strong selection for
population-genetic processes, but reasonable for selection for
drug-resistance.
Regarding the distribution of relapses, we assume a situation
in which 1 year corresponds to 10 transmission cycles. Relapses
often occur in periodic patterns (White, 2011). We first assume a
pattern which resembles the relapse pattern described by
(Hankey et al., 1953) in temperate zones of Korea. Namely,
let vbe the probability that a malaria episode relapses, i.e., R
0
=
1−v. We assume the first relapse can occur after 10 transmission
cycles, and all further relapses after 4 further transmission cycles
for a maximum delay of D= 90. More precisely, Rdv
21 for d=
10, 14, 18, 22, ..., 90 and R
d
= 0 else. As a comparison we assume
a simple second pattern of relapses, in which relapses occur
4–50 generations after the initial infection with equal probability,
i.e., Rdv
43 for d=4,... , 50. Compared with the first pattern,
relapses occur more frequently and earlier.
The evolutionary dynamics are illustrated in Figure 4.
Without relapses v= 0, the resistance-conferring allele spreads
in approximately 110 generations, which corresponds to 11 years,
under the assumed number of 10 transmission cycles per year.
Relapses substantially slow down the spread of resistance. The
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reason is that relapses act like ‘seed banks’which retain the
frequency distribution of previous generations. For the first
pattern (Figure 4A), 5% relapses already substantially delay
the spread of resistance to about 400 generations or 40 years.
With 20% relapses, the frequency of the mutant allele is just 75%
after 1,000 generations corresponding to 100 years. For the
second pattern (Figure 4B), the results are qualitatively
similar, but relapses have a less profound effect, because they
occur with shorter delay after the original infection.
These results provide formal evidence that drug resistance
spreads faster in P. falciparum, where no relapses occur, than in
P. vivax, where relapses are common. In fact, while drug
resistance is a major concern in P. falciparum, it is less
common in P. vivax (Schneider and Escalante, 2013).
The pattern of relapses depends on 1) genetic factors
mediating the frequency of their occurrence; 2) transmission
intensities determining the number of malaria generations
(transmission cycles per year); 3) the fractions of new
infections and relapses; and4)treatmentpolicies.
Particularly, if a drug is partnered with primaquine (PQ) or
tafenoquine (TQ) for radical cure, the fraction of relapses
reduces, accelerating the spread of resistance to the primary
treatment. However, since PQ or TQ also act on gametocytes,
they prevent transmission and reduce the selective advantage of
drug resistance (cf. 23).
3.2.2 Prevalence
Next consider the prevalences corresponding to the
evolutionary dynamics illustrated in Figure 4. The
evolutionary dynamics are determined by the average fitnesses
across the groups of hosts and the distribution of relapses.
Consequently, it was not necessary to specify the groups
explicitly. However, prevalence given by (Collins and Jeffery,
2007) depends on the generating functions of MOI in the
different groups. In the simplest case, which we consider here,
the whole population consists of only one group (S= 1).
Furthermore, we assume that the MOI distribution does not
change over time, and follows a conditional Poisson distribution
(cf. Eq. (6)) with parameter λ. The generating function of this
distribution is given by
Ux
()
exp λx
()
−1
exp λx
()
−1(29)
(cf. 17).
The prevalence of the resistance-conferring allele is obtained
from (Collins and Jeffery, 2007) by assuming that haplotypes are
characterized by a single locus. Hence,
qt
()
21−
D
d0
RdU1−pt−d
()
2


D
d0
Rd
1−exp −λpt−d
()
2

1−exp −λ
(). (30)
The prevalences corresponding to the dynamics illustrated in
Figure 4A, are depicted in Figure 5, assuming different values of
the Poisson parameter λ, corresponding to different transmission
intensities.
The case λ= 0, implies that only ‘single-infection’(one
infective event) occurs, in which case prevalence and
frequency coincide. As shown in (Schneider, 2021) prevalence
always exceeds frequency in the case in which no relapses occur
(Figures 5A,F,K). This is intuitive, because the likelihood to
observe a parasite variant in an infection increases as the
average number of super-infections increase. If transmission
intensities are intermediate to high (λ≥1), prevalence is
considerably higher than frequency (Figure 5F). If the
frequency of the resistance-conferring allele is small, the
difference between frequency and prevalence is small in
absolute terms, but high in relative terms (compare Figure 5F
with Figure 5K).
