A Study of Within-Host Dynamics of Dengue Infection incorporating both Humoral and Cellular Response with a Time Delay for Production of Antibodies

: a. Background: Dengue is an acute illness caused by a virus. The complex behaviour of the virus in human body can be captured using mathematical models. These models helps us to enhance our understanding on the dynamics of the virus. b. Objectives: We propose to study the dynamics of within-host epidemic model of dengue infection which incorporates both innate immune response and adaptive immune response (Cellular and Humoral). The proposed model also incorporates the time delay for production of antibodies from B cells. We propose to understand the dynamics of the this model using the dynamical systems approach by performing the stability and sensitivity analysis. c. Methods used: The basic reproduction number (R 0 ) has been computed using the next generation matrix method. The standard stability analysis and sensitivity analysis were performed on the proposed model. d. Results: The critical level of the antibody recruitment rate( q ) was found to be responsible for the existence and stability of various steady states. The stability of endemic state was found to be dependent on time delay( τ ). The sensitivity analysis identified the production rate of antibodies ( q ) to be highly sensitive parameter. e. Conclusions: The existence and stability conditions for the equilibrium points have been obtained. The threshold value of time delay ( τ 0 ) has been computed which is critical for change in stability of the endemic state. Sensitivity analysis was performed to identify the crucial and sensitive parameters of the model.


Introduction
Dengue is a mosquito-borne viral disease transmitted by female mosquitoes mainly of the species Aedes aegypti and, to a lesser extent, Ae. albopictus. It is an acute illness of sudden onset that usually follows a benign course with symptoms such as headache, fever, exhaustion, severe muscle and joint pain, swollen lymph nodes (lymphadenopathy), and rash. Dengue virus has four serotypes denoted by DEN I, DEN II, DEN III and DEN IV. When a person is infected rst time by a dengue virus, the immune system against the dengue virus is activated. The infection of dengue virus for the second time by a di erent serotype leads to secondary infection.
Mathematical models play a crucial role in understanding and controlling the dengue virus. Many vector-host models are developed to study the dengue virus at population level [2,4,9]. There are very few works performed to understand the dengue viral dynamics in human body [10,15]. But, not one of the current models [6,8,12,13] have considered the role of innate immune response in the defence to the dengue viral attack until recently [5] introduced innate immune response and show that it can reproduce the characteristic features of the primary infection.
Innate immune response is the rst line of defense mechanism in the human body to ght against any invading pathogen. This is non-speci c to invading pathogen. The innate immune system thus produces a type I interferon(IFN), with main interferon's being IFNα/β. The IFN produced induce resistance in infecting the neighboring uninfected cells thus, limiting the quick spread of the virus [11,14,16]. Not only this, the IFN has ability to active the natural killer cells(NK) during the primal stages of the infection, that have ability to destroy infected cells [14,3,17]. In dengue, we can detect IFN generally in 24-48 hours after the infection and correlating with the virus titer peak [1]. Thus, IFN produced can act as early indicator for the infection. Dengue patients have also shown early activation of the NK cells [5,7].
In the second stage of infection, the viral pathogen will be carried to the thymus to activate the adaptive immune response which is more powerful and more speci c to the invading pathogen. The adaptive immune system takes longer to respond but it provides a long term immunity against the virus [8,12]. The adaptive immune system consists of both cellular and humoral responses. In Cellular response, T-cells will clear the virus and in humoral response, B-cells will mature into plasma cells and secrete antibodies that neutralises the antigen-presenting viruses.
In this paper, a non-linear model is proposed which incorporates innate and adaptive (both cellular and humoral) immune responses. The equilibrium points and the stability analysis is carried out. The time delay has been considered during the process of antibody secretion and its further impact on stability of steady states have been discussed. The paper is organized as follows: In section 2, the mathematical model has been formulated. In section 3, model analysis has been performed. In section 4, the sensitivity analysis has been performed. In section 5, discussion has been given.

