mathematical modeling of dengue viral infection modeling of dengue viral infection ryan nikin-beers...

Mathematical Modeling of Dengue Viral Infection

Ryan Nikin-Beers

October 1, 2014

Ryan Nikin-Beers Mathematical Modeling of DVI October 1, 2014 1 / 45

1 IntroductionBiological Background

2 Within-Host ModelPrimary InfectionSecondary InfectionDiscussion

3 Epidemiological Model

4 Future Work

Introduction

4 Future Work

Introduction Biological Background

4 Future Work

Immune System

Innate vs Adaptive

B-cells vs T-cells

Neutralizing Antibodies

Non-neutralizing Antibodies

Immune System

Innate vs Adaptive

B-cells vs T-cells

Immune System

Innate vs Adaptive

B-cells vs T-cells

Immune System

Innate vs Adaptive

B-cells vs T-cells

Dengue Viral Infection

Mosquito-borne disease

50 million infections

500,000 hospitalizations

Tropical regions

Four serotypes (strains)

Primary infection leads to dengue fever (DF)

DF characterized by rash, fever, headache

Lifelong immunity from serotype

Secondary infection with heterologous serotype

Increased risk of dengue hemorrhagic fever (DHF) or dengue shocksyndrome (DSS)

DHF characterized by high blood platelet count, bleeding, and liverdamage

DSS characterized by high blood pressure, internal bleeding, andpossible shock

Antibody Dependent Enhancement

Neutralizing antibodies bind to virus and attach to Fc-receptors onhost cell

Cells with Fc-receptors not normally infected since antibodyneutralizes virus beforehand

Fc-receptor cells subsequently ingested in a process known asphagocytosis

Long-lived antibodies remain from primary infection

These antibodies unequipped to neutralize virus

Fc-receptor cells infected

Leads to more cells being infected, thus more virus produced

More virus correlates with more severe infection

Antibody Dependent Enhancement1

1Whitehead, et al. ”Prospects for a dengue virus vaccine.” Nature, 2007.Ryan Nikin-Beers Mathematical Modeling of DVI October 1, 2014 11 / 45

Vaccination

Currently no vaccine for dengue

Must avoid making population susceptible to more severe disease

Must protect against all serotypes at once

Within-Host Model

4 Future Work

Within-Host Model

Develop a model for primary infection based on known dynamics

Develop a model for secondary infection with heterologous serotype

Fit secondary infection model to known data

Infer from these results the underlying causes of more severe disease

Within-Host Model

Within-Host Model Primary Infection

4 Future Work

Primary Infection Model3

dt= s − dTT − βTV

1 + ηA,

1 + ηA− δI ,

dt= pI − cV − γVA,

dt= −αBV − dBB,

dt= αBV − kBaV − dBaBa,

dt= rP

(1 − P

)+ kBaV ,

dt= NP − dAA.

3Nikin-Beers, R and Ciupe, S. “The role of antibody in dengue viral infection.” MathBiosciences, submitted 2014.

Steady States

System (1) has four steady states. The disease-free steady state is given by

dT, 0, 0, 0, 0, 0, 0

The virus persistence in the absence of antibody responses is given by

S2 = (T2, I2,V2, 0, 0, 0, 0).

The virus persistence in the presence of antibody responses is given by

S3 = (T3, I3,V3, 0, 0,KP ,KA).

The antibody-induced virus clearance is given by

dT, 0, 0, 0, 0,KP ,KA

Stability

Steady state S2 exists when R0 = βpsdT cδ

> 1Steady states S1 and S2 are unstable.Steady state S3 is locally asymptotically stable if Rp

0 = R0

(1+ηKA)(1+γcKA)

Steady state S4 is locally asymptotically stable if Rp0 < 1.

Primary Infection Dynamics

Primary infection viremia peak 3-4 days after infection

Primary infection viremia clear at 7-10 days

Viremia always clears (S4 is locally asymptotically stable)

Antibodies above limit of detection between one to two weeks afterinfection

0 20 40 60 80 100104

Day After Infection

Target Cells

0 2 4 6 8 10

Day After Infection

Infected Cells

0 2 4 6 8 10

Day After Infection

0 2 4 6 8 1010 -8

Day After Infection

B Cells

0 10 20 30 40 50

Day After Infection

Plasma Cells

0 10 20 30 40 50

Day After Infection

Antibodies

Within-Host Model Secondary Infection

4 Future Work

Secondary Infection Model

The new model is

dt= s − dTT − β1TV1

1 + ηA1− β2TV2

1 + ηA2,

=βiTVi

1 + ηAi− δIi ,

dt= p1I1 − (c + γ1A1)V1,

dt= p2I2 − (c + γ2A2 − γEA1)V2,

dt= −αBV1 − αBV2 − dBB,

dt= αBVi − kBaiVi − dBaBai ,

dt= rPi (1 − Pi

KP) + kBaiVi ,

dt= NPi − dAAi ,

where i = 1, 2.

Steady States

Assume S4 is stable in original model. Then we only consider four steadystates of model 2.

S5 = (s

dT, 0, 0, 0, 0, 0, 0, 0, 0,KP , 0,KA, 0),

S6 = (T6, 0, I6, 0,V6, 0, 0, 0,KP , 0,KA, 0),

S7 = (T7, 0, I7, 0,V7, 0, 0, 0,KP ,KP ,KA,KA),

S8 = (s

dT, 0, 0, 0, 0, 0, 0, 0,KP ,KP ,KA,KA).

