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Graph Theoretic Technique to the Global Stability Analysis of the Discrete Age Structured SVEIR Model with Application to Ebola Virus Disease

Discrete age structured models provide a better approach of discussing the spread of infectious diseases with infectivity, mortality and recovery being age dependent. Measures such as vaccination are carried out in discrete time or applied to individuals in certain age-groups. In the past Ebola Virus disease (EVD) outbreaks, infections in children under 5 years was associated with high mortality rates. In this work, a general discrete SVEIR epidemic model with age structure is formulated. The study establishes the existence of the endemic equilibrium and further shows that it is globally stable using the graph theoretic approach on the method of Lyapunov functions. The model is then applied to EVD dynamics to study the impact of vaccination on the age structured population. The numerical simulation of the discrete age case scenario demonstrates that vaccination of children under 5 years of age against EVD has a great impact in reducing their susceptibility because of the active immune system as compared to the older population who have a poor response to vaccine immunity. For the older population, vaccination is not very eﬀective in reducing their susceptibility.


Introduction
Mathematical models are instrumental in providing guidance regarding the future projections of ongoing public health crises, and in assessing the potential impact various intervention strategies may have towards the transmission control. Age has been connected to symptom severity and mortality in several diseases like Ebola Virus Disease (EVD) and Corona virus (COVID-19) [1]. Age is also a clear modifier of survival probability, with very high values of case fatalities in young children and the elderly.
An Ebola vaccines working group meeting of April 2019 recommended the vaccination of expectant women, lactating women and children below 5 years [2]. The report further suggested that manufacturers of candidate Ebola vaccines should prioritize the development of a strategy to generate safety, immunogenicity and efficacy data for children aged under 5 years [2].
Disease transmission models with age dependent contact rates are more realistic [3], which makes the models with age dependent susceptibility relevant for vaccination planning [1]. Factors in an immunization program such as target groups and the number of doses recommended can be actualized easily in an age-structured framework. Age structure in disease models has been studied using continuous and discrete approaches in literature and they include ordinary differential equations for discrete age groups [4] and for continuous ages [5]. The challenge for ordinary differential equations models lies in the high dimensionality and scale of the system. This makes the establishment of the uniqueness and global stability of the endemic equilibrium when the effective reproduction number is greater than one very difficult. Discrete age structured models are regarded as nonlinear differential equations coupled on transmission networks.
With regard to transmission dynamics of a given infection there has to be interaction among the age groups and the subgroups and transition among the subgroups, which is analogous to the directed oriented paths in Graph Theory. A graph G can be drawn within age groups and between subgroups, because of the strong connectivity, which guarantees the existence of an oriented path. From a given age group, a unicyclic sub-graph can be formed by adding the infection arcs to the corresponding tree T which is equivalent to the total population N. Therefore the total number of infections can be considered as a sum over all the unicyclic graphs representing the interactions.
The study adapts a discrete age structured SVEIR epidemic model to analyze EVD data from the Democratic Republic of Congo and examine the effects of vaccination for children under 5 years. The paper is organized as follows; in section 2 the discrete age structured SVEIR model is developed and the effective reproduction number derived. In section 3, the global stability of the endemic equilibrium is discussed. In section 4, a model with three age groups is developed to analyze the impact of vaccination on the various age groups.

Description and Formulation of the Model
In this section, a general model is developed that subdivides the human population into classes of susceptible S(t) , vaccinated V (t), exposed E(t), infected I(t) and recovered R(t) with variations with respect to age. The susceptible population is recruited at the rate Λ and are vaccinated at the rate α and acquire infection at the rate β kj , the vaccinated individuals acquire infection at the rate β kj upon interaction with the infected and recover at the rate θ, infected individuals die as a result of infection at the rate δ and recover at the rate π and natural death occurs in all classes at the rate µ. The total population is given by; The resulting model is represented by the flow diagram below; The system of differential equations describing the dynamics of the model is as follows; where S k , V k , E k , I k , R k denote the k th population segregated in terms of age that are susceptible, vaccinated, exposed, infected and recovered respectively.

Positivity and boundedness of solutions
Since the model describes human population, the state variables of model (1) can be shown to be non-negative and ultimately bounded in Γ(R 5 ). Therefore, the model is biologically meaningful and mathematically well posed.

The effective reproduction ratio and equilibrium
Definition 3.1. The Effective reproduction number (Rv) is the average number of secondary infections due to a single infectious individual introduced in a fully susceptible population over the course of the infectious period in which vaccination is introduced as an intervention measure. If Rv < 1 it means that on average, an infected individual produces less than one new infected individual while Rv > 1 means each infected individual produces more than one new infection on average.
Rv for model (2) is computed as the spectral radius of the next generation matrix [6] and is given by;

The endemic equilibrium E *
This is a state where the disease persists in a given population.
Proof. To prove the existence of the endemic equilibrium, it is shown that dI dt > 0 whenever Rv > 1. From equation (4) of system (2); but from equation (3) of system (2), Therefore substituting for E k equation (5) into equation (4) yields, Simplifying equation (6) using equation (3) gives; Theoretically, if Rv < 1 the disease dies out and if Rv > 1, then the disease persists in a given population.

Global stability of the endemic equilibrium
In this section results from [7] and [8] are reviewed to study the stability analysis of system (2).  (i) n = 1 and B = 0 where B1 and B3 are square matrices and Q T is the transpose of matrix Q. Otherwise, B is called irreducible.

