Virus antibody dynamics in primary and secondary dengue infections

Tanvi P. Gujarati; G. Ambika

doi:10.1007/s00285-013-0749-4

Journal of Mathematical Biology

December 2014, Volume 69, Issue 6–7, pp 1773–1800 | Cite as

Virus antibody dynamics in primary and secondary dengue infections

Authors
Authors and affiliations

Tanvi P. Gujarati
G. AmbikaEmail author

Article

First Online: 04 January 2014

Received: 02 February 2012

Revised: 12 October 2013

Abstract

Dengue viral infections show unique infection patterns arising from its four serotypes, (DENV-1,2,3,4). Its effects range from simple fever in primary infections to potentially fatal secondary infections. We analytically and numerically analyse virus dynamics and humoral response in a host during primary and secondary dengue infection for long periods using micro-epidemic models. The models presented here incorporate time delays, antibody dependent enhancement, a dynamic switch and a correlation factor between different DENV serotypes. We find that the viral load goes down to undetectable levels within 7–14 days as is observed for dengue infection, in both cases. For primary infection, the stability analysis of steady states shows interesting dependence on the time delay involved in the production of antibodies from plasma cells. We demonstrate the existence of a critical value for the immune response parameter, beyond which the infection gets completely cured. For secondary infections with a different serotype, the homologous antibody production is enhanced due to the influence of heterologous antibodies. The antibody production is also controlled by the correlation factor, which is a measure of similarities between the different DENV serotypes involved. Our results agree with clinically observed humoral responses for primary and secondary infections.

Keywords

Humoral immune response Antibody dependent enhancement Correlation factor among serotypes Time delay Stability analysis

Mathematics Subject Classification (2010)

37N25 34A34 92B05 92C60

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Notes

Acknowledgments

We would like to thank Dr. Hedgewar Hospital, Aurangabad, Maharashtra for providing the ELISA data on Dengue infections. We thank the anonymous reviewers for their valuable comments.

Appendix

Details of the Jacobian and the characteristic equation

The Jacobian describing the primary infection in terms of $S^{*}, V^{*}, B^{*}$ and $A^{*}$ which are the equilibrium values of healthy cells, virus count, B-cells and antibody count respectively is given as,

$$\begin{aligned} J = \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} -(\alpha +aV^{*}) &{} 0 &{} -aS^{*} &{} 0 &{} 0\\ aV^{*} &{} -\beta &{} aS^{*} &{} 0 &{} 0\\ 0 &{} k &{} -(\gamma +pA^{*}) &{} 0 &{} -pV^{*}\\ 0 &{} 0 &{} cB^{*} &{} -(\delta -cV^{*}) &{} 0\\ 0 &{} 0 &{} -qA^{*} &{} fe^{-\lambda \tau _{2}} &{} -(\kappa +qV^{*})\\ \end{array} \right] \end{aligned}$$

(17)

where;

$$\begin{aligned} S^{*}&= \frac{\mu }{(\alpha + aV^{*})} \end{aligned}$$

(18a)

$$\begin{aligned} I^{*}&= \frac{aS^{*} V^{*}}{\beta } \end{aligned}$$

(18b)

$$\begin{aligned} B^{*}&= \frac{\eta }{(\delta - cV^{*})}\end{aligned}$$

(18c)

$$\begin{aligned} A^{*}&= \frac{f B^{*}}{( qV^{*} + \kappa )} \end{aligned}$$

(18d)

The characteristic equation for the Jacobian $J$ with eigenvalues $\lambda $ can be written as:

$$\begin{aligned} G(\lambda )&= \lambda ^{5} + G_{4}\lambda ^{4} + G_{3}\lambda ^{3} + G_{2}\lambda ^{2} + G_{1}\lambda ^{1} + G_{0} + H_{2}e^{-\lambda \tau _{2}}\lambda ^{2}\nonumber \\&\quad +\,H_{1}e^{-\lambda \tau _{2}}\lambda + H_{0}e^{-\lambda \tau _{2}}\nonumber \\&= 0 \end{aligned}$$