If relapses occur, the pattern is similar, however, prevalence
can be lower than frequency (see Figures 5F–J). The reason is that
prevalence is also determined by the frequency distribution of
past generations. This occurs only if the average number of
super-infections is small (λslightly larger than 0) and is
increasingly pronounced if relapses are more frequent. In
general, the difference between frequency and prevalence
becomes smaller in absolute and relative terms as the fraction
FIGURE 4
Effect of relapses on the evolutionary dynamics. Shown is the frequency of the resistance-conferring allele as a function of time assuming
different proportions, vof relapses (colors) for the first (A) and second (B) patterns of relapses.
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FIGURE 5
Prevalence. Panels (A–E) show the prevalence of the resistance-conferring allele corresponding to the dynamics in Figure 4A for different
values of the Poisson parameter λ(colors). Panels (A–E) correspond to the dynamics with 0%, 5%, 10%, 15%, and 20% relapses, respectively. Panels
(F–J) show the corresponding difference between prevalence and frequency, and panels (K–O) show the corresponding relative difference
(prevalence minus frequency divided by frequency) in percent.
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of relapses increase. If this fraction is high (v= 0.15 or v= 0.2) the
particular pattern of relapses leads to oscillations in the relative
difference between prevalence and frequency, if the frequency of
the resistance-conferring allele is low (see Figures 5N,O).
4 Discussion
We introduced a general framework to model evolutionary-
genetic processes in malaria, which is flexible enough to capture the
characteristics of all human-pathogenic Plasmodium species. Such a
framework is justified since standard population-genetic theory can
only be approximately applied to malaria. The reason is rooted in the
malaria transmission cycle, which involves one step of sexual
reproduction in the mosquito vectors. A high degree of selfing
occurs during this step, because only parasites which descend
fromthesamehumanhostcanrecombine(cf. Figure 3). The
framework extends the one introduced in (Schneider and Kim,
2010;Schneider and Kim, 2011;Schneider, 2021), which is only
applicable to P. falciparum, because it ignores relapses from dormant
liver stages as they occur in P. vivax and P. ovale sp., and
recrudescence form prolonged blood stage parasites as they occur
in P. malariae. These previously widely neglected species are resilient
because of relapses and recrudescence, and hence are gaining more
importance in the context of malaria eradication. We demonstrated
the importance of relapses/recrudescence by contrasting drug
resistance-evolution in P. vivax and P. falciparum.
The necessity to extend the population-genetic framework
towardothermalariaspeciesisclearlyjustified by the results
presented here. Even in the simplest case of resistance being
determined by a single locus, relapses have a profound effect on
the evolutionary dynamics, when assuming the same hypothetical
drug pressure in both species. Namely, relapses substantially delay
the spread of resistance, because they are equivalent—at least in the
idealization of the model—to infections with regard to past parasite
frequency distributions. In other words, relapses act as seed banks.
Dormancy by seed banks is known in evolutionary biology as a bet-
hedging strategy that allows organisms to survive through sub-
optimal conditions (Shoemaker and Lennon, 2018)—in the case of
malaria the absence of the vector. Seed banks are also known to slow
down evolutionary processes and influence recombination (Živković
and Tellier, 2012;Koopmann et al., 2017;Tellier, 2019). This is no
exception in malaria. Although exploring the effect of relapses/
recrudescence on recombination was beyond the scope of this work,
the effect is rather obvious. Because relapses/recrudescence slow
down the evolutionary dynamics, more genetic variation is
maintained, leading to a higher level of recombination. In fact, in
P. vivax higher levels of genetic variations than in P. falciparum are a
common empirical observation (e.g. (Pacheco et al., 2020)).
Our results have to be understood in a qualitative rather than a
quantitative context. Namely, the pattern of relapses have a substantial
influence on the evolutionary dynamics. Hence, for adequately predict
the spread of resistance, good empirical estimates on the pattern of
relapses are necessary. However, empirically distinguishing re-
infections (consecutive independent infectious), recrudescence (a
rebound of parasitaemia due to incomplete clearance of
merozoites), and relapses are notoriously difficult. With more
advanced molecular methods becoming available to produce deep-
sequencing data (e.g. (Zhong et al., 2018;Gruenberg et al., 2019)),
heuristic methods to distinguish recrudescence from reinfections have
been proposed (Lin et al., 2015). Also haplotype-based statistical
models have been proposed (e.g. (Plucinski et al., 2015)). In principle
the framework here can be used to further develop statistical methods
to distinguish reinfections from relapses.