Mathematical Model
The present model is an extension of the model in [10]. Let S, I and V be the number of healthy target cells(monocytes, macrophages or dendritic cells etc), infected cells and dengue virions respectively. To model the innate immune response on dengue viral dynamics, let F denote the interferons (IFN) released by the infected cells. Let Z be the number density of T immune cells due to adaptive immune response (cellular immune response). The T cells activate B cells (humoral immune response) that again di erentiate to plasma cells to produce antibodies. Let B and A be the density of B cells and neutralizing antibodies respectively. Let τ be the time delay in the production of antibodies from B cells. The antibodies bind the virions and neutralize them.
The dynamics of S, I, V, Z, B, A, F are described in the following system of di erential equations.
The values for parameters are given in table 1. Using the parameter values in table 1, the dynamics of all the compartments (S, I, V, Z, B, A, F) from the initial point (S,I,V,Z,B,A,F) = (200,50,100,6,50,0,0) are given in the gure 1.

Positivity of Solutions
For any mathematical model it is fundamental to show that the system of equations considered are positive and has bounded solutions. We now show that if the initial conditions of the system (1) -(7) are positive, then the solution remain positive for any future time. Using the equations (1) -(7), we get, Thus all the above rates are non-negative on the bounding planes (given by S = 0, I = 0, V = 0, Z = 0, B = 0, A = 0, F = 0) of the non-negative region of the real space. So, if a solution begins in the interior of this region, it will remain inside it throughout time t. This happens because the direction of the vector eld is always in the inward direction on the bounding planes as indicated by the above inequalities. Hence, we conclude that all the solutions of the the system (1) -(7) remain positive for any time t > 0 provided that the initial conditions are positive. Next we will show that the solution is bounded with each of the bounded control variables.
Therefore the biologically feasible region is given by the following set,

Basic reproductive number
The basic reproductive number for the model (1) - (7) is de ned as the average number of infected target cells generated by a single infected cell placed in an uninfected target cell population [13]. Basic reproductive number computed by next generation approach is obtained as . Equilibrium states and their stability Three equilibrium solutions are obtained for the dengue model (1) - (7).

. . Infection-free state(E )
The equilibrium densities for the state E is given as Now, for the stability analysis, the jacobian matrix (J) for the system (1) - (7) is computed as The characteristic equation about infection free state is given as where, From the above equation 10, the following observations are made: a. It is independent of time delay. b. When R < 1, all the eigenvalues are negative. Therefore, disease-free state is locally asymptotically stable. c. When R > 1, the two eigenvalues will have negative real parts provided Thus, even for R > 1, there is the critical level of antibodies(qc) over which infection will die out and the E becomes locally asymptotically stable. If q ≤ qc, then E will be unstable.

. . Ine ective cellular immune response state(E )
The expression for this state E = S, I, V , , B, A, F, are obtained as: The existence condition for E becomes : R > 1. This is because only then we can have I ≥ 0,which is our initial condition.
The characteristic equation about the state E is given as: where, Now it is observed from the above characteristic equation that all the eigenvalues are found to be negative provided q > qc.

. . Endemic equilibrium state(E * )
The expression for this state E * = S * , I * , V * , Z * , B * , A * , F * are obtained as: The endemic equilibrium point (E * ) exists for the following conditions to be satis ed, and Further, to analyze the stability of endemic state,the characteristic equation about endemic state is given as: where, In the absence of delay,the characteristic equation (12) becomes Since the terms are very lengthy and complex, Routh-Hurwitz criteria for determining the sign of λ is of little use. Hence, we numerically show the stability of E * in the absence of delay. This is shown in gure 3. When delay is present, the characteristic equation is modi ed by substituting λ = iω as where, For this equation to have at least one positive root, E < 0. Thus, there will exist a positive root ω , which will satisfy the equation (14). It is given as: where, The corresponding critical delay with the parameters as in table 1 is computed as τ = 12.3168. We now show numerical simulations in gure 4 that the virus count converges to E * from the initial point P = ( , , , , , , ) and exhibits a stable behaviour when τ < τ . But, the system admit periodic solution or more complex behavior when τ > τ . Thus, the system undergoes Hopf-bifurcation with the critical delay given by equation 15.