(19)

where:

$$\begin{aligned} G_{4}&= ( \alpha + \beta + \gamma + \delta + \kappa + V^{*}(a + \kappa -c) + pA^{*}); \nonumber \\ G_{3}&= ( \alpha \beta + \alpha \gamma + \gamma \beta + \alpha \delta + \beta \delta +\gamma \delta + \alpha \kappa + \beta \kappa + \gamma \kappa + \delta \kappa \nonumber \\&\quad + V^{*}(a( \beta + \gamma + \delta + \kappa ) + q( \alpha + \beta + \gamma + \delta ) - c( \alpha + \beta + \gamma + \kappa )) \nonumber \\&\quad + V^{*2}(qa-ac-cq)+ A^{*}V^{*}p(a-c) + A^{*}p(\alpha + \beta + \delta + \kappa ) - akS^{*}); \nonumber \\ G_{2}&= ( \alpha \beta \gamma + \alpha \beta \delta + \alpha \gamma \delta + \beta \gamma \delta + \alpha \beta \kappa + \alpha \gamma \kappa + \beta \gamma \kappa + \alpha \kappa \delta + \beta \delta \kappa + \gamma \delta \kappa \nonumber \\&\quad \!+ V^{*}(a( \beta \gamma \!+\! \beta \delta \!+\! \gamma \delta \!+\! \beta \kappa \!+\! \gamma \kappa \!+\! \delta \kappa ) \!+\! q(\alpha \beta \!+\! \alpha \gamma \!+\! \beta \gamma \!+\! \alpha \delta \!+\! \beta \delta \!+\! \gamma \delta ) \nonumber \\&\quad -\, c(\alpha \beta + \alpha \gamma + \beta \gamma + \alpha \kappa + \beta \kappa + \gamma \kappa )) \nonumber \\&\quad + V^{*2}(aq\beta + aq\gamma + aq\delta - c(a\beta + a\gamma + q\alpha + a\kappa + q\beta + q\gamma )) - aqcV^{*3} \nonumber \\&\quad + A^{*}p(\alpha \beta \!+\! \alpha \delta \!+\! \beta \delta \!+\! \gamma \kappa \!+\! \beta \kappa \!+\! \delta \kappa ) \!+\! A^{*}V^{*}p(a(\beta \!+\! \delta \!+\! \kappa ) - c(\alpha \!+\! \beta \!+\! \kappa )) \nonumber \\&\quad -\, apcA^{*}V^{*2} - akS^{*}(\kappa + \alpha + \delta + qV^{*} -cV^{*})); \nonumber \\ G_{1}&= ( \alpha \beta \gamma \delta + \alpha \beta \gamma \kappa + \alpha \beta \delta \kappa + \alpha \gamma \delta \kappa + \beta \gamma \delta \kappa \nonumber \\&\quad + V^{*}(a(\beta \gamma \delta + \beta \gamma \kappa + \beta \delta \kappa + \gamma \delta \kappa ) + q(\alpha \beta \gamma + \alpha \beta \delta + \alpha \gamma \delta + \beta \gamma \delta ) \nonumber \\&\quad - c(\alpha \beta \gamma + \alpha \beta \kappa + \alpha \gamma \kappa + \beta \gamma \kappa )) \nonumber \\&\quad + V^{*2}(aq(\beta \gamma \!+\! \beta \delta \!+\! \gamma \delta ) \!-\! cq(\alpha \beta \!+\! \alpha \gamma \!+\! \beta \gamma ) \!-\! ac(\beta \gamma \!+\! \beta \kappa \!+\! \gamma \kappa ))\nonumber \\&\quad -aqcV^{*3}(\beta \!+\! \gamma ) \nonumber \\&\quad + A^{*}p(\alpha \beta \delta \!+\! \alpha \beta \kappa \!+\! \alpha \delta \kappa \!+\! \beta \delta \kappa ) \!+\! A^{*}V^{*}p(a(\beta \delta \!+\! \beta \kappa \!+\! \delta \kappa ) \!-\! c(\alpha \beta \!+\! \alpha \kappa \!+\! \beta \kappa )) \nonumber \\&\quad - A^{*}V^{*2}apc(\beta \!+\! \kappa ) \!-\! akS^{*}(\alpha \kappa \!+\! \delta \kappa \!+\! \alpha \delta \!+\! V^{*}(q\alpha \!+\! q\delta \!-\!c\kappa \!-\!c\alpha \!-\!cqV^{*}))); \nonumber \\ G_{0}&= (\alpha \beta \gamma \delta \kappa + V^{*}(q\alpha \beta \gamma \delta + k\alpha \beta \gamma \delta - c\alpha \beta \gamma \kappa ) \nonumber \\&\quad + V^{*2}(aq\beta \gamma \delta -cq\alpha \beta \gamma -ac\beta \gamma \kappa ) - V^{*3}acq\beta \gamma + A^{*}p\alpha \beta \kappa \delta \nonumber \\&\quad + A^{*}V^{*}p(q\alpha \beta \delta + a\beta \delta \kappa - c\alpha \beta \kappa - q\alpha \beta \delta ) - A^{*}V^{*2}apc\beta \kappa \nonumber \\&\quad - akS^{*}(\alpha \delta \kappa + a\alpha \delta V^{*} - c\alpha \kappa V^{*} - cq\alpha V^{*2})); \nonumber \\ H_{2}&= (cfpB^{*}V^{*}); \nonumber \\ H_{1}&= cfp(\alpha B^{*}V^{*} + \beta B^{*}V^{*} + aB^{*}V^{*2}); \nonumber \\ H_{0}&= cfp\beta (\alpha B^{*}V^{*} + aB^{*}V^{*2}) \end{aligned}$$

