Can we spend our way out of the AIDS epidemic?

(1)

Robert Smith?

Department of Mathematics and Faculty of Medicine

The University of Ottawa

Can we spend our way out

of the AIDS epidemic?

(2)

Using math to solve real problems

Biological problem

(3)

Using math to solve real problems

Biological

problem Mathematical

model

(4)

Using math to solve real problems

Biological

model

Mathematical analysis

(5)

Using math to solve real problems

Biological

model

Mathematical conclusion

(6)

Using math to solve real problems

Biological

model

Mathematical conclusion Biological

conclusion

(7)

Using math to solve real problems

Biological

model

Mathematical conclusion Biological

conclusion Compare with data

(8)

Models

• From simple assumptions, we can make models that might be simple, or might be complicated

(9)

Models

• Mathematical modelling is like map-making

(10)

Models

• We need to decide which factors are

important and which we can safely ignore

(11)

Models

• We need to decide which factors are

important and which we can safely ignore

“All models are wrong... but some are useful”

- George Box.

(12)

The basic reproductive ratio

• One of the fundamental concepts in mathematical biology

(13)

The basic reproductive ratio

• One of the fundamental concepts in mathematical biology

• Defined as “the average number of

secondary infections caused by a single infectious individual during their entire infectious lifetime.”

(14)

A brief history of R

0

• Originally developed for demographics (1886)

(15)

A brief history of R

0

• Independently studied for malaria (1911,1927)

(16)

A brief history of R

0

• Now widely used for infectious disease (1975+)

(17)

A brief history of R

0

• Now widely used for infectious disease (1975+)

“One of the foremost and most valuable ideas that mathematical thinking has

brought to epidemic theory”

(Heesterbeek & Dietz, 1996).

(18)

Definition of R

0

• Expected number of secondary individuals produced by an individual in its lifetime

(19)

Definition of R

0

• However, “secondary” depends on context:

(20)

Definition of R

0

- mean lifetime reproductive success (demographics and ecology)

(21)

Definition of R

0

- number of individuals infected within a single infected individual’s entire infectious lifetime (epidemiology)

(22)

Definition of R

0

- number of individuals infected within a single infected individual’s entire infectious lifetime (epidemiology)

- number of newly infected cells produced by a single infected cell

(in-host dynamics).

(23)

A threshold criterion

• If R0<1, each individual produces, on average, less than one new infected individual...

(24)

A threshold criterion

...and hence the disease dies out

(25)

A threshold criterion

• If R0>1, each individual produces more than one new infected individual...

(26)

A threshold criterion

...and hence the disease is able to invade the susceptible population

(27)

A threshold criterion

...and hence the disease is able to invade the susceptible population

•

This allows us to determine the effectiveness of control measures.

(28)

Total: 33.4 million (31.1 -35.8 million)

Adults and Children estimated to be living with HIV, 2008

Sub-Saharan Africa

22.4 million

[20.8–24.1 million]

Latin America

2.0 million

[1.8–2.2 million]

Caribbean

240 000

[220 000–260 000]

North America

1.4 million

[1.2–1.6 million]

Middle East and North Africa

310 000

[250 000–380 000]

Western and Central Europe

850 000

[710 000–970 000]

Oceania

59 000

[51 000–68 000]

East Asia

850 000

[700 000–1.0 million]

South and South-East Asia

3.8 million

[3.4–4.3 million]

Eastern Europe and Central Asia

1.5 million

[1.4–1.7 million]

(29)

Spending on HIV/AIDS

• Recent increase in the available money for HIV/AIDS

(30)

Spending on HIV/AIDS

• The Gates foundation alone has over $60 billion

(31)

Spending on HIV/AIDS

• Present plans are to hold the money in reserve and spent it slowly over 20 years

(32)

Spending on HIV/AIDS

• Present plans are to hold the money in reserve and spent it slowly over 20 years

• What if we spend it all at once?

(33)

How do we eradicate AIDS?

• Current tools:

(34)

How do we eradicate AIDS?

• Current tools:

– Condoms

(35)

How do we eradicate AIDS?

• Current tools:

– Condoms – drugs

(36)

How do we eradicate AIDS?

• Current tools:

– education

(37)

How do we eradicate AIDS?