Sensitivity Analysis Sensitivity Analysis of Rate of Infection(β )
To analyze the sensitivity of the model with respect to the rate of infection (the rate at which healthy cells are converted to infected cells due to interaction with virus particles (β ), we solve the model numerically for randomized values of β . Using the method mentioned in [15], we can analyze how sensitive the system reacts to uctuations in β . Here we use initial conditions (S, I, V, Z, B, A, F) = (192,0.1,0.5,6,240,1400,6) and the parameter values from table 1. Figures 5, 6 contains two subplots each, where the rst subplot the time series for virus count at random values of β in the respective ranges. The second subplot gives the mean value of virus count in the range of β . From gure 5, the mean value of virus count reaches peak around ve days and declines slowly till 10 days. From gure 6, the mean value of virus count reaches peak value initially and declines very rapidly in the short span of 5-10 days. Further, by comparing gure 5, gure 6 it is observable that, for small values of β , the infection takes longer time to clear when compared with the larger values of β . Thus the model is stable with respect to the infection rate β .

Sensitivity Analysis of Burst Rate of Virus Particles(k)
Here the variation dynamics of model (1) -(7) with respect to the burst rate of virus particles, k, is considered. The time series for virus count at random values of k and the mean value of the virus count is given in the two subplots in gure 7. This plot means that for various values of k, the mean virus count declines faster before 5 days very rapidly. From this plot, it can be inferred that the burst rate of particles has a predictable e ect on the behaviour of the virus count. Figures 8, 9 contains two subplots each, where the rst subplot the time series for virus count at random values of q in the respective ranges. The second subplot gives the mean value of virus count in the range of q. From gure 8, the mean virus count remains low till 5 days and starts increasing after that. As the value of q increases, from gure 9, the mean virus count starts decreasing from rst day and reaches to the minimum by 15 days. Further, by comparing gure 8, gure 9 it is observable that, for small values of q, the mean virus increases over time. As the value of q increases from 0.4, the mean virus count starts decreasing and reaches to the minimum. Thus we can infer that the system is highly sensitive for changes in the parameter q.

Results
From the above mathematical analysis it can be seen that the critical level of the antibody recruitment rate(q) has been found to be responsible for the existence and stability of various steady states. The stability of endemic state was found to be dependent on time delay(τ). The critical value of time delay(τ ) was obtained from Hopf bifurcation analysis.
Findings from the sensitivity analysis of several key parameters such as β , k, q that are thought to play a signi cant role in dengue infection are the following: • The production rate of antibodies (q) was found to be highly sensitive.
• It was observed that the burst rate of virus particles (k) has a predictable behaviour and the rate of infection(β ) is sensitive only in some interval range.

Conclusions
In this study, a conceptual model was developed using a system of non-linear ordinary di erential equations to understand the dynamics of dengue virus. The ordinary di erential equations describe the dynamical behaviour of healthy cells, infected cells, virus count, T-cells, B-cells, antibodies and interferons. The model incorporates the e ects of innate and adaptive(humoral and cellular) immune responses to present the mechanism of viral infection. Also, a delay in the B-cell production has been considered.
The existence and stability conditions for the equilibrium points have been obtained. The threshold value of time delay (τ ) has been computed which is critical for change in stability of the endemic state.
It has been observed that that innate response co-relates with the virus titer in the early stages of the infection which can help in an early detection of a serious disease.
Sensitivity analysis was performed to identify the crucial and sensitive parameters of the model. It was see that the burst rate of virus particles and the the rate of production of antibodies are the sensitive parameters in our system.
In the end we conclude by stating that the human immune system is very complex and the works presented in this paper could enhance our understanding of this complex immune response.