A theoretical model of the West Nile Virus survival data
 James K. Peterson^{1}Email author,
 Alison M. Kesson^{2, 3} and
 Nicholas J. C. King^{4}
https://doi.org/10.1186/s128650170206z
© The Author(s) 2017
Published: 21 June 2017
Abstract
Background
In this work, we develop a theoretical model that explains the survival data in West Nile Virus infection.
Results
We build a model based on three cell populations in an infected host; the collateral damage cells, the infected dividing cell, and the infected nondividing cells. T cellmediated lysis of each of these populations is dependent on the level of MHC1 upregulation, which is different in the two infected cell populations, interferongamma and free virus levels.
Conclusions
The model allows us to plot a measure of host health versus time for a range of initial viral doses and from that infer the dependence of minimal health versus viral dose. This inferred functional relationship between the minimal host health and viral dose is very similar to the data that has been collected for WNV survival curves under experimental conditions.
Keywords
Background
Viruses in the family Flaviviridae are singlestranded, plussense RNA viruses, principally transmitted by mosquitoes and ticks. They are either viscerotropic, causing diseases such as dengue and yellow fever, or neurotropic, causing central nervous system disease, like West Nile virus encephalitis, Japanese encephalitis or Saint Louis encephalitis, all of which may be fatal. These viruses are found worldwide and the diseases they cause pose a significant public health burden. A careful discussion of the many facets of flavivirus infections can be found in [1], and we therefore confine our introductory remarks to the features salient to this paper.
Virusinfected cells process virus proteins into peptide fragments in the proteasome and these are bound to class I major histocompatibility complex molecules (MHCI) in the endoplasmic reticulum. With the transport of MHC molecules to the cell surface, viral peptides displayed in the context of MHC on the infected cell can be recognized by virusspecific cytotoxic T lymphocytes (CTL). Such CTL kill infected cells by a variety of lytic effector molecules prior to the release of mature virus progeny from infected cells, thereby progressively reducing virus numbers and ultimately eradicating virus from surviving hosts (reviewed in [2]).
Paradoxically, flaviviruses such as West Nile (WNV) and others, directly induce increased expression of MHCI, as well as MHCII, and several adhesion molecules involved in immune recognition by CTL. This increased expression results in a marked increase in the efficiency of recognition and killing of infected cells by WNVspecific CTL ([3, 4]), because although the affinity of individual T cell receptor (TcR)MHCvirus peptide interaction is unchanged, the multiple intermolecular interactions increases the avidity of interaction of virusspecific CTL with the infected cell. This increased avidity also enables the functional interaction with MHC ^{ h i }, infected cells by CTL clones of low MHCvirus peptide affinity, i.e., clones previously below the recognition threshold. Some of these lowaffinity CTL clones are likely to be selfreactive [4] or even able to recognize MHC without peptide specificity [5]. Thus, the increased avidity brought about by high MHC expression enables lowaffinity, selfreactive clones, not normally involved in antiviral immune responses, to lyse both infected and uninfected target cells [6]. Adhesion molecules such as ICAM1, as nonspecific accessory molecules upregulated by WNV infection, also increase the avidity of CTLtarget cell interactions, further lowering the affinity threshold for T cell recognition and target cell lysis [7]. In addition, interferon γ (IFN γ), released by CTL on recognition of their cognate ligand, strongly increases MHC and ICAM1 expression on neighboring target cells, further contributing to the progressive increase in avidity of interaction between CTL and target cells [8]. In this context, the stage of the cell cycle in which a cell is infected is also important; cells infected by WNV in G _{0} (resting) increase MHCI expression by 610fold, compared to a 23fold increase observed in cells infected during the cell cycle (G _{1}, S, G _{2}+M) [3]. WNVinfected G _{0} cells are approximately 10fold more susceptible to CTL lysis than infected cycling cells exposed to the same CTL [3]. Thus, WNVinfected cycling cells are less easily recognized, while the avidity of interaction between CTL and infected G _{0} cells is significantly enhanced by the higher levels of MHCI and ICAM1. Notably, WNV replicates significantly better in the poorly recognized cycling cells than in G _{0} cells. In vivo, most cells are in G _{0}, presumably presenting an easy CTL target once infected, but a small population of productively infected cycling cells maintaining a low immunological profile, could substantially increase the probability of virus transmission to the next host [4]. As indicated above, the release of IFN γ upon target cell recognition by CTL would also increase MHC and ICAM1 expression on uninfected cells in the vicinity of virus infected cells, making high MHCIexpressing (uninfected) cells a potential target for lysis by low affinity virusspecific and/or selfreactive (i.e., crossreactive) CTL clones. The collateral destruction of uninfected cells by lowaffinity clones of this kind would cause substantial additional damage to the brain, with corresponding increases in morbidity and mortality [9]. We have developed a model of the collateral damage caused by a West Nile Virus infection, which is supported by simulation results [10]. A discussion of the underlying simulation code can be found in [11]. We have also used these simulation results to explain the unusual ragged survival data seen in West Nile Virus infections [12]. The previous work, in focusing on very low level details, based on first principles, could be regarded as a micro model. Here we develop a macro model, based on a much higher level approach and ideas. Each model has its pros and cons but we believe each brings complementary insights into a very complicated problem. Our focus here is therefore on developing a macro level theoretical model of how the survival of a host depends on the level of initial viral dose. We believe this approach provides an abstract focus which can also be directed towards more general models of immunopathology and we will briefly mention those connections at the end. These more general models of autoimmune response are the focus of additional work we are doing.
