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Probabilistic models for pathogen evolution within and between hosts
Shishi Luo UC Berkeley
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Pathogen evolution
• Influenza
• HIV
• General two-level selection
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Antigenic drift of influenza (with K Koelle, JC Mattingly, MC Reed)
…
Vaccinate
…
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Viral dynamics and evolutionPassage experiments
initial infection mutation &
replication
naive
vaccinated
Within-host influenza viral dynamics (colors are different studies)
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ODEs with Poisson mutation
˙C = ��CVr
˙Vr = �CVr � �rVr
Vr(0) = V0 C(0) = C0
mutations occur at rate µVr(t)
��CVm
˙Vm = �CVm � �mVm
Viral load
(×109)/ Target
cells
(×2×108)
0
1
2
0 4 8
16
0
Time since infec=on (days)
0 4 8
105
0
Viral load
(×109)/ Target
cells
(×2×108)
0
1
2
Luo, Reed, Mattingly, Koelle Interface 2012
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Results: Within hosts
Increasing
immunity to
resident strain
Probability
of immune
escape
Par6ally
immune hosts
drive
influenza
evolu6on
Simulations
Analytical formula
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Results: Population level
2 4 8 6 0
0.2
0.4
Cluster persistence (yrs)
Proportion of clusters
(for influenza) Probability density
5 10 15
Time until escape occurs (yrs)
×10-2
4
0
8
Koelle et al Epidemics 2009
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Broadly neutralizing HIV antibodies (with A Perelson)
Data from UNAIDS Global fact sheet 2012
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Kwong & Mascola Immunity 2012
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Broadly neutralizing HIV antibodies are highly mutated
Antibodies in a single lineage in patient CH505
Data from: Wrammert et al J Exp Med 2011, Kwong & Mascola Immunity 2012, hiv.lanl.gov
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‘Bitstring’ model
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Conserved region is less accessible
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Binding properties
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Infection with single strain
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Vaccine strategiesStrains selected uniformly at random Strains are nearest neighbor mutants
Bars show +/-1 SE
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Multilevel selection in host-pathogen systems (with JC Mattingly)
• myxomatosis-rabbit
• human papillomavirus
• malaria
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Suppose:
•
•
!
•
!
•
!
m groups, n individuals per group
red beats blue in each group,
1 + s vs 1
bluer groups beat redder groups,
1 + r kn , were
kn is fraction of blue
w = rate of group-level events
rate of individual-level events
Luo J Theor Biol 2014
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0 1 2 3
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Recall: blue types beneficial at group level
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• Let Xjt be the position of ball j on the (rescaled) 1-d lattice {0, 1
n , . . . , 1}at time t.
• Define the measure-valued process:
µm,nt =
1m
mX
j=1
�Xjt
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µm,nt has transition rate matrix R = R1 + wR2 where:
R1
⇣v, v +
1m (� j
n� � i
n)
⌘=
8<
:
mv( in )i
�1� i
n
�(1 + s) if j = i� 1, i < n
mv( in )i
�1� i
n
�if j = i+ 1, i > 0
0 otherwise
R2
⇣v, v +
1m (� j
n� � i
n)
⌘= mv( i
n )v(jn )(1 + r j
n )
s: individual-level selection, r: group-level selection, w: relative rate of events, m: number of
groups, n: number of individuals in each group
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The deterministic limit of µ
m,n
t
:
@
@t
µ(t, x) =s
@
@x
[x(1� x)µ(t, x)] + wrµ(t, x)
x�
Z 1
0yµ(t, y)dy
�
The stochastic (Fleming-Viot) limit:
@
t
⌫
t
= �@
x
[x(1� x)⌫
t
] + @
xx
[x(1� x)⌫
t
]
+ w↵⇢⌫
t
·x�
Z 1
0y⌫
t
(y)dy
�+ w↵
p2⌫
t
(1� ⌫
t
)
˙
W
t
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Long-term behavior of PDE
Suppose µ0 is a density and k
⇤is such that
µ
(k)0 (1) = 0 for all integers k < k
⇤and µ
(k⇤)0 (1) 6= 0
If �� 1 > k
⇤,
µ(t, x) ! Beta(�� k
⇤ � 1, k
⇤+ 1) as t ! 1
If �� 1 k
⇤,
µ(t, x) ! �0(x) as t ! 1
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� = 1.5 � = 4
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Properties of particle systemRelative rate of replication acts like group-level selection
@
@t
µ(t, x) =s
@
@x
[x(1� x)µ(t, x)] + wrµ(t, x)
x�
Z 1
0yµ(t, y)dy
�
� := wrs
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What if m and n are finite?
@
@t
µ(t, x) =s
@
@x
[x(1� x)µ(t, x)] + wrµ(t, x)
x�
Z 1
0yµ(t, y)dy
�
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