key: cord-0828920-w2fleeg9 authors: Zhdanov, Vladimir P.; Jackman, Joshua A. title: Analysis of the initiation of viral infection under flow conditions with applications to transmission in feed date: 2020-06-09 journal: Biosystems DOI: 10.1016/j.biosystems.2020.104184 sha: 648c1ea549181dceb3f75548109c3c83df7e9bb6 doc_id: 828920 cord_uid: w2fleeg9 While kinetic models are widely used to describe viral infection at various levels, most of them are focused on temporal aspects and understanding of corresponding spatio-temporal aspects remains limited. In this work, our attention is focused on the initial stage of infection of immobile cells by virus particles (“virions”) under flow conditions with diffusion. A practical example of this scenario occurs when humans or animals consume food from virion-containing sources. Mathematically, such situations can be described by using a model constructed in analogy with those employed in chemical engineering for analysis of the function of a plug-flow reactor with dispersion. As in the temporal case, the corresponding spatio-temporal model predicts either the transition to a steady state or exponential growth of the populations of virions and infected cells. The spatial distributions of these species are similar in both of these regimes. In particular, the maximums of the populations are shifted to the upper boundary of the infected region. The results illustrating these conclusions were obtained analytically and by employing numerical calculations without and with the dependence of the kinetic parameters on the coordinate. The model proposed has also been used in order to illustrate the effect of antiviral feed additives on infection towards curbing disease transmission. The mechanisms of initiation and spread of viral infections are inordinately diverse and complex, and one of the ways to clarify them systematically is based on the development of theoretical models (Bocharov et Herein, we focus on developing a mathematical framework to describe spatiotemporal aspects of the initial phase of viral infection within the scope of category (ii), i.e., taking virus and cell populations into account. At this level, existing spatio-temporal models are usually aimed at human infections and are constructed by complementing the corresponding temporal models with terms describing the Fickian diffusion of virions in combination with the Neumann (no flux) boundary conditions that do not account for flow in the medium. The first models of this category were, to our knowledge, proposed fifteen years ago by Funka et al. (2005) and Wang and Wang (2007) . More recent treatments are described in articles by The minimal "standard" (or "basic") temporal model of viral infection operates with the concentrations (or populations) of target and infected cells and virions, C * (t), C(t), and c(t) (Perelson, 2002; Smith, 2018) . Mathematically, it can be formulated as where w and J are the supply rates, µ, κ, and γ are the death (or elimination) rate constants, k is the infection rate constant, and r is the virion-production rate constant. A slightly extended version of this model can be obtained by dividing the population of infected cells into two subpopulations staying in the eclipse and active phases, respectively (Smith, 2018) . In reality, viral infection is accompanied by the activation of the immune system. This means that the virion-elimination rate constant, γ, increases with increasing time. In the standard model, γ is considered to be constant. This approximation corresponds to the initial phase of infection. We are interested in this phase and accordingly accept the standard model. Aiming at this phase, we can simplify the model further by assuming C(t) C * (t) and neglecting the dependence of C * on time in Eq. (2), i.e., by replacing C * by C * (0). Including then C * (0) into k, i.e., replacing kC * (0) by k, we rewrite Eq. (2) as In this approximation, Eq. (1) is not needed, and we can operate only with Eqs. (3) and (4). The types of kinetics predicted by the standard model for the initial phase of infection can be clarified by illustrating its prediction in the steady-state case. In this case, Eq. (4) yields With this relation, Eq. (3) is reduced to The latter expression for the virion concentration indicates that the steady-state regime is possible provided γκ > kr. If this condition is not fulfilled, the growth of the populations of infected cells and virions is predicted to be exponential. This can be proved by calculating the eigenvalues corresponding to Eqs. (3) and (4), J o u r n a l P r e -p r o o f The kinetics predicted by Eqs. (3) and (4) contain exp(λ 1 t) and exp(λ 2 t). If γκ < kr, we have λ 1 > 0, and accordingly the growth is exponential. Bearing in mind the initiation of viral infection occurring via consumption of contaminated feed, we add the spatial ingredients into the model described above c(x, t) (x is the coordinate along the intestine). More specifically, we consider that the cells are immobile and describe infected cells by using the equation apparently identical to (4), i.e., The difference between