Mathematical models on Malaria with multiple
strains of pathogens
Yanyu Xiao
Department of Mathematics
University of Miami
CTW: From Within Host Dynamics to the Epidemiology of Infectious Disease
MBI, Columbus, Ohio
Outline
Background
Within-host Level
Between-host Level
Discussion and Future Work
Geographic Distribution of Malaria
WHO, World Malaria Report 2010, December 2010.
The Pathogen of Malaria
I
Malaria is a mosquito-borne infectious disease caused by
Malaria parasites.
I
Malaria parasites are members of eukaryotic protists of the
genus Plasmodium.
I
In general, there are five kinds of plasmodiums associated
with human malaria infections.
Multiple Strains
Different Characters of Multiple Strains
Some comparative characters of the five human malaria
parasites:
P. falciparum
P.vivax
P.ovale
P.malaria
P. knowlesi
5.5
8
9
14-15
8-9
30 000
10 000
15 000
15 000
48
48
50
72
10
8-9
12-14
14-15
Duration of primary exoerythrocytlc
cycle (days)
Number of exoerythrocytlc
merozoites
Duration of erythrocytic
cycle (hours)
Duration of mosquito cycle
at 27◦ C (days)
(Source: http://www.malariasite.com/malaria/MalarialParasite.htm)
24
Multiple Strains
The Facts
I
Newly transmitted P. falciparum infections were
suppressing patient infections (either new or latent) with P.
vivax. —- K. Maitland, et al. (Parastitol Today 1997)
I
On the Thai-Burma border, pregnant women whose first
attack of malaria during pregnancy was caused by P. vivax
had a significantly lower risk of developing P. falciparum
later in the pregnancy. —- M. Mayxay, et al. (Trend
Parasitol 2004)
I
Authors have detected ..., including the co-occurrence of all
4 species in populations in Madagascar and New Guinea.
—- F. E. McKenzie and W. H. Bossert (J Parasitol 1997)
The Facts
Another fact:
I
There is no obvious cross-immunity between two species.
—- S.L. Hoffman (J Infect Dis 2002), K. Jangpatarapongsa
(PLoS One 2012)
This work answers the question by using mathematical model.
To this end, we need model
I
at within-host level, and
I
at population level.
Parasites Life Cycle
Single Strain Within-host Level
l
T
d
k
T*
m( p )
p
VI
VM
d1
VM
d1
ec
(1- e)c
d1
Ṫ
= λ − dT − kVM T ,
Ṫ ∗
= kVM T − µ(p)T ∗ ,
V̇I
= pT ∗ − d1 VI − cVI ,
V̇M
V̄˙
= cVI − d1 VM ,
M
= (1 − )cVI .
Single Strain Within-host Level
Basic reproduction number (the number of secondary cases
one case generates on average over the course of its infectious
period, in an otherwise uninfected population):
Z ∞
λk c
pe−µ(p)a da
R0 =
N, where N =
d(d1 + c)d1
0
I
If R0 < 1, the parasites will be cleaned up in the host cells;
I
if R0 > 1, the parasites will establish a stable steady state
inside of the host cells globally.
Double Strains Within-host Level
d1
m( p1 )
T1*
l
k1
p1
V I1
e1 c1
VM 1
d1
T
d
k2
T2*
m( p2 )
p2
VI 2
e 2 c2
VM 2
d2
d2
Ṫ
Ṫ1∗
Ṫ2∗
V̇I1
V̇I2
V̇M1
V̇M2
=
=
=
=
=
=
=
λ − dT − k1 VM1 T − k2 VM2 T ,
k1 VM1 T − µ(p1 )T1∗ ,
k2 VM2 T − µ(p2 )T2∗ ,
p1 T1∗ − d1 VI1 − c1 VI1 ,
p2 T2∗ − d2 VI2 − c2 VI2 ,
1 c1 VI1 − d1 VM1 ,
2 c2 VI2 − d2 VM2 .
Double Strains Within-host Level
The basic reproduction number R0 = maxi (R1 , R2 ):
Ri =
λki ci pi
.
dµ(pi )(di + ci )di
Theorem
I
If R0 < 1, the infection free equilibrium E0 is G-A-S.
I
If R0 > 1,
(i) If R1 > 1, and R2 < R1 , E1 exists and is G-A-S.
(ii) If R2 > 1, and R1 < R2 , E2 exists and is G-A-S.
(iii) If R1 = R2 > 1, there are infinitely many co-infection
equilibria.
where E1 and E2 are boundary equilibrium for species 1 and 2,
respectively.
Principle of Competitive Exclusion, Hardin science 1960; Iggidr et. al. SIAP 2006;
Double Strains Within-host Level
Double Strains Within-host Level
Double Strains Within-host Level
Single Strain Between-host Level
SH0 = bH NH − dH SH − ac1
SH
IM + βRH ,
NH
SH
IM − dH IH − γIH ,
NH
RH0 = γIH − dH RH − βRH ,
IH0 = ac1
0
SM
= bM NM − dM SM − ac2 SM
0
IM
= ac2 SM
IH
− dM IM .
NH
IH
,
NH
Single Strain Between-host Level
Set n = NNMH and nondimensionalize the system, we have the
basic reproduction number:
s
a 2 c1 c2 n
R0 =
dM (dH + γ)
The stability of disease free equilibrium (DFE)
E0 = (1, 0, 0, 1, 0) is fully determined by R0 :
Theorem (Stability)
If R0 < 1, E0 is G-A-S;
if R0 > 1, it is unstable.
Single Strain Between-host Level
When R0 > 1, there is a unique endemic equilibrium (EE)
∗ , I ∗ ), and
E ∗ = (SH∗ , IH∗ , RH∗ , SM
M
Theorem (Stability)
Assume R0 > 1, the EE E ∗ is G-A-S, provided that
dH + dM − max {−β, β − γ} > 0.
Double Strains Between-host Level
bH
S
b1
dM
ae
11
I
b2
H
dH
I
M1
I
g1
dH
H1
dH
ae
R2
I
M2
12
M2
22
21
H1
ae
ae
ae
R
dM
I
dM
S
H2
R
g2
dH
dH
M
bM
I
H2
M1
ae
R1
Double Strains Between-host Level




