WNV survival data
The CDN model
Next, we do a standard tangent plane approximation on the nonlinear dynamics functions F _{1}, F _{2} and F _{3} to derive approximation dynamics.
Linearization details
where we now use a standard subscript scheme to indicate the partials. Now let’s add IFN γ, upregulation and virus level.
The CDN IFN γ, upregulation and virus model
Next, we again perform a tangent plane approximation on the nonlinear dynamics functions H _{1}, H _{2} and H _{3} just as we did in Section “Linearization details” for the F _{1}, F _{2} and F _{3} we use for the CDN model.
Linearization details
where we now use a standard subscript scheme to indicate the partials.
The algebraic signs of the linearization matrix
Next hold everything constant except the virus level a and increase a to a+δ a. What happens? Let’s think of the virus increase δ a as giving rise to an increase in the amount of virus stored inside a dividing cell or a non dividing cell. Now if the amount of virus in the cell goes up, that means when the cell is lysed, there is more virus available to infect cells which means more cells will be infected in later times. An increase in virus means an increase in collateral damage in general, so \(H_{1\boldsymbol {a}}^{o} = +\). At a given time then, A is the virus level. We can write A=A _{ F }+A _{ D }+A _{ N } where A _{ F } is the free virus, A _{ D } is the virus inside the dividing cells and A _{ N } is the virus inside the nondividing cells. So if A _{ D } goes up, we expect the amount of A and A _{ F } to remain constant. Hence, if A _{ D } goes up, A _{ N } goes down.
Now we need to estimate i, u and a.
The IFN γ, upregulation and free virus model
The approximation to the i, u and a model is handled in a way that is very similar to the previous two expansions which were explained in some detail in “Linearization details” section.
Linearization details
The algebraic signs of the linearization matrix
Oscillations in upregulation and free virus
where A, R, \(G_{1\boldsymbol {i}}^{o}\), β and δ determine a given model.
A health model
however, the UT integration are more complicated.
The UT calculation
Integration details
Model results
The AT calculation
Building the health model
These parameters depend in complex ways on the initial virus dose S _{ 0 } and it is very difficult to tease out the details. The only data we have to help fit this model is the survival data, so the next step of seeing how this model of health gives rise to the experimental survival data requires additional analysis.
Results and discussions
but then the real part of the eigenvalues would be negative and we would have to model the exponential term as \(e^{r_{2} \boldsymbol {S_{0}} t}\phantom {\dot {i}\!}\). The induced oscillations would then be damped and the potential for the WNV survival curve would vanish.
Collateral damage
Conclusions
and the chance of oscillation between the cellular population groups is lost. Hence, we can note some consequences and predictions due to our model.

The crucial assumption here is that the viral infections effect on the host divides into two parts. For a WNV infection, these two cell populations are the dividing and nondividing infected cells, D and N, respectively. We can envision other infectious agents or triggers that give rise to such a split response which then in principle could engender a similar collateral damage response which we interpret as an autoimmune reaction. So there is hope that this approach could perhaps give us insight into more general autoimmune responses. Note that Fig. 11 shows there is collateral damage that oscillates due to the infectious agent which here is WNV. It is clear that other triggering events, another virus or bacteria or even an environment toxin, could give rise to this behavior as well.