Eqs. (4) and (9) is that the former is temporal (i.e., c depends only on time) while the latter is spatio-temporal (i.e., c depends on time and the coordinate along the intestine). To describe virions, we omit J in Eq. (3) and complement it by the terms taking medium flow and virion diffusion into account, i.e., where v is the flow rate, and D is the diffusion coefficient. The rate constants in these equations have the same meaning as in (3) and (4). Concerning the concentrations, C and c, used in Eqs. (3) and (4) By analogy with macromolecules (e.g., protein) in feed, one can expect that the virions function primarily at a certain length scale, i.e., at 0 ≤ x ≤ L, because further on (at x > L) their degradation is rapid and their concentration is negligible. Following this line, we solve Eqs. (9) and (10) at 0 ≤ x ≤ L. Mathematically, the structure of Eqs. (9) and (10) where c • is the inlet concentration in the feed. The meaning of the former boundary condition is obvious. Concerning the latter boundary condition, it can be valid e.g. in the situations when the virion degradation at x > L rapid and their concentration is there negligible. Alternatively, one can employ at x = 0 the Danckwerts boundary condition implying that the total virion flux, 7 J o u r n a l P r e -p r o o f is fixed. The discussion of this boundary condition in the context of chemical engineering is given by one of us (Zhdanov, 2020) . In our present context, as we have already noticed, the role of diffusion is expected to be less important compared to flow. In this limit, the Dirichlet and Danckwerts boundary conditions are nearly equivalent. With this reservation, we use the Dirichlet boundary conditions in our analysis below, because the corresponding analytical results are somewhat simpler. In particular, we operate with constant inlet concentration. Practically, this means that the feed-mediated supply of virions takes place over a long period. For short periods, the use of the viral dose (or, in other words, viral uptake) might be preferable. The suitable initial conditions for Eqs. (9) and (10) For analytical and numerical calculations, it is convenient to use the dimensionless coordinate, normalize the flow velocity and diffusion coefficient, and rewrite Eq. (10) and conditions (11) and (13), respectively, as c(0, t) = c • and c(1, t) = 0, C(x, 0) = 0 and c(x, 0) = 0. Then, Eqs. (9) and (15) can be solved analytically or numerically provided the rate constants are independent of x or numerically provided the rate constants depend where and If β v 2 /4D, η 1 and η 2 are complex and can be represented as With this specification, expression (19) can be rewritten as The stability of the steady-state solution [(19) and (24)] of Eqs. (9) and (15) can be identified by calculating the corresponding eigenvalues. In particular, our analysis indicates that the solution is stable provided β 0.6, than in the case when k is constant ( Fig. 1 with r/v > 0.36). In addition, we performed calculations in the case when both γ and k depend on x as described above. As expected, the corresponding kinetics are close to those shown in Fig. 4 (not shown). The model under consideration can be easily extended in various directions. In At the simplest level, one can e.g. consider that the viral elimination rate constant depends on the concentration of the additive, c a , where γ • is the value corresponding to the additive-free case, and A is a constant. Then, the model can be used directly by considering c a (or Ac a /v) to be a governing parameter. This choice is reasonable provided that an additive is used for a long period (for short periods, the additive dose can be employed). Alternatively, one can consider that c a depends on the coordinate and complement Eqs. (9) and (10) [or (15) ] by the corresponding equation that can be constructed by analogy with Eq. (10). In this section, we use the former approach including Eqs. (9) , (15) , and (28) . To describe the system without feed additive (c a = 0), we choose the coordinateindependent parameters (k/v = 0.1, κ/v = 0.01, r/v > 0.5, γ • /v = 0.1, and D/v = 0.1) so that the populations of virions and infected cells grow exponentially (Fig. 5) . Then, Ac a /v is employed as a governing parameter. The corresponding kinetics were calculated for Ac a /v = 0.01, 0.1, 1, and 10 (Fig. 5) . For Ac a /v = 0.01, the effect of an additive on the evolution of infection is predicted to be practically negligible, so that the corresponding kinetics are not distinguishable from those obtained with c a = 0 (not shown). If Ac a /v = 0.1 and 1, the effect of an additive on the kinetics is relatively weak and strong, respectively, and, in both cases, the growth remains exponential. For Ac a /v = 10, the system reaches a steady state with small populations of virions and infected cells. In other words, this means that the exponential growth is suppressed, and the infection is under control. The initiation of viral infection can be described in the framework of the stan- The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 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