Infectious Diseases of Humans Infectious Diseases of Humans SeminarSeminar

The basic modelThe basic model

Zhuobin LiZhuobin Li

OutlineOutline

Framework of the basic modelFramework of the basic model Basic reproduction number, Basic reproduction number, RR00 Average age of infectionAverage age of infection Transmission parameter,Transmission parameter,

Frame work of the basic modelFrame work of the basic model

AssumptionAssumption• Previous modelPrevious model

Age-specific host death rate, per capitaAge-specific host death rate, per capita

Age-specific recovery rateAge-specific recovery rate

Age-specific disease-induced death rateAge-specific disease-induced death rate

‘ ‘force of infection’ at time tforce of infection’ at time t

Frame work of the basic modelFrame work of the basic model

AssumptionAssumption• X(a),Y(a),and Z(a) are time independentX(a),Y(a),and Z(a) are time independent

• Birth rate and death rate are exactly balancedBirth rate and death rate are exactly balanced

• Ignore mortality associated with the infectionIgnore mortality associated with the infection

is assumed to be zerois assumed to be zero

Frame work of the basic modelFrame work of the basic model

ModelModel

Frame work of the basic modelFrame work of the basic model

Boundary conditionBoundary condition

Frame work of the basic modelFrame work of the basic model

Solve the modelSolve the model

where Survivorship functionwhere Survivorship function

Frame work of the basic modelFrame work of the basic model

Frame work of the basic modelFrame work of the basic model

Basic reproductive number RBasic reproductive number R00

RR00 for a microparasite for a microparasite• The basic reproductive number RThe basic reproductive number R00, is essentially the averag, is essentially the averag

e number of successful offspring that a parasite is intrinsicae number of successful offspring that a parasite is intrinsically capable of producing.lly capable of producing.

• For a microparasite, RFor a microparasite, R00 is defined as the average number of is defined as the average number of secondary infections produced when one infected individuasecondary infections produced when one infected individual is introduced into a host population where everyone is susl is introduced into a host population where everyone is suscetible.cetible.

• R R 00 >1 is the condition of being capable of invading and est >1 is the condition of being capable of invading and establishing itself within a host populationablishing itself within a host population

Basic reproductive rate RBasic reproductive rate R00

RR00 for a microparasite for a microparasite• At the equilibrium, the rate of the susceptible being infecteAt the equilibrium, the rate of the susceptible being infecte

d is balanced against a rate of newly susceptible individuals d is balanced against a rate of newly susceptible individuals appearing.appearing.

• At the equilibrium, each infection will be on average produAt the equilibrium, each infection will be on average produce exactly one secondary infection; ce exactly one secondary infection;

R=RR=R0 0 xx*=1*=1

where where x*x* is the fraction of the host population that is suscep is the fraction of the host population that is susceptible at equilibriumtible at equilibrium

Basic reproductive rate RBasic reproductive rate R00

RR00 for a this model for a this model• It seems to more plausible that the net rate of acquisition of It seems to more plausible that the net rate of acquisition of

new infectious is proportional to than to .new infectious is proportional to than to .

““doubling the number of susceptibles in a school is arguably more likdoubling the number of susceptibles in a school is arguably more likely to double the incidence of infection than is doubling the number oely to double the incidence of infection than is doubling the number of infectious individual”f infectious individual”

• Basic reproductive numberBasic reproductive number

Basic reproductive number RBasic reproductive number R00

Two types of mortalityTwo types of mortalityType I:Type I:

Type II:Type II:

where L is life expectancy.where L is life expectancy.

Basic reproductive number RBasic reproductive number R00

Two types of mortalityTwo types of mortalityType I:Type I:

Type II:Type II:

Survivor ship function:Survivor ship function:

Basic reproductive number RBasic reproductive number R00

Two types of mortalityTwo types of mortalityType I:Type I:

Type II:Type II:

Basic reproductive number RBasic reproductive number R00

Two types of mortalityTwo types of mortalityType I:Type I:

Type II:Type II:

Approximation of LApproximation of L

““The difference between L and G can be significant, especially in some The difference between L and G can be significant, especially in some developing country. For an example, life expectancy L of India is developing country. For an example, life expectancy L of India is around 40 years, while births are around 40 per 1000 per annum around 40 years, while births are around 40 per 1000 per annum which corresponds to G~25 years”which corresponds to G~25 years”

Average age at infectionAverage age at infection

The average age at which individuals acquire infection, A.The average age at which individuals acquire infection, A.

Type IType I

Type IIType II

Average age at infectionAverage age at infection

The average age at which individuals acquire infection based oThe average age at which individuals acquire infection based on the proportions susceptible.n the proportions susceptible.

If If λλ is age-dependent, then is age-dependent, then

Average age at infectionAverage age at infection

Come back to basic reproductive numberCome back to basic reproductive number

Type I: Type I:

Transmission parameter,Transmission parameter,

Transmission parameter Transmission parameter • Combines a multitude of epidemiolagical, environmental, and social Combines a multitude of epidemiolagical, environmental, and social

factors that affect transmission ratesfactors that affect transmission rates

is “ force of infection”, the per capita rate of acquisition of infection. is “ force of infection”, the per capita rate of acquisition of infection. If If is time dependent, is time dependent, (t)(t)t represents the probability that a given t represents the probability that a given susceptible host will become infected in the small time interval susceptible host will become infected in the small time interval t. t.

Transmission parameter,Transmission parameter,

Type I:Type I:

Type II:Type II: