Download Pathogen–Host Models “SIR” Models of Direct Transmission

Pathogen–Host Models
“SIR” Models of Direct Transmission
A classic way of modeling the dynamics of infectious disease assumes the host population is
constant, and tracks the dynamics of the portions of the host population which are Susceptible,
Infected, and Recovered (or Resistant). These models, with modifications to incorporate variant
forms of transmission, time-delays, etc., have proved effective for understanding the major
features of epidemics of many human diseases.
As will be seen later, these models resemble in form the simplest models of multi-species
metapopulations, which track the dynamics of the portion of habitat patches which are occupied
by various of the interacting species.
Basic assumptions
The SIR models all share several core assumptions:
1. Hosts are born susceptible.
2. The host population is constant; this implies that the dynamics of the disease are much faster
than those of the host population or that the disease has negligible impact on host birth and
death rates.
3. Transmission is “direct”: by contact between an infected host and a susceptible host.
4. The degree of infection of an individual host — its pathogen load — has no effect on transmission.
The basic SIR models to be described below make additional assumptions, which have been
eliminated or modified in other, more complex SIR models. Among these are that:
5. The host population is well mixed, so that encounters between hosts of various categories are
proportional to their frequencies in the population.
6. There is no delay between becoming infected and becoming infective.
No recover (S and I only)
First let’s assume all infected individuals remain infected. We’ll also assume the host population
is constant.
Model
• S: the number of susceptible hosts
• I: the number of infected hosts
• N: the total number of hosts; N = S + I
•
a: the “transmission coefficient” (describing the rate of encounters among individuals,
together with the probability of infection when a susceptible encounters an infected); its
units are numbers per infected host per susceptible host per unit time.
The preceding assumptions about transmission, and encounters among hosts, cause the rate of
new infections, which convert hosts from S to I, to be proportional to the product of the
abundances of S and I.
The model therefore is
d
I = aIS = aI ( N – I )
dt
(Note that since S + I = N, a constant, only one equation is needed.). This equation can be divided
by N, converting I and S to proportions of the total population (which I’ll denote by i and s). The
model then is
d
i = ai ( 1 – i )
dt
This model has exactly the form of the logistic model of density dependent population growth,
here with K = 1. Therefore the infection will spread (I or i will increase) following the sigmoid
logistic curve, until finally the entire population is infected.
Recovery (SIR), no host dynamics
What happens to the disease if infected individuals eventually recover, losing the infection and
becoming resistant to reinfection? This leads to the full SIR model with three categories of hosts.
Assume the total host population is constant with no births or deaths. This implies that the
dynamics of the disease are much faster than of births and deaths of hosts. Assume also that
recovered hosts remain resistant forever.
Model
• S:
• I:
• R:
• N:
• a:
• v:
the number of susceptible hosts
the number of infected hosts
the number of recovered hosts, which are neither infective or susceptible to re-infection
the total number of hosts; N = S + I + R
the “transmission coefficient,” as before
the rate of recovery of infected hosts; its units are numbers per infected host per unit time.
As before, the rate of new infections will be proportional to S times I. The subpopulation of
infecteds will decrease due to recoveries, which will be proportional to the abundance of
infecteds.
The model therefore is
d
S = – aSI
dt
d
I = aSI – vI
dt
d
R = vI
dt
(Note that since S + I + R = N, a constant, only two of these three equations are needed; the
dynamics of the third category are entirely determined by the two others.)
Equilibria?
This model is unlike any we have seen this semester: the only equilibrium is with I = 0! To see
this, notice that from the first equation, S always decreases, unless either S or I is 0, in which case
S is constant. Then in the second equation, I is constant only if
d
S = 0
dt
( aS – v )I = 0
which will be true if either I = 0 or S = v / a > 0. The only way both S and I can be constant
therefore is for I to be 0: for the disease to be absent from the population. If this is true S then can
be at any level between 0 and 1.
As will be seen in the next model below, the present model does not allow an equilibrium level of
infections — an “endemic” level of disease — because it does not have any source of new
susceptible hosts.
Conditions for an epidemic?