Consider the linear system
: : : : : : be the laplacian matrix of the directed graph G(B) associated to matrix B.
The matrix B = β kj and B 1 = β 1 kj codes patterns of interaction and transmission among groups that are built into the model. A directed graph L = G(B)(G(B 1 )) whose vertex k represents the k th group, k=1,2,....,n associated to B(B 1 ) = β kj (β kj ) can be constructed with vertices 1,2,...,n and a directed arc (k,j) from k to j, if and only if β kj (β kj ) > 0. Since matrix B(B 1 ) is irreducible then G(B)(G(B 1 )) is strongly connected, this implies that any two distinct vertices are joined by an oriented path. Biologically, any two age groups k and j have an indirect or direct transmission route which is analogous to the oriented path. (1) The spectral radius ρ(B) is a simple eigenvalue of B and B has a positive eigenvector w = (w1, w2, ...., wn) corresponding to ρ(B).
(3)If F is a diagonal and positive matrix, then BF is irreducible. Proof. See [10] for the proof.
Proposition 3.2. Suppose B = (β kj )n×n is irreducible and R0 > 1, and assume that; then system 2 is uniformly persistent and there exists an endemic equilibrium E * and it is globally asymptotically stable in Γ(R 5 ).
Consider the following Lyapunov function; Computing the derivatives of F with respect to t along solutions of system (2) gives; Substituting the first four terms of system (2) into equation (9) dF Expanding equation (10) yields; Factorizing equation (11) yields; Using the relation 2 − S k S * k − S * k S k ≤ 0, for all S k > 0 and assumption (8) gives; Equation (13) can be expressed as; Equation (16) can be written as; Substituting equation (17) in the second equation of system 2 gives; Comparing equation (18) with the second equation of system (2), the two equations are equal if and only if a = 1. Therefore It follows from Bv = 0 and Bv 1 = 0, that; holds. By lemma (3.2), {v1, v2, ..vn} = {C11, C22, ..., Cnn}. Consequently each term in v k β kj is a product of n β ij whose sub indices define transformations and are represented by directed arcs in a unicyclic graph Q. v k = C kk represents the sum of all rooted spanning directed subtrees T of G of vertex k. The directed arc from a vertex k to j forms a unicyclic subgraph Q of G. Therefore each term v k β kj represets the weights of a unicyclic subgraph. The double sum in M2 is considered as the sum over all arcs in the cycle of all unicyclic subgraphs.
Therefore; ln(Π T ∈T i (kj)∈E(CQ) This as well applies to, ln(Π T ∈T i (kj)∈E(CQ) Since M1 and M2 = 0, it therefore follows that the only compact invariant subset of the set where dF dt = 0 is the singleton E * . By the La Salle's invariance principle [11,12], E * is globally stable in Γ. Theoretically, this implies persistence of the disease in the population.

Application to Ebola Virus Disease
In this section, the study shows the applicability of the main theoretical results. Moreover, effects of vaccination and age on the transmission dynamics of EVD are discussed. Consider the system with n = 3, structured according to age under 5 years (group 1), above 6 -49 years (group 2) and above 50 years (group 3).: where β kj = β k c kj , β k is the probability of transmission between a susceptible individual in age group k with an infected individual and c kj is the average number of contacts from individuals in age groups j with individuals in age group k.
Other parameters have the same meaning like in the general model.

Numerical results and analysis
In this section the numerical results of model (19) and its analysis are presented. The parameter values used in the simulation are either obtained from literature and others were from assumptions to better understand the impacts vaccination has on different age groups. Matlab software was used to illustrate the numerical results describing the theoretical results.  Fig. 2 illustrates how recoveries of children below 5 years would be in the absence of vaccination. The previous lack of vaccination as an intervention strategy for this age group might have been the reason for the high case fatalities as compared to other age groups, since at the onset of the infection the clinical symptoms of EVD are similar to those of other childhood infections. This together with the rapid disease progression are the causes of high fatality rates.
From Fig. (3) and Fig. (4) it can be deduced that the higher the rate of vaccination the lower the number of infections to age groups 1 and 2. This indicates that to reduce the number of EVD infections, vaccination rate should be near perfect, i.e the target population should be vaccinated at nearly 100%.

Discussion
An individuals age is important in the disease dynamics and intervention measures. With a set of differential equations the dynamics of EVD was described by a system of general first order ordinary differential equations. Qualitative analysis was done for the general model. A threshold R0 which determines whether the disease persists or dies out was computed. The disease free equilibrium is proved to be globally stable if R0 < 1 and the Fig. 4. Impact of Vaccination on reducing infections in age group 3 endemic equilibrium is unique whenever R0 > 1 and globally stable as tested using a graph theoretic approach on the method of Lyapunov functions. An example of a model with three age groups, under 5 years (group 1), above 6-49 years (group 2) and above 50 years (group 3) is given. Simulation was done with data from published literature on DRC. Analysis of different scenarios was done and demonstrates that children under 5 years should also be considered for a vaccination program in the fight against EVD. The simulation results also indicate that with effective vaccination α = 0.5 − 0.9 the exposed population drop to nearly zero after some given number of days that is, they tend to the disease free equilibrium. For the older population vaccination is not very effective in reducing their susceptibility.

Conclusion
EVD is a disease that causes a great economic impact and has caused significant disease mortality among children under 5 years. Following its recurrence especially in Uganda and the DRC, the WHO suggests the need to include children under 5 years as case contacts. Due to the severity of the disease among children below 5 years and development of a vaccine that is safe for this population, it is important to consider vaccinating this age group to reduce disease mortality. The numerical simulation of a discrete age case scenario demonstrates that vaccination of children under 5 years against EVD has a great impact in reducing their susceptibility to Ebola virus disease, this may be, because of the active immune system as compared to the older people who may have weak immunity and therefore response to vaccine immunity is poor.

Data Availability
All data used to support the findings of the study are included within this article and can be obtained from the corresponding author upon request.

Competing Interests
Authors have declared that no competing interests exist.