To obtain quantitative predictions it is also important to estimate
other model parameters. In the context of drug resistance, this
includes fitness parameters, metabolic costs for resistance, and the
proportion of asymptomatic or untreated infections. The latter can be
achieved by routine diagnostics using reliable methods such as ultra-
sensitive PCR (e.g. (Gruenberg et al., 2020)). However, also the
transmission potential, determined by the abundance of gametocytes
has to be determined (cf. 9). Selection parameters of drug-resistant
haplotypes can be determined from longitudinal molecular data by a
linear regressions in P. falciparum (McCollum et al., 2012;Schneider,
2021). Disentangling the fitness parameters into metabolic costs and
selective advantages of resistance is more difficult. Namely, costs and
selectiveadvantagesasfoundinvitro studies (cf. Cortese and Plowe,
1998) do not linearly scale with in vivo observations. In principle,
costs can be achieved by contrasting different populations with
different drug usage. Comparing such results with in vitro studies
helps to identify the functional relationship between in vitro
measurements and in vivo observations. Notably, fitness estimates
from a linear regression apply mainly to P. falciparum. For other
malaria species the estimates have to be adapted to the evolutionary
dynamics which account for relapses/recrudescence.
Note that the application to modelling drug resistance here
had only the purpose of contrasting the absence and presence of
relapses. Therefore, only a simplistic model was assumed for drug
resistance, i.e., resistance was assumed to be determined by a
single biallelic locus. The examples here did not exhibit the full
flexibility of the model. If drug resistance occurs in a stepwise
fashion as it is found in sulfadoxine-pyrimethamine resistant P.
falciparum haplotypes (Cortese and Plowe, 1998), where
resistance is caused by mutations at several codons in the
Pfdhfr and Pfdhps loci. To capture this situations, resistance-
conferring haplotypes have to be modelled by two mulltiallelic
loci, where each two-locus haplotype is associated with its own
metabolic costs and fitness advantage. Moreover, the mutation
haplotypes have to be introduced into the model at different time
points. A simple example can be found in (Schneider, 2021).
Relapses are irrelevant in P. falciparum, and recrudescences
can be neglected, because they occur shortly after the initial
infection and do not need to be modeled explicitly. Nevertheless,
if transmission intensities are high, which is mainly relevant for
P. falciparum, the assumption of non-overlapping generations
(transmission cycles) are questionable. In the extended
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framework, relapses can be reinterpreted to mimic overlapping
generations. This explains, at least partially, why drug resistance
in P. falciparum does not necessarily spread first in areas of high
transmission (as they occur in Africa) with many more
transmission cycles per year.
Reinterpreting relapses in the framework is also important when
applied to P. knowlesi, which is primarily pathogenic to non-human
primates, but became the dominant human-pathogenic malaria
species in some endemic areas (Sutherland, 2016). The zoonotic
animal-host reservoir renders P. knowlesi resilient. Different
transmission dynamics between humans and animal hosts can
mediate the duration of a transmission cycle. If the number of
transmission cycles per year differs among human and non-human
primate hosts, this discrepancy can be compensated by modeling
overlapping generations by relapses.
We also discussed the differences of frequency and prevalence of
parasite haplotypes. The former is the relative abundance of a
haplotype in the parasite population, the latter the likelihood that
the haplotype occurs in an infection. Studying the haplotype
frequency distribution over time is the aim of evolutionary
genetics. From a clinical or epidemiological point of view,
prevalence is more relevant. The latter is determined by the
haplotype frequency distribution and the distribution of super-
or co-infections. This was already emphasized in the context of
seasonal malaria transmission in (Schneider, 2021)forP. falciparum.
Itwasshownthattheprevalenceofahaplotypealwaysexceedsits
frequency. This changes if relapses/recrudescence occur and was
exemplified here by the hypothetical dynamics of drug-resistance
evolution.
The applications of the framework introduced here are
manifold. For instance, in the context of drug resistance, the
framework allows to investigate the evolution of multi-drug
resistance determined by several loci and changing drug-
treatment policies. Also patterns of selection, e.g., genetic
hitchhiking, can be studied using this framework. The illustrated
applications were only under the simplest assumptions, e.g., of no
intra-host competition and super- but no co-infections.
Intra-host competition plays an important role in the spread of
HRP2/3 gene deletions associated with false-negative malaria rapid
diagnostic tests (RDTs) (Gamboa et al., 2010). Namely, if the
treatment guidelines require to verify suspected infections by
RDTs before treatment with artemisinin combination therapies
(ACTs), as recommended by the WHO (World Health
Organization, 2017), false-negative results can lead to delayed
treatment. Similarly intra-host competition seems relevant when
considering selection on merozoite surface proteins (Goh et al., 2021).