SH0
= bH NH − dH SH −ae11 NSHH IM1 − ae12 NSHH IM2 + β1 RH1 + β2 RH2 ,
0
IH1
= ae11 NSHH IM1 − dH IH1 − γ1 IH1 +aeR1 RNH2
IM1 ,
H
0
RH1
=
0
IH2
= ae12 NSHH IM2 − dH IH2 − γ2 IH2 +aeR2 RNH1
IM2 ,
H

0


RH2







0

SM








0

IM1






 0
IM2
=
IM2 − dH RH1 − β1 RH1 ,
γ1 IH1 − aeR2 RNH1
H
γ2 IH2 − aeR1 RNH2
IM1 − dH RH2 − β2 RH2 ,
H
= bM NM − dM SM − ae21 SM NIH1H − ae22 SM NIH2H ,
=
ae21 SM NIH1H − dM IM1 ,
=
ae22 SM NIH2H − dM IM2 .
Double Strains Between-host Level
Rescale the system
 0
SH
= dH − dH SH − ae11 nSH IM1 − ae12 nSH IM2 + β1 RH1 + β2 RH2 ,







0

IH1
= ae11 nSH IM1 − dH IH1 − γ1 IH1 + aeR1 nRH2 IM1 ,







0
 RH1
= γ1 IH1 − aeR2 nRH1 IM2 − dH RH1 − β1 RH1 ,







0

= ae12 nSH IM2 − dH IH2 − γ2 IH2 + aeR2 nRH1 IM2 ,
 IH2

0

RH2







0


SM






0


IM1





 0
IM2
=
γ2 IH2 − aeR1 nRH2 IM1 − dH RH2 − β2 RH2 ,
=
dM − dM SM − ae21 SM IH1 − ae22 SM IH2 ,
=
ae21 SM IH1 − dM IM1 ,
=
ae22 SM IH2 − dM IM2 .
where n =
NM
NH
is the number of mosquitoes per person.
Double Strains Between-host Level
The basic reproduction number for species i in the absence of
species j, j 6= i is:
R̄i =
a2 e1i e2i n
, i = 1, 2.
dM (dH + γi )
Further,
R̄0 = max R̄1 , R¯2 ,
The system has a DFE Ē0 = (1, 0, 0, 0, 0, 1, 0, 0).
Theorem (Stability)
If R̄0 < 1, Ē0 is G-A-S;
if R̄0 > 1, Ē0 becomes unstable.
Double Strains Between-host Level
When R̄i > 1, i = 1, 2, there are two boundary equilibria:
I
∗ , R ∗ , 0, 0, S ∗ , I ∗ , 0).
If R̄1 > 1, Ē1 = (SH∗ , IH1
H1
M M1
I
∗∗ , R ∗∗ , S ∗∗ , 0, I ∗∗ ).
If R̄2 > 1, Ē2 = (SH∗∗ , 0, 0, IH2
H2
M
M2
The stabilities of Ē1 and Ē2 are not simply decided by
R̄i , i = 1, 2.
Double Strains Between-host Level
Define R̄ji as the species i-mediated reproduction number for
species j by
R̄21 =
∗ +a2 e e nS ∗ R ∗
a2 e12 e22 nSH∗ SM
22 R2
M H1
,
dM (dH +γ2 )
R̄12 =
∗∗ +a2 e e nS ∗∗ R ∗∗
a2 e11 e21 nSH∗∗ SM
21 R1
M
H2
.
dM (dH +γ1 )
I
R̄ij measures the number of secondary infections caused
by an individual infected by species i, assuming the
species j has been settled at Ēj .
I
R̄ji can be considered as the threshold parameter for
invasion of species j to residence species i.
Double Strains Between-host Level
R̄ji can be considered as the threshold parameter for invasion
of species j to residence species i.
Theorem (Stability)
I
(i) If R̄1 > 1, R̄21 < 1 and
(a) dH + dM − max (−β1 , β1 − γ1 ) > 0, then Ē1 is L-A-S;
I
(ii) If R̄2 > 1, R̄12 < 1 and