Despite the model not allowing an equilibrium with positive I, it is interesting to consider the
dynamics of the disease before it disappears from the population. A first question is whether the
disease can ever increase: can a temporary epidemic occur? The second equation in the model
says that I does increase whenever
aS – v > 0
or
aS
R 0 = ------ > 1
v
The quantity aS/v is very important in epidemiology, and so has its own notation, R0, and a name:
the basic reproductive number for the disease. (In fact its more common name is the basic
reproductive rate of the disease, but it is not actually a rate; it is dimensionless.) To interpret this
parameter R0, it helps to realize that since v is the rate of recovery of infected hosts, 1 / v is the
average duration of an infection. Since aS is the rate at which a single infected host causes new
infections, R 0 = aS ⁄ v = aS ⋅ ( 1 ⁄ v ) is the average number of new infections caused by a
single infection over its duration. It therefore makes sense that R0 must be greater than 1 —
each infection must do more than replace itself — for the level of infection to increase.
This result for spread of the disease also shows that there is a threshold size of the population,
below which the disease cannot spread. The condition for increase in I, that R0 > 1, can be
rearranged to be S > v/a. Obviously the number of susceptibles, S, cannot exceed the total
population N. So if N < v/a, the disease cannot spread (I cannot increase).
After the epidemic?
If N does exceed v/a and a disease enters the population (all initially susceptible), it will spread.
Eventually, though, as the number of susceptibles to which it can be transmitted decreases, the
rate of increase of the disease will drop to 0. An obviously important question is whether this will
happen before all the susceptibles have become infected. It can be shown, by methods beyond this
course, that an epidemic will end long before all susceptibles get the disease.
The fraction of the total population remaining susceptible at the end of the epidemic — that is, the
fraction that never got the disease — is a decreasing function of the ratio of the population size to
the minimum population size for an epidemic to occur, i.e. of N / (v/a). When this ratio is close to
1 (the population is barely large enough for the epidemic to spread) nearly all hosts will get
infected before the disease disappears, but when the population is much larger, a larger fraction
escape getting infected.
Host births and deaths but constant population size
Over a longer time span there will of course be births of new hosts, which will be susceptible. For
many diseases also, resistance of recovered hosts to reinfection is only temporary, so new
susceptibles also arise by loss of resistance. It turns out that when new susceptibles appear an
equilibrium is possible with a positive number of infected hosts, i.e. a persistent or endemic
disease.
In the following I will consider only the effect of birth of new (susceptible) hosts, omitting loss of
resistance. For simplicity I will still assume the total host population is constant, and also that .
The most natural way to do this is to assume the birth rate and death rate are exactly equal, and
that there is no affect of the disease on mortality or fecundity.
Model
• S:
• I:
• R:
• N:
the number of susceptible hosts
the number of infected hosts
the number of recovered hosts, which are neither infective or susceptible to re-infection
the total number of hosts; N = S + I + R
•
•
•
a: the “transmission coefficient,” as before
b: the per capita birth rate (and also death rate)
v: the rate of recovery of infected hosts.
The model is
d
S = bN – bS – aIS
dt
d
I = aIS – bI – vI = ( aS – b – v )I
dt
d
R = vI – bR
dt
Equilibria:
From the equation for the dynamics of the I subpopulation, we can see that at equilibrium
d
( I = 0 ), either I = 0, or aS – ( b + v ) = 0
dt
which gives
S∗ = ( b + v ) ⁄ a
The equilibrium number of susceptible hosts therefore equals the ratio of the rate at which
infected hosts are removed from the population and the transmission rate. For simplicity, let d
denote the rate of removal of infected (= b + v), so that S* = d / a.
From the equation for the S subpopulation, at equilibrium,
d
S
dt
0 = bN – bS – aIS
aSI = b ( N – S )
( N – S∗ )I∗ = b----------------------aS∗
0 =
( N – d ⁄ a )= b--------------------------d–c
( N – d ⁄ a )= b--------------------------d
From this (and the fact that all the rates a, b, v, and thus d, are positive), we see that for the disease
to persist in the population, i.e. for I* to be positive,
d
N – --- > 0
a
d
N > --a
Na
------- > 1
d
The quantity Na/d — the product of the population size times the ratio of the rate of removal of
the infection to the transmission rate — is the basic reproductive number of the disease for this
model, again denoted R0. In this model the rate at which infecteds are removed is b + v = d, so 1/d
is the average duration of a single infection. Thus the quantity R0 = aN(1/d) is again the expected
number of new infections caused by a single infection during its “lifespan”.