Intra-host dynamics enter the model via the definition of fitness.
It is not necessary to define an evolutionary-genetic model which
captures two timescales, the evolutionary dynamics in terms of
generations of transmission cycles, and the timescale of an infectious
episode in the same model, as it was done, e.g., in (Kim et al., 2014).
Rather, the framework can be used in a multi-scale model, which
takes input from a separate intra-host model.
Similarly, the framework does not require to model the mosquito
dynamics explicitly. They rather enter via the distribution of super-
and co-infections. Considering only super-infections has the
conceptional advantage, that it is a well-defined model. It is
frequently used in statistical approaches to estimate haplotype
frequency distributions and MOI (cf. e.g. Hill and Babiker, 1995;
Stephens et al., 2001;Li et al., 2007;Hastings and Smith, 2008;Wigger
et al., 2013;Schneider, 2018;Hashemi and Schneider, 2021). Ignoring
co-infectionsisjustified if the distribution of haplotypes in the
mosquitoes is uncorrelated or when considering only few loci.
However, if one aims to include genetic relatedness, it is
important to specify a model for co-infections. This becomes
increasingly popular as more high-quality genomic data is
becoming available in malaria, which has enough resolution to
study genetic relatedness (cf. Nkhoma et al., 2012;Wong et al.,
2018;Zhu et al., 2019;Nkhoma et al., 2020;Dia and Cheeseman,
2021;Neafsey et al., 2021).
Although the framework is very general, it also has several
limitations. First, it ignores mutations. This is not a strong
restriction, because in many applications one is interested in
de novo mutations which occur at discrete time points. This is
captured by the model, by introducing new haplotypes
(i.e., extending the model) at certain times. However, constant
mutation rates, e.g., to study mutation-selection balance, can be
easily introduced. Another limitation is the deterministic nature
of the framework. When aiming to study stochastic effects such
as genetic drift, it is rather straightforward to develop a stochastic
version of the framework. Third, the model ignores mitotic
recombination during merozoite production inside the host.
This plays an important role in some applications, particularly
in the structural rearrangement of Var genes (Claessens et al.,
2014). These hypervariable genes are responsible to
generate important antigen profiles for parasite-host
interactions (Warimwe et al., 2009). In any case the
framework introduced here allows studying manifold
evolutionary-genetic aspects of malaria. Importantly, it allows
us to specify benchmark scenarios. More empirical evidence is
required to refine relevant parametrizations of the framework.
Data availability statement
The original contributions presented in the study are
included in the article/Supplementary Material, further
inquiries can be directed to the corresponding author.
Author contributions
KS conceptualized the work, developed the mathematical
model, performed the mathematical analysis, produced all
figures, and wrote the manuscript. CS assisted to
conceptualize the work and wrote the manuscript.
Frontiers in Genetics frontiersin.org14
Schneider and Salas 10.3389/fgene.2022.1030463

Funding
This study was funded by the Armed Forces Health Surveillance
Division (AFHSD), Global Emerging Infections Surveillance (GEIS)
Branch, ProMIS ID P0082_22_N6. This work was also supported by
grantsoftheGermanAcademicExchange(DAAD;https://www.
daad.de/de/; Project-ID 57417782, Project-ID: 57599539), the
Sächsisches Staatsministerium für Wissenschaft, Kultur und
Tourismus and Sächsische Aufbaubank—Förderbank (SMWK-
SAB; https://www.smwk.sachsen.de/;https://www.sab.sachsen.de/;
project “Innovationsvorhaben zur Profilschärfung an Hochschulen
für angewandte Wissenschaften”, Project-ID 100257255; project
“Innovationsvorhaben zur Profilschärfung 2022”,Project-ID:
100613388), the Federal Ministry of Education and Research
(BMBF) and the DLR (Project-ID 01DQ20002; https://www.
bmbf.de/;https://www.dlr.de/).
Acknowledgments
The authors gratefully acknowledge the constructive
comments of the two reviewers and the editor.
Conflict of interest
The authors declare that the research was conducted in the
absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Publisher’s note
All claims expressed in this article are solely those of the
authors and do not necessarily represent those of their affiliated
organizations, or those of the publisher, the editors and the
reviewers. Any product that may be evaluated in this article, or
claim that may be made by its manufacturer, is not guaranteed or
endorsed by the publisher.
Supplementary material
The Supplementary Material for this article can be found
online at: https://www.frontiersin.org/articles/10.3389/fgene.
2022.1030463/full#supplementary-material
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