(20)

For $V^{*} = 0$, we get the coefficients of the characteristic Eq. (19) as:

$$\begin{aligned} G_{4}&= ( \alpha + \beta + \gamma + \delta + \kappa + pA^{*}); \nonumber \\ G_{3}&= ( \alpha \beta + \alpha \gamma + \gamma \beta + \alpha \delta + \beta \delta +\gamma \delta + \alpha \kappa + \beta \kappa + \gamma \kappa + \delta \kappa \nonumber \\&\quad + A^{*}p(\alpha + \beta + \delta + \kappa ) - akS^{*}); \nonumber \\ G_{2}&= ( \alpha \beta \gamma + \alpha \beta \delta + \alpha \gamma \delta + \beta \gamma \delta + \alpha \beta \kappa + \alpha \gamma \kappa + \beta \gamma \kappa + \alpha \kappa \delta + \beta \delta \kappa + \gamma \delta \kappa \nonumber \\&\quad + A^{*}p(\alpha \beta + \alpha \delta + \beta \delta + \gamma \kappa + \beta \kappa + \delta \kappa ) - akS^{*}(\kappa + \alpha + \delta )); \nonumber \\ G_{1}&= ( \alpha \beta \gamma \delta + \alpha \beta \gamma \kappa + \alpha \beta \delta \kappa + \alpha \gamma \delta \kappa + \beta \gamma \delta \kappa \nonumber \\&\quad + A^{*}p(\alpha \beta \delta + \alpha \beta \kappa + \alpha \delta \kappa + \beta \delta \kappa ) - akS^{*}(\alpha \kappa + \delta \kappa + \alpha \delta )); \nonumber \\ G_{0}&= (\alpha \beta \gamma \delta \kappa + A^{*}p\alpha \beta \kappa \delta - akS^{*}(\alpha \delta \kappa )); \nonumber \\ H_{2}&= 0; \nonumber \\ H_{1}&= 0; \nonumber \\ H_{0}&= 0 \end{aligned}$$

(21)

On substituting the above coefficients in Eq. (19) and simplifying after using Eq. (18) we get the characteristic equation for $V^{*} = 0$ as

$$\begin{aligned} G(\lambda )&= (\alpha + \lambda )(\kappa + \lambda )(\delta + \lambda )\left( \lambda ^{2} + \lambda \left( \gamma \left( 1 + \frac{fp\eta }{\delta \gamma \kappa }\right) + \beta \right) \right. \nonumber \\&\left. \quad + \beta \gamma \left( 1 + \frac{fp\eta }{\delta \gamma \kappa } - R_{0}\right) \right) \end{aligned}$$

(22)

where;

$$\begin{aligned} R_{0}=\frac{a\mu k}{\alpha \beta \gamma } \nonumber \end{aligned}$$

We carry out further analysis in the paper by considering the coefficient values given in this Appendix.

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Authors and Affiliations

Tanvi P. Gujarati
- 1
G. Ambika
- 2
Email author

1.Indian Institute of Science Education and Research, TVMThiruvananthapuramIndia
2.Indian Institute of Science Education and Research, PunePuneIndia

Virus antibody dynamics in primary and secondary dengue infections

Abstract

Keywords

Mathematics Subject Classification (2010)

Notes

Acknowledgments

References

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