Probabilistic models for pathogen evolution within and between hosts

Shishi Luo UC Berkeley

Pathogen evolution

• Influenza

• HIV

• General two-level selection

Antigenic drift of influenza (with K Koelle, JC Mattingly, MC Reed)

…

Vaccinate

…

Viral dynamics and evolutionPassage experiments

initial infection mutation &

replication

naive

vaccinated

Within-host influenza viral dynamics (colors are different studies)

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

Results: Within hosts

Increasing 

immunity to 

resident strain 

Probability 

of immune 

escape 

Par6ally 

immune hosts 

drive 

influenza 

evolu6on 

Simulations

Analytical formula

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

Broadly neutralizing HIV antibodies (with A Perelson)

Data from UNAIDS Global fact sheet 2012

Kwong & Mascola Immunity 2012

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

‘Bitstring’ model

Conserved region is less accessible

Binding properties

Infection with single strain

Vaccine strategiesStrains selected uniformly at random Strains are nearest neighbor mutants

Bars show +/-1 SE

Multilevel selection in host-pathogen systems (with JC Mattingly)

• myxomatosis-rabbit

• human papillomavirus

• malaria

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

0 1 2 3

Recall: blue types beneficial at group level

• 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

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

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

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

� = 1.5 � = 4

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

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