• Current tools:

– education

• Future tools:

(38)

How do we eradicate AIDS?

• Current tools:

– education

• Future tools:

– vaccines

(39)

How do we eradicate AIDS?

• Current tools:

– education

• Future tools:

– vaccines

– microbicides

Evaluating the introduction of vaginal microbicides

Robert Smith?

Departments of Mathematics and Faculty of Medicine

(40)

How do we eradicate AIDS?

• Current tools:

– education

• Future tools:

– vaccines

– microbicides – etc.

Evaluating the introduction of vaginal microbicides

Robert Smith?

Departments of Mathematics and Faculty of Medicine

(41)

When have we done enough?

• When R⁰ <1

(42)

When have we done enough?

• When R⁰ <1

• R⁰ is a threshold parameter that determines whether a disease will remain endemic or be eradicated

(43)

When have we done enough?

• When R⁰ <1

• R⁰ is a threshold parameter that determines whether a disease will remain endemic or be eradicated

• (The value calculated by mathematical models is an eradication threshold, not necessarily the average number of

secondary infections).

(44)

Some issues

• How do we measure R⁰?

(45)

Some issues

– It’s difficult

(46)

Some issues

• If the R⁰ for your country is less than 1, is that sufficient?

(47)

Some issues

– Clearly not

(48)

Some issues

– Clearly not

• If R⁰ <1 for all countries, is that sufficient?

(49)

Some issues

– Clearly not

• If R⁰ <1 for all countries, is that sufficient?

– Surprisingly, no.

(50)

• But maybe we don’t need R⁰ exactly

Simplifying the problem

(51)

• To model HIV is incredibly complicated

Simplifying the problem

(52)

• However, a simple model might be sufficient...

Simplifying the problem

(53)

• However, a simple model might be sufficient...

• ...if it has the same eradication threshold as the more complicated model.

Simplifying the problem

(54)

Infectives

• Let’s look at infected people only

(55)

Infectives

• People in a region (country, continent, etc) change their infection status when they’re

(56)

Infectives

– infected

(57)

Infectives

– infected

– die (of the disease, or other causes)

(58)

Infectives

– infected

– die (of the disease, or other causes) – relocate (immigration/emigration).

(59)

• πi=β×(total uninfected population in ith region)

Infection model

(60)

• Thus, for p regions, in the ith region, we have

Infection model

(61)

dI_i

dt = _iI_i d_iI_i +

p

j=1

m_ijI_j

p

j=1

m_jiI_i

Infection model

(62)

dI_i

dt = _iI_i d_iI_i +

p

j=1

m_ijI_j

p

j=1

m_jiI_i

Infection model

Infection rate

(63)

dI_i

dt = _iI_i d_iI_i +

p

j=1

m_ijI_j

p

j=1

m_jiI_i

Infection model

Infection rate

Death rate

(64)

dI_i

dt = _iI_i d_iI_i +

p

j=1

m_ijI_j

p

j=1

m_jiI_i

Infection model

Infection rate

Death rate

Immigration to the ith region from

the jth

(65)

dI_i

dt = _iI_i d_iI_i +

p

j=1

m_ijI_j

p

j=1

m_jiI_i

Infection model

Infection rate

Death rate

Immigration to the ith region from

the jth

Emigration to the jth region from

the ith.

(66)

• This is a linear model

Linear infection model

(67)

• This is a linear model dI

dt = KI(t)

Linear infection model

Ii=# infectives in region i πi=influx of infectives di=death rate mij=migration rate

(68)

where

dI

dt = KI(t)

Linear infection model

(69)

I = (I₁, I₂, . . . , I_p)

K_ij = m_ij (i ⇥= j) K_ii = _i d_i

p

j=1

m_ij

where

dI

dt = KI(t)

Linear infection model

(70)

I = (I₁, I₂, . . . , I_p)

p

j=1

m_ij

where

dI

dt = KI(t)

Linear infection model

(71)

I = (I₁, I₂, . . . , I_p)

p

j=1

m_ij

where

dI

dt = KI(t)

Linear infection model

.