Specific to the WNV model, we assume that \(G_{2\boldsymbol {u}}^{\boldsymbol {S_{0}}} = +\), \(G_{3\boldsymbol {u}}^{\boldsymbol {S_{0}}} = +\), \(G_{2\boldsymbol {a}}^{\boldsymbol {S_{0}}} = \), \(G_{3\boldsymbol {a}}^{\boldsymbol {S_{0}}} = +\), which then says the coefficient matrix of the linearized upregulation and free virus model has the form$$\begin{array}{@{}rcl@{}} \left[\begin{array}{ll} G_{2\boldsymbol{u}}^{\boldsymbol{S_{0}}} & G_{2\boldsymbol{a}}^{\boldsymbol{S_{0}}}\\ G_{3\boldsymbol{u}}^{\boldsymbol{S_{0}}} & G_{3\boldsymbol{a}}^{\boldsymbol{S_{0}}}\\ \end{array}\right] &=& \left[\begin{array}{ll} + & \\ + & +\\ \end{array}\right] \end{array} $$This algebraic sign pattern itself can give rise to complex eigenvalues for the linearized nonlinear interaction model and we have not explored this more general problem. We have noted in our discussion in Section “Results and discussions” that if we did not have \(G_{2\boldsymbol {u}}^{\boldsymbol {S_{0}}} = +\), we could still have oscillatory behavior but it would be damped and therefore it would not explain the data we see in the survival experiments. Here, we have posited specific relations that give rise to clearcut oscillations. We have assumed \(G_{2\boldsymbol {u}}^{\boldsymbol {S_{0}}} = G_{3\boldsymbol {a}}^{\boldsymbol {S_{0}}}\) and \(G_{3\boldsymbol {u}}^{\boldsymbol {S_{0}}} = G_{2\boldsymbol {a}}^{\boldsymbol {S_{0}}}\) which gives rise to the characteristic coefficient matrix$$\begin{array}{@{}rcl@{}} \left[\begin{array}{cc} G_{2\boldsymbol{u}}^{\boldsymbol{S_{0}}} & G_{3\boldsymbol{u}}^{\boldsymbol{S_{0}}}\\ G_{3\boldsymbol{u}}^{\boldsymbol{S_{0}}} & G_{2\boldsymbol{u}}^{\boldsymbol{S_{0}}} \end{array}\right] &=&\left[\begin{array}{cc} \alpha^{\boldsymbol{S_{0}}} & \beta^{\boldsymbol{S_{0}}}\\ \beta^{\boldsymbol{S_{0}}} & \alpha^{\boldsymbol{S_{0}}} \end{array}\right] \end{array} $$

The assumptions above then give rise to a general model of how the minimal health changes with varying initial virus dose, which appears to allow us to approximate the data we can measure.
The exact replication of the data found in the biological situation is unlikely to occur. Indeed, a standard dose response curve of survival does not usually repeat exactly from experiment to experiment, except in the form of it, even when using genetically identical animals. The ragged dose response of mortality seen in WNV infection is similarly subject to biological variability. Clearly, with no virus there will be 100% survival, and with a large amount of virus there will be 100% death, as expected in a standard dose response curve. In between these two doses, however, the response to infection is subject to probability, which affects the outcome (survival or death). Thus, if the experiment were undertaken several times it would show the ragged form on each occasion, but not exactly the same percentage survival at each dose used. This implies that small biological differences at the starting point of infection, albeit in genetically identical mice, may subtend a large range of endpoints. It is of interest to note that bypassing the early initiation of the adaptive immune response, by inoculating virus intracranially, that the standard graded dose response occurs with WNV. This is because the replication of the virus overtakes the animal before an effective immune response can be generated, emphasising the role of the immune system in generating this ragged survival curve [12].
Conclusions
We have shown that we can build a reasonable model of how WNV infects a host’s cell in such a way that damage to the host can decrease, even though the inoculating viral dose increases.
Methods
Our model is a macro one and we believe it provides insight as to how we can model more general autoimmune reactions. We propose that for an infectious agent or trigger to cause oscillations in health it is required that the trigger causes alterations in two distinct cell populations. Then, if the nonlinear interactions between these two populations satisfies the conditions for damped oscillatory response we have mentioned here, we should see oscillations in the host health. We consider this work essentially a theoretical model and as we explore what we can do to extend the results to more general autoimmune settings, we hope that we can generate greater mechanistic insight into autoimmune disease.
Declarations
Funding
Publication of this article was funded by National Health and Medical Research Council Project Grants, Numbers, 512413, 1030897, to NJCK and Australian Research Council Discovery Project Number DP0666152 to AK and NJCK.
Availability of data and materials
Not applicable.
Authors’ contributions
All authors were equally responsible for the ideas and development of the work with JP responsible for the mathematical and computational details. All authors have read and approved the manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics and consent to participate
Not applicable.
About this Supplement
This article has been published as part of BMC Immunology Volume 18 Supplement 1, 2017. Systems Immunology & ImmunoInformatics. The full contents of the supplement are available online https://0bmcimmunolbiomedcentralcom.brum.beds.ac.uk/articles/supplements/volume18supplement1.
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Authors’ Affiliations
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