(b) dH + dM − max (−β2 , β2 − γ2 ) > 0, then Ē2 is L-A-S.
Double Strains Between-host Level
Theorem (Persistence)
I
Species 1 is uniformly persistent if
(C1) R̄1 > 1 and R̄2 < 1; or (C2) R̄2 > 1, R̄12 > 1 and (b)
exists.
I
Species 2 is uniformly persistent if
(C3) R̄2 > 1 and R̄1 < 1; or (C4) R̄1 > 1, R̄21 > 1 and (a)
exists.
Double Strains Between-host Level
Theorem (Persistence)
If one of the three holds,
(i) R̄1 > 1, R̄2 < 1, R̄21 > 1 and (b);
(ii) R̄2 > 1, R̄1 < 1, R̄12 > 1 and (a); or
(iii) R̄1 > 1, R̄2 > 1, R̄12 > 1, R̄21 > 1 and (a), (b) hold;
both species are uniformly persistent.
Double Strains Between-host Level
Double Strains Between-host Level
Conclusions
We modeled the transmission of Malaria in both within- and
between- host level.
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At within-host level: co-infection (super-infection) is
generically impossible (unless R1 = R2 > 1). Parasites will
compete with each other until only one species survives.
I
At population level: co-existence of two species in a region
is possible, as they not only compete but also benefit each
other!
Remark: Both within- and between- host level models can be
extended to scenarios with more than two strains, but conditions are
more compicated at between-host level.
Future Work
I
Explore the special case,’R1 = R2 ’, for the within-host
model (super-infection);
I
More strains of pathogens involved;
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Disease latency within host and vector;
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Spatial impacts.
Some References
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C. Castillo-Chavez and H. R. Thieme, Asymptotically
autonomous epidemic models, in Mathematical Population
Dynamics: Analysis of Heterogeneity, I. Theory of Epidemics, O.
Arino, D. E. Axelrod, and M. Kimmel, eds., Wuerz, Winnepeg,
Canada, 1995, pp. 33-50.
I
P. van den Driessche and J. Watmough, Reproduction numbers
and sub-threshold endemic equilibria for compartmental models
of disease transmission, Math. Biosci., 180 (2002), pp. 29-48.
I
A. Iggidr, J.C. Kamgang, G. Sallet, and J.J. Tewa, Global
analysis of new malaria intrahost models with a competitive
exclusion principle, SIAM J. Appl. Math., 67 (2006), pp. 260-278.
I
M. Y. Li and J. S. Muldowney, A geometric approach to
global-stability problems, SIAM J. Math. Anal., 27 (1996), pp.
1070-1083.
Thank you!