(72)

• This system is easy to analyse

An eradication threshold

K=travel matrix

(73)

• The only equilibrium is the disease-free equilibrium, (0,0,...,0)

An eradication threshold

K=travel matrix

(74)

• This equilibrium is stable if s(K)<0, where s(K) is real part of the largest eigenvalue of K

An eradication threshold

K=travel matrix

(75)

• It’s unstable if s(K)>0

An eradication threshold

K=travel matrix

(76)

• Thus, we have an eradication threshold:

An eradication threshold

K=travel matrix

(77)

• Thus, we have an eradication threshold:

T₀ = e^s(K)

An eradication threshold

K=travel matrix

.

(78)

Is the model too simplistic?

• The dynamics of HIV are not linear

(79)

Is the model too simplistic?

• They depend, at a minimum, on the behaviour of susceptibles and their interaction with those infected

(80)

Is the model too simplistic?

• They depend, at a minimum, on the behaviour of susceptibles and their interaction with those infected

• Thus, this model does not capture the transient dynamics of infection and

interaction.

(81)

Stability properties

• However, it does serve as a predictor for eradication

(82)

Stability properties

• If the disease-free

equilibrium is unstable, then trajectories will

increase without bound

(83)

Stability properties

equilibrium is stable, it’s globally stable

(84)

Stability properties

equilibrium is stable, it’s globally stable

• In this case, the disease will be eradicated.

(85)

SI model

• Consider the two-dimensional SI model

(86)

SI model

• Consider the two-dimensional SI model dS

dt = SI µS

dI

dt = SI µI ⇥I

S=susceptibles I=infectives Λ=birth rate β=infection rate µ=background death rate γ=disease death rate

(87)

SI model

• This model has equilibria dS

dt = SI µS

dI

dt = SI µI ⇥I

(88)

SI model

• This model has equilibria dS

dt = SI µS

dI

dt = SI µI ⇥I

µ , 0

⇥

and µ + ⇥

, µ

µ + ⇥ ⇥ .

(89)

Jacobian

• The Jacobian is

(90)

Jacobian

• The Jacobian is

J =

⇥ I µ S

I S µ ⇥

⇤

J

(S,I)=( /µ,0)

=

⇥ µ /µ

0 /µ µ ⇥

⇤

Λ=birth rate µ=background death rate β=infection rate γ=disease death rate

(91)

Jacobian

• The Jacobian is

J =

⇥ I µ S

I S µ ⇥

⇤

J

(S,I)=( /µ,0)

=

⇥ µ /µ

0 /µ µ ⇥

⇤

(92)

Jacobian

• The Jacobian is

• Thus, the eigenvalues are

J =

⇥ I µ S

I S µ ⇥

⇤

J

(S,I)=( /µ,0)

=

⇥ µ /µ

0 /µ µ ⇥

⇤

(93)

Jacobian

• The Jacobian is

• Thus, the eigenvalues are

J =

⇥ I µ S

I S µ ⇥

⇤

J

(S,I)=( /µ,0)

=

⇥ µ /µ

0 /µ µ ⇥

⇤

⇤ = µ,

µ µ ⇥ .

(94)

Influx of infectives

• It follows that

(95)

Influx of infectives

• It follows that

R_0,SI =

µ(µ + ⇥)

Λ= birth rate β=infection rate µ=background death rate γ=disease death rate

(96)

Influx of infectives

• It follows that

• Since S ≤ Λ/µ, the influx of infectives is less than

R_0,SI =

µ(µ + ⇥)

(97)

Influx of infectives

• It follows that

R_0,SI =

µ(µ + ⇥)

(total population without infection)

(98)

Influx of infectives

• It follows that

• This is πi in our linear model.

R_0,SI =

µ(µ + ⇥)

(total population without infection)

(99)

I-only model

• Thus, the model

(100)

I-only model

• Thus, the model

dI

dt =

µ µ ⇥

⇥ I

(101)

I-only model

• Thus, the model

will always overestimate the epidemic

dI

dt =

µ µ ⇥

⇥ I

(102)

I-only model

• Thus, the model

• This model also has eradication threshold

dI

dt =

µ µ ⇥

⇥ I

(103)

I-only model

• Thus, the model

dI

dt =

µ µ ⇥

⇥ I

R_0,I =

µ(µ + ⇥)

(104)

I-only model

• Thus, the model

• It follows that there will be eradication in the linear model if and only if there is eradication in the SI model.

dI

dt =

µ µ ⇥

⇥ I

R_0,I =

µ(µ + ⇥)

(105)

BMC Public Health 2009, 9(Suppl 1):S15 http://www.biomedcentral.com/1471-2458/9/S1/S15

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(page number not for citation purposes)

will always overestimate the epidemic. We shall refer to this as the I-only model.