The condition for persistence of the disease then is simple: the basic reproductive number R0 must
be greater than 1. The difference from the analogous condition for the preceding model without
host births and deaths is that now N has taken the place of S and the sum b+v has replace v in the
denominator.
This criterion for persistence of the disease again can be re-arranged easily to show that if the
population size N is less than d / a, the disease will disappear from the population. That is, there is
a threshold population size below which the disease cannot persist:
d
b+v
N T = --- = -----------a
a
(Note that this NT is simply the equilibrium abundance of susceptibles, S*. Thus the requirement
that the total population must be greater than NT for the disease to persist is equivalent to the
intuitively obvious result that the population must be large enough to include the equilibrium
number of susceptibles plus some infecteds.)
This result directly quantifies the effect of crowding on the prevalence of infectious diseases.
Indirect Transmission
Many important diseases — of plants and other animals as well as of humans — are transferred
between host individuals by individuals of a different species, often biting or sucking insects (e.g.
transmission of malaria by mosquitoes). The dynamics of these diseases therefore depend on the
abundances of both vector and host populations, as well as the fractions of each that are infected.
The model presented here was the first model proposed for vector-borne disease, and is intended
to represent (in very simplified form) the dynamics of malaria; other diseases have been modeled
by variants on this basic model.
Assumptions
1. The densities of host and vector populations are constant (this can be interpreted as implying
that the dynamics of disease spread is much faster than the dynamics of population change in
either species).
2. Infected hosts do not suffer mortality from the disease, and eventually recover.
3. Vectors which are infected (and thus infective to hosts) do not recover (thus remain infective).
4. Encounters between vectors and hosts occur randomly, independently of either’s infection
status.
5. The rate of host encounters per vector individual (per unit time) is constant, independent of
the density of either population.
(For the case of malaria in humans, “hosts” are humans, “vectors” are mosquitoes, “encounters”
are bites.)
Model
Notation:
• V: the fraction of vectors that are infected
• H: the fraction of hosts that are infected
• a: the “transmission coefficient”
• r: the rate of recovery of infected hosts
• b: the encounter rate per vector
• c: the proportion of encounters (of infected vectors with uninfected hosts) that result in
transmission of the pathogen from vector to host
• v: the proportion of encounters (of uninfected vectors with infected hosts) that result in
transmission of the pathogen from host to vector
• NV: the population density of the vectors
•
NH:the population density of the hosts
•
u: the death rate of vectors (assumed to be independent of infection status)
Equations:
The probability an encounter will result in a new infection will be proportional to the probability
the vector is infected (V) and the probability the host is not already infected (1 − H), with the
proportionality constant being a, the “transmission coefficient.” Therefore
d
H = aV ( 1 – H ) – rH
dt
The transmission coefficient a — which is in units of infections per vector per host — can in turn
be modeled as the product of the encounter rate per vector individual, b, the probability an
encounter results in transmission, b, and the number of vectors per host individual, which is equal
to the ratio NV / NH. The equation for the dynamics of the infected hosts therefore becomes
NV
d
H = ⎛⎝ bc -------⎞⎠ V ( 1 – H ) – rH
dt
NH
Similarly, the rate of new infections of vectors depends on the rate of encounters between
uninfected vectors with infected hosts, which will be the product of the rate of vector–host
encounters times the fraction of vectors which are uninfected (thus available to become infected)
and the fraction of hosts which are infected. This then is multiplied by the probability such an
encounter will result in host-to-vector transmission of the pathogen. The rate of change in the
fraction of vectors which are infected then equals the rate of new infections minus the death rate
of vectors:
d
V = bvH ( 1 – V ) – uV
dt
Equilibria
This model is most easily analyzed by considering the zero-growth isoclines for the two state
variables, V and H. From the first model equation, the zero-growth isocline for H is
d
H = 0
dt
aV ( 1 – H ) – rH = 0
aV ( 1 – H ) = rH
1.0
zero-growth isocline for H (dH/dt = 0)
0.8
V
0.6
rH V = -------------------a(1 – H)
0.4
0.2
As shown to the right, on this isocline V increases
as an accelerating function of increasing H (the
numerator increases proportionally to H while
the denominator decreases as H increases).