Stability

Steady state S6 only exists when R02 = β2p2sdT cδ

> 1.Steady states S5 and S6 are unstable.Steady state S7 is locally asymptotically stable whenRs0 = R02

(1+ηKA)(1+γ2−γE

Steady state S8 is locally asymptotically stable when Rs0 < 1.

4Wang, et al. “Slower Rates of Clearance of Viral Load and Virus-ContainingImmune Complexes in Patients with Dengue Hemorrhagic Fever.” Clin Infect Dis, 2006.

Secondary DF vs DHF

Higher viral peak in DHF

Slower viral clearance in DHF

Similar viral peak time

Secondary DF vs DHF

Data Fitting

Assume only neutralizing antibodies play a role in enhancingsecondary infection

Implies higher β2 in more severe infection since more Fc-receptorbearing cells will be infected

So,γE = 0, p2 = p1, γ1 = γ2, and β2 = (1 + ξKA)β1However, from fitting the data we get that βDHF

2 < βDF2 < β1.

We also get that the peak of DF is higher than DHF, which is inconsistentwith the dynamics of secondary DF and DHF.This implies there is not higher infectivity rate β2 in more severe infection.So, we set β2 = β1.

Data Fitting

Assume only neutralizing antibodies play a role in enhancingsecondary infection

Implies higher β2 in more severe infection since more Fc-receptorbearing cells will be infected

So,γE = 0, p2 = p1, γ1 = γ2, and β2 = (1 + ξKA)β1However, from fitting the data we get that βDHF

2 < βDF2 < β1.

We also get that the peak of DF is higher than DHF, which is inconsistentwith the dynamics of secondary DF and DHF.This implies there is not higher infectivity rate β2 in more severe infection.So, we set β2 = β1.

Data Fitting

Now we fit p2 > p1, γ2 and γE , which correspond to the effectnon-neutralizing antibodies have on secondary infection.We get results more consistent with known viral dynamics of secondaryinfection.

Secondary DF vs DHF

0 2 4 6 8 10

Day After Infection

Secondary DF

0 2 4 6 8 10

Day After Infection

Secondary DHF

Primary vs Secondary Infections

Higher viral peak in secondary infections

Earlier viral peak time in secondary infections

Faster viral clearance time in secondary infections

Primary vs Secondary DF

0 5 10 500 505 510

Time After Primary Infection

Primary vs Secondary DF

Table : Comparison of Primary DF and Secondary DF Viral Dynamics

Disease Level Viral Peak (RNA/ml) Viral Peak Time (days) Clearance Time (days)

Primary 7.7 × 107 3.9 8.4DF 5.8 × 108 1.8 6.1

Primary vs Secondary DHF

0 5 10 500 505 510

Time After Primary Infection

Primary vs Secondary DHF

Table : Comparison of Primary DF and Secondary DHF Viral Dynamics

Disease Level Viral Peak (RNA/ml) Viral Peak Time (days) Clearance Time (days)

Primary 7.7 × 107 3.9 8.4DHF 9.6 × 108 1.7 8.3

Within-Host Model Discussion

4 Future Work

Model Dynamics

Model of primary infection consistent with known dynamics

Model of secondary infection can distinguish between secondary DFand secondary DHF

Model of secondary infection can distinguish between primary andsecondary infection

Model Dynamics

Biological Explanation

Neutralizing antibodies do not increase infectivity rate as described byADE

Clearance rate of virus by non-neutralizing antibodies is decreased insecondary infection

Non-neutralizing antibodies from primary infection bind to virus fromsecondary infection, thus not allowing the virus to be cleared byantibodies specific to secondary infection

Epidemiological Model

4 Future Work

Model with Two Strains

I1p R1 I2s

I2p R2 I1s

Model Equations

dt= µ− β1I1pS − β1φ1I1sS − β2I2pS − β2φ2I2sS − µS

dI1pdt

= β1I1pS + β1φ1I1sS − σI1p

dI2pdt

= β2I2pS + β2φ2I2sS − σI2p

dI1sdt

= β1I1pR2 + β1φ1I1sR2 − σI1s

dI2sdt

= β2I2pR1 + β2φ2I2sR1 − σI2s

dt= σI1p − β2I2pR1 − β2φ2I2sR1 − µR1

dt= σI2p − β1I1pR2 − β1φ1I1sR2 − µR2

dt= σI1s + σI2s − µR12

Epidemiological Data 5

5Ferguson, et al. ”The effect of antibody-dependent enhancement on thetransmission dynamics and persistence of multiple-strain pathogens.” PNAS USA, 1999.

Model Dynamics

Future Work

4 Future Work

Future Work

Original Antigenic Sin

Another explanation for results from within-host model may beoriginal antigenic sin

Effector cells specific to primary infection unable to clear virus easily

Determine whether OAS model can explain data

Future Work

Multi-scale Modeling

Transmission of virus is dependent on severity of infection (β vs βφ)

Determine effect of enhancement from within-host model

Use these results in epidemiological model

Future Work

Questions

Questions?

mathematical modeling of dengue viral infection modeling of dengue viral infection ryan nikin-beers...

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