This model has only the trivial equilibrium I = 0. If the trivial equilibrium is stable, then all trajectories will approach it. If the trivial equilibrium is unstable, then solutions will increase without bound.

Although the total population without infection is not constant, it should nevertheless be noted that the I-only model has the eradication threshold

which is the same eradication threshold as the SI model (as expected). It follows that there will be eradication in the I-only model if and only if there is eradication in the SI model.

To illustrate, we simulated two cases: R_0,SI< 1 (Figure 1) and R_0,SI> 1 (Figure 2). Parameters used in the simulations

were ! = 20 people·years^-1, " = years^-1, # = years^-

1, with $ = 0.00007 people^-1years^-1(Figure 1) and $ = 0.0002 people^-1years^-1(Figure 2). Despite the fact that the transient dynamics are vastly different in the two models, the linear approximation has the same eradication threshold as the more accurate SI model and, furthermore, always overestimates the epidemic. It follows that, for eradication purposes, the simple, linear model should determine whether our control methods will be sufficient.

We now extend the SI model to a metapopulation model with p regions. For 1 % i % p, we can write

S_i= susceptible individuals in the ith region;

I_i= infected individuals in the ith region;

!_i= the rate of appearance of new susceptible individuals in the ith region;

"_i= background death rate in the ith region;

#_i= death rate due to disease in the ith region;

n_ij= migration rate of susceptible individuals from jth region to ith region;

dI

dt =  − − I

 



$

"^Λ " #

R_{o I}_,

( ) ,

= +

$

" " #

Λ (1)

701 1

10

When R₀< 1, both models lead to eradication Figure 1

When R₀< 1, both models lead to eradication. When R₀< 1, both models lead to eradication. The SI model (solid curve) is always below the I-only model (dashed curve). In this case, R₀= 0.8575.

0 50 100 150 200 250 300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

time (years)

population (millions)

Eradication (R

0

=0.8575)

I-only model

SI model

(106)

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Page 5 of 17

(page number not for citation purposes)

m_ij= migration rate of infected individuals from jth region to ith region.

With these assumptions, we have n_ii! m_ii! 0. Then, for i = 1,..., p,

If the system (2) is well-posed, we can find a bound for the population of susceptibles in each patch. We can then extend the same idea to obtain the linear system of I-equa- tions only. (See Appendix for details.)

From Theorem 1 in the Appendix, we have

Thus, the total population of susceptible individuals in the ith region is limited. So we can write

"_i= # × (total population in the ith region without infection)

d_i= $_i+ %_i,

where d_irepresents the total death rate in the ith region.

Thus, for i = 1,..., p, the equations of the I-only model are

This can be rewritten as

where I(t) = (I₁(t), I₂(t),..., I_p(t))^Tand

Analysis

A two-region example

The simplest nontrivial version of model (2) is the case when p = 2, which can be presented as follows:

dSi

dt S I S n S n S

dIi

dt S I

i i i i i ij j ji i

j p j

p

i i i

= − − + −

= − +

=

∑

Λ # $

# $ %

1 1

( _ii _i _{ij j} _{ji i}

j p j

I p m I m I

) + − .

=

∑

¹ ¹

(2)

S t_i( ) ≤ N t( ) ≤ L*, for i =1, , .! p

dIi

dt _{i i}I d I_{i i} m I_{ij j} m I_{ji i}

j p

= − + −

=

∑

"

1 1

. (3)

d

dt I( )t = KI( ),t

K =

− +

=

∑

"

1 1 1 1 12 1

21 2 2 1 2 2

1 2

( )

d m m m

m d m m

m m

j j p

p

j j p

p

p p

"

# # $ #

"

" "_p _p _jp

j

d p m

− +

















∑ =

( )

.