0.0
0.0
0.2
0.4
0.6
H
0.8
1.0
Similarly, the isocline for V is
0.8
= 0
0.7
0.6
=
=
=
=
0
uV
uV
( u + bvH )V
bvH V = ------------------u + bvH
0.5
V
d
V
dt
bvH ( 1 – V ) – uV
bvH ( 1 – V )
bvH – bvHV
bvH
0.9
0.4
0.3
zero-growth isocline for V (dV/dt = 0)
0.2
0.1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
H
This isocline also has V increasing as H increases, but at a decelerating rate (the denominator as
well as the numerator increases with H).
If, however, the H isocline is steeper than the V
isocline even at very low levels of H, the
isoclines meet only at the origin: the only
equilibrium is with the pathogen absent from the
system. This equilibrium also is stable: if the
disease is initially present, it will die out.
1.0
0.8
V
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
H
1.0
0.8
0.6
V
Combining the two isoclines allows
determination of whether an equilibrium exists
with the disease present in both populations. As
shown in the upper graph to the right, if the H
isocline (solid line, accelerating upward) has a
lower slope at low values of H than does the V
isocline (dashed line, decelerating), there will be
an interior equilibrium with both V and H greater
than 0; this equilibrium will be stable, as the
vectors indicate.
From the preceding results the conditions under
which the disease will persist can be determined.
The slope of the H isocline at the origin is r / a,
and the slope of the V isocline at the origin is
bv / u. Existence of an equilibrium with the
disease present in the population therefore requires that
0.4
0.2
0.0
0.0
0.2
bv
------ > --ru a
Using the expression above for a = bcNV / NH, this condition becomes
0.4
0.6
H
0.8
1.0
bv
r
------ > -----------------------u bcN V ⁄ N H
( N V ⁄ N H )b 2 cv
----------------------------------- > 1
ru
z0 > 1
where z0, the expression to the left of the inequality, is the basic reproductive number of the
disease: the ratio of transmission (in both directions) to the rate of loss of the disease. As was the
case for directly transmitted pathogens in the SIR model, persistence of this vector-borne disease
requires its basic reproductive number z0 to be greater than 1. Attempts to control such diseases
therefore aim to reduce z0 below 1, by reducing transmission — reducing NV, b, c, and v — and
increasing the rate of removal of infectious vectors — increasing r and u.
Diseases Affecting Host Abundance
If the pathogen inflicts mortality or reduced fecundity on the host which is not compensated for,
the disease will affect the population growth, and abundance, of the hosts. This occurs with
human diseases such as AIDS which have high prevalence and pathogenicity, as well as with
many diseases of non-human populations. Understanding the dynamics of such diseases requires
that the dynamics of the host population, and not only of the fractions of hosts in various
categories, must be considered.
Assumptions
The simplest model for pathogen–host dynamics, with the host abundance a dynamic variable, is
a variant of the classic SIR model, adopting all its assumptions except the second one, that the
host population is constant. This model makes one additional assumption, that hosts which
recover from the infection become susceptible, not resistant; this probably valid at least for many
invertebrate hosts, and simplifies the model by eliminating the third category of hosts:
No free-living infective stage
The basic model assumes that the pathogen can only be transmitted by contact between hosts, i.e.
that it cannot live outside a host.
Model
Although this model is closely related to the SIR model presented above, different notation for the
two dynamical variables will be used to emphasize that this model changes one central
assumption of the SIR model.
•
X: the abundance of susceptible hosts
•
•
•
•
•
•
•
Y: the abundance of infected hosts
N: the total abundance of hosts
a: the “transmission coefficient” (describing the rate of encounters among individuals,
together with the probability of infection when a susceptible encounters an infected)
b: the per capita rate of “natural” (i.e. not related to the disease) mortality, assumed to be the
same for all categories of hosts
c: the per capita rate of additional mortality of infected hosts, due to the disease,
v: the rate of recovery of infected hosts.
f: the per capita birth rate of the hosts (assumed to be unaffected by infection)
The differences between this model and the SIR model are that
•
•
•
the birth of new susceptibles depends on the abundance of hosts, and is not assumed to equal
the total death rate
infected hosts have additional mortality
recovering hosts become susceptible.
The model therefore is
d
X = f ( X + Y ) – bX – aXY + vY
dt
d
Y = aXY – ( b + c + v )Y = [ aX – ( b + c + v ) ]Y
dt
Alternatively, one of these equations can be replaced by one giving the dynamics of the total host
population:
d
N = ( f – b )N – cY
dt
Equilibria
From the equation for the dynamics of the infected population (Y) the equilibrium value for the
susceptible population (X*) is seen to be
+ c + v- = d--- = N
X∗ = b-------------------T
a
a
(Note that here the total rate of loss of infecteds, d, includes the additional mortality rate c, which
was not present in the SIR models presented earlier.)
Setting the growth rate of the susceptible population to 0, substituting in the value here for X*,
and solving for Y (details are shown at the end of this section) gives the equilibrium abundance of
infected hosts:
b+c+v
( f – b ) ⎛ ---------------------⎞
⎝
a ⎠
Y∗ = -----------------------------------------c – (f – b)
rN T
= ---------c–r
where r = f − b is the intrinsic rate of increase of uninfected hosts.
Summing these two equilibria gives the equilibrium abundance for the total population:
N∗ = X∗ + Y∗
rN T
= N T + ---------c–r
r -⎞
= N T ⎛ 1 + ---------⎝
c – r⎠
c
= N T ⎛ -----------⎞
⎝ c – r⎠
Implications
From these equilibrium results for infected hosts and total hosts, i.e. Y* and N*, it is evident that
positive (and finite) equilibrium levels require that the extra mortality inflicted by the pathogen
exceeds the intrinsic rate of uninfected hosts:
c>r
This makes sense: if the disease isn’t pathogenic enough for infected hosts to have a negative per
capita rate of increase (r − c < 0), the host population will continue growing even if all the hosts
are infected, and the host population will not be regulated by the disease.
These equilibrium expressions also show that, just as in the SIR model, the disease will not persist
unless the total host population exceeds NT, or equivalently, unless R0 = aN / d exceeds 1.
Long-lived free-living infective stage
Many pathogens can persist for long periods of time outside the host, and remain infective. The
spores of many bacteria, protozoans, and fungi allow this, and many viruses also can persist either
as free virus particles or in capsules or polyhedra. The models for this situation are too complex
for this course, but one interesting result deserves mention.
The free-living infective stages constitute a reservoir of infective pathogens even when the host
population is at low abundance. Therefore if the host population increases, the infection can
quickly spread. It turns out that these dynamics can produce stable limit cycles.
Cyclic dynamics, rather than a stable stationary equilibrium, will occur when the free-living
infective stage is produced (from infected hosts) in large numbers and is long-lived: that is, when
the reservoir of free-living infectives is large. Cycles also are more likely when the host rate of
increase (r = f − b) is low. The frequency of the cycles varies, with high host reproductive rates
and/or high rates of death of infectives (in the free-living stage) producing high frequency cycles
(periods of a few years) and low host rates of increase and low infective death rates producing
very low frequency cycles (on the scale of decades).
solving for Y* in the model with host dynamics:
d
X = 0
dt
f ( X∗ + Y ) – bX∗ – aX∗ Y + vY = 0
f ( N T + Y ) – bN T – aN T Y + vY = 0
fY – aN T Y + vY + fN T – bN T = 0
fY – aN T Y + vY = bN T – fN T
( f – aN T + v )Y = ( b – f )N T
( b – f )N T
Y∗ = -------------------------f – aN T + v
( b – f )N T
Y∗ = ---------------------------------------------b+c+v
f – a ⎛ ---------------------⎞ + v
⎝
a ⎠
( b – f )N
Y∗ = -----------------------Tf – (b + c)
( f – b )N
Y∗ = -----------------------T(b + c) – f
( f – b )N T
Y∗ = ----------------------c – (f – b)
rN T
Y∗ = ---------c–r