A discrete stochastic model of the transmission cycle of the tick borne encephalitis virus

A discrete stochastic model of the transmission cycle of the

tick borne encephalitis virus

Luca Ferreri Mario Giacobini

Computational Epidemiology Group Department of Veterinary Sciences

Universit`a degli Studi di Torino

luca.ferreri@unito.it

Tick Borne Encephalitis (TBE)

• no consequences for animals

Tick Borne Encephalitis (TBE)

Transmissions

Systemic Transmission:

Transmissions

Networks - Lattice

frequency

number of connections 1

Networks - Random Graph

frequency

number of connections 1

Networks - Scale Free Network

frequency

number of connections 1

Compartmental Models on Networks

Susceptible-Infectious-Susceptible (SIS) model: d ik d t = −µik + βk (1 − ik) 1 hki X h hphih

the epidemic threshold:

β µ >

hki hk2_i

Compartmental Models on Networks

Susceptible-Infectious-Susceptible (SIS) model: d ik d t = −µik + βk (1 − ik) 1 hki X h hphih

the epidemic threshold:

β µ >

hki hk2_i

The model

The model

ik,y prevalence forlarvaethat encounter k nymphs at the end of year

The model

The model

a strong assumption(!):

ik,y = 1 − (1 − Ay)k

where Ay is the probability that an edge is transmitting the disease

Ay = β hkiN→L y X h h · p_h,yN→L·ih,y −1 therefore Ay = β − β hkiN→L y X h h · p_h,yN→L· (1 − Ah_{y −1})

ik,y = 1 − (1 − Ay)k

where Ay is the probability that an edge is transmitting the disease

Ay = β hkiN→L y X h h · pN→L_h,y · j_h,y therefore Ay = β − β hkiN→L y X h h · p_h,yN→L· (1 − Ah_{y −1})

ik,y = 1 − (1 − Ay)k

where Ay is the probability that an edge is transmitting the disease

Ay = β hkiN→L y X h h · p_h,yN→L·ih,y −1 therefore Ay = β − β hkiN→L y X h h · p_h,yN→L· (1 − Ah_{y −1})

ik,y = 1 − (1 − Ay)k

where Ay is the probability that an edge is transmitting the disease

Ay = β hkiN→L y X h h · p_h,yN→L·ih,y −1 therefore Ay = β − β hkiN→L y X h h · p_h,yN→L· (1 − Ah_{y −1})

The Epidemic Threshold

If we can assume that year y and year y − 1 are ecologically comparable then we check when Ay = Ay −1= x .

This corresponds to x = F (x ) = β − β hkiN→L y X h · p_h,yN→L· (1 − x)h but F0(x ) = β hkiN→L y X h2pN→L_hy (1 − x )h−1 and F00(x ) = − β hkiN→L y X h2(h − 1)pN→L_hy (1 − x )h−2

The Epidemic Threshold

If we can assume that year y and year y − 1 are ecologically comparable then we check when Ay = Ay −1= x .

This corresponds to x = F (x ) = β − β hkiN→L y X h · p_h,yN→L· (1 − x)h but F0(x ) = β hkiN→L y X h2pN→L_hy (1 − x )h−1 and F00(x ) = − β hkiN→L y X h2(h − 1)pN→L_hy (1 − x )h−2

The Epidemic Threshold

If we can assume that year y and year y − 1 are ecologically comparable then we check when Ay = Ay −1= x .

This corresponds to x = F (x ) = β − β hkiN→L y X h · p_h,yN→L· (1 − x)h but F0(x ) = β hkiN→L y X h2pN→L_hy (1 − x )h−1 and F00(x ) = − β hkiN→L y X h2(h − 1)pN→L_hy (1 − x )h−2

The Epidemic Threshold

If we can assume that year y and year y − 1 are ecologically comparable then we check when Ay = Ay −1= x .

This corresponds to x = F (x ) = β − β hkiN→L y X h · p_h,yN→L· (1 − x)h but F0(x ) = β hkiN→L y X h2pN→L_hy (1 − x )h−1 and F00(x ) = − β hkiN→L y X h2(h − 1)pN→L_hy (1 − x )h−2

x F0(x = 0) > 1 x F (x ) x F0(x = 0) < 1 x F (x ) β > hki N→L y hk2_iN→L y

x F0(x = 0) > 1 x F (x ) x F0(x = 0) < 1 x F (x ) β > hki N→L y hk2_iN→L y

Some recurrent ecological cycles are reported from real world data. Hence, sometimes, instead of the assumption Ay = Ay −1, it could be

more suitable Ay = Ay −n. In that case the epidemiological threshold

would be: βn> hki N→L y hk2_iN→L y · . . . · hki N→L y −n+1 hk2_iN→L y −n+1