1

When R₀> 1, both models lead to endemic disease Figure 2

When R₀> 1, both models lead to endemic disease. When R₀> 1, both models lead to endemic disease. The I-only model (dashed curve) always overestimates the SI model (solid curve). In this case, R₀= 2.45.

0 100 200 300 400 500

0 50 100 150 200 250 300

time (years)

population (millions)

0 50 100

0 5 10

Persistence (R

0

=2.45)

SI model I-only model

(107)

• For i=1,...,p,

A patch model with p regions

(108)

• For i=1,...,p,

A patch model with p regions

Λ=birth rate µ=background death rate β=infection rate γ=disease death rate nik=migration rate (susceptibles) mik=mmigration rate (infectives)

dS_i

dt = _i _iS_iI_i µ_iS_i +

p

j=1

n_ijS_j

p

j=1

n_jiS_i

dI_i

dt = _iS_iI_i (µ_I + ⇥_i)I_i +

p

j=1

m_ijI_j

p

j=1

m_jiI_i

(109)

• For i=1,...,p,

• We can bound

A patch model with p regions

dS_i

p

j=1

n_ijS_j

p

j=1

n_jiS_i

dI_i

p

j=1

m_ijI_j

p

j=1

m_jiI_i

(110)

• For i=1,...,p,

• We can bound

A patch model with p regions

dS_i

p

j=1

n_ijS_j

p

j=1

n_jiS_i

dI_i

p

j=1

m_ijI_j

p

j=1

m_jiI_i

S_i(t) L , for i = 1, . . . , p

(111)

• For i=1,...,p,

• We can bound where

A patch model with p regions

dS_i

p

j=1

n_ijS_j

p

j=1

n_jiS_i

dI_i

p

j=1

m_ijI_j

p

j=1

m_jiI_i

S_i(t) L , for i = 1, . . . , p

(112)

• For i=1,...,p,

• We can bound where

A patch model with p regions

dS_i

p

j=1

n_ijS_j

p

j=1

n_jiS_i

dI_i

p

j=1

m_ijI_j

p

j=1

m_jiI_i

S_i(t) L , for i = 1, . . . , p

L = ¹ + · · · + _p

min{µ₁, · · · , µ_p} .

(113)

The linear model is an overestimate

• The linear model has the same eradication threshold as the more accurate SI model

(114)

The linear model is an overestimate

• The linear model always overestimates the epidemic

(115)

The linear model is an overestimate

• The linear model always overestimates the epidemic

• For eradication purposes, the linear model should

determine whether our control methods will be sufficient.

(116)

A two-region example: the flow chart

(117)

Case 1: Two isolated regions

dS_i

dt = _i S_iI_i µ_iS_i dI_i

dt = S_iI_i (µ_i + ⇥_i)I_i

(118)

which has the eradication condition

Case 1: Two isolated regions

dS_i

(119)

which has the eradication condition R⁽⁰⁾_0,i = ⁱ

µ_i(µ_i + ⇥_i) for i = 1, 2

Case 1: Two isolated regions

dS_i

(120)

which has the eradication condition

• This is the same as the previous SI model.

R⁽⁰⁾_0,i = ⁱ

µ_i(µ_i + ⇥_i) for i = 1, 2

Case 1: Two isolated regions

dS_i

(121)

Case 2: Only susceptibles travel

dS₁

dt = ₁ S₁I₁ µ₁S₁ + n₁₂S₂ n₂₁S₁ dS₂

dt = ₂ S₂I₂ µ₂S₂ + n₂₁S₁ n₁₂S₂ dI₁

dt = S₁I₁ (µ₁ + ⇥₁)I₁ dI₂

dt = S₂I₂ (µ₂ + ⇥₂)I₂

(122)

• The basic reproductive ratio is

Case 2: Only susceptibles travel

dS₁

dt = ₁ S₁I₁ µ₁S₁ + n₁₂S₂ n₂₁S₁ dS₂

dt = ₂ S₂I₂ µ₂S₂ + n₂₁S₁ n₁₂S₂ dI₁

dt = S₁I₁ (µ₁ + ⇥₁)I₁ dI₂

dt = S₂I₂ (µ₂ + ⇥₂)I₂