SEIS MODEL WITH MULTIPLE LATENT STAGES AND TREATMENT IN AN EXPONENTIALLY GROWING POPULATION

An SEnIS epidemiological model with vital dynamics in an exponentially growing population is discussed. Without treatment three threshold parameters R0,R1 and R2 determine the dynamic of compartments sizes and that of the fractions. With the treatment the dynamics of the population and that of the epidemic depend on three other threshold parameters RT ,R1T and R2T . We made a link between the models with one latent stage and the models with multiple latent stages by defining and deriving the ”effective” activation rate and the ”effective” treatment rate for the latent individuals. We defined the treatment force, the relative treatment force and deduced the critical treatment force needed to eradicate the disease. The theoretical results are validated by numerical simulations.


Introduction
Bame et al. studied SEIS models with n Latent classes in [1] assuming that an infected individual passes through n latent classes before becoming infectious.In [2] [11].For these models an individual can become infectious from any latent class.Jabbari et al. studied a two-strain TB model with multiple latent stages in [9].In [10] Meng et al. studied the dynamics of an SEIS model with therapeutic strategies.For these models the population has a constant recruitment rate and the incidence is the simple mass action.
In this paper we study an SEIS model with n latent classes.The population has an individual birth rate b and an individual death rate µ such that b > µ, that is the population is growing exponentially without the epidemic.With the introduction of the disease, the population is split into n + 2 compartments.The compartment S of susceptible individuals, the compartments E 1 , ..., E n of individuals in n different latent stages, and the compartment I of infectious individuals.S, E 1 , • • • , E n , I denote the numbers of individuals in the corresponding compartments.We consider serial latent stages that is individual pass successively in the compartments E 1 ,...,E n .We assume that an individual in the latent class E m can become infectious with rate k m or be transferred into E m+1 with rate ν m for m = 1, • • • , n − 1.The contact rate is c and the probability of infection during a contact of an infectious with a susceptible is β .Thus the effective contact rate is cβ and the incidence is cβ SI/N, where N denotes the total population size (N = S + E 1 + ... + E n + I).An infectious individual recovers with rate δ .In that case, he becomes susceptible again.The disease induces an additive death rate d to the infectious individuals.We assume afterward that there is a treatment for latent individuals and infectious individuals.With the treatment a new compartment T of the individuals under treatment is added.The treatment rate for an individual in the latent class E m is r m , m = 1, • • • , n and that for an infectious individual is r.A treated individual recovers with rate θ .All the parameters listed above are positive.The summary of the notations used in this paper is given in Table 1.Our model is different from that listed above since we consider a population that grows exponentially in absence of the disease, instead of a population with a constant recruitment rate.Furthermore we have a standard incidence instead of the simple mass incidence.This model fits for tuberculosis.
Our paper is organized as follow.In Section 2 we study the dynamic of the model without treatment.Section 3 is dedicated to the study of the model with treatment.We validate the Table 1.Summary of notations theoretical results by simulations in Section 4. Thereafter we conclude the paper and discuss some perspectives in Section 5.

The model without treatment
In this section we study the dynamic of the model without treatment.We give the system of the model, then we derive its basic reproduction number and compare hence the multiple latent stages model to the unique latent stage model, give conditions for endemic equilibria, therefore we study the dynamic of the fractions and hence deduce the asymptotic behaviour of the compartments sizes.The diagram of the SE n IS model is given in Figure 1.

The model
The model without treatment is given by the following system of ordinary differential equations (ODE). (2.1)

The basic reproduction number
The basic reproduction R 0 is defined as the average number of new cases of an infection caused by one typical infected individual, in a population consisting of susceptibles only [4].
Theorem 2.2.The basic reproduction number R 0 of the epidemic defined by System (2.1) is given by Proof.R 0 is the product of the contact rate c, the probability to transmit the disease during a contact β , the average infectious time 1/(µ + d + δ ) and the probability for the newly ex- µ+ν l +k l .The latter is the sum of the probabilities to become infectious from the different latent compartments.More precisely µ+ν l +k l is the probability for the individual to become infectious while being in the compartment E m .
If R 0 ≤ 1, then the epidemic cannot invade the population.But if R 0 > 1, then the epidemic will invade the population.
Then it is enough to show that (2.4) Let's assume first that n = 2 then Let's assume now that the result is true for n = p for a given integer p ≥ 2, that is Thus we have Thus we have shown by induction that R 0 satisfies (2.3) when

Equilibria
If I = 0 then dN/dt = (b − µ)N.Therefore the population will grow if the disease dies out as we assume that b > µ.Thus, System (2.1) does not admit a disease free equilibrium.The following result gives necessary and sufficient conditions for endemic equilibria.
Theorem 2.6.Let X(t) = (S(t), E 1 (t), • • • , E n (t), I(t)) be a solution of System (2.1).X(t) is constant if and only if the parameters satisfy and the initial values satisfy (2.7) Proof.By using successively the derivatives of E n , E n−1

The dynamic of the fractions
For epidemics in populations that grow exponentially in the absence of the disease, it is common to study the fractions of individuals in the different compartments [3,7,12,5].
Remark 2.8.As for other models the natural death rate µ does not appear in System (2.8).
That is understandable since all the individuals of the population have the same natural death rate.Thus the natural death does not affect the fractions.
If e 1 (t) = 0 at a given time t ≥ 0, then If e m (t) = 0 at a given time t ≥ 0, then Thus all solution of System (2.9) starting in D remains in D for all t > 0.
In the following we use the method of the next generation matrix described in [6] to derive a threshold parameter R 1 with threshold value equals 1 for System (2.9).It is obvious that ).After some calculus, one gets Remark 2.10.One gets R 1 by substituting µ by b in the expression of R 0 in Equation (2.2).
Following Odo et al. [5, Page 89], we refer to R 1 as the relative basic reproduction.
Theorem 2.11.The disease free equilibrium (0, Proof.By Theorem 2 in [6], the disease free equilibrium is asymptotically stable if R 1 < 1 and unstable if R 1 > 1.To show its global stability, let L be the function defined on D by By Equation (2.11) we have W is an affine function.Thus, it reaches its maximum on the extreme points of the closed set D. We have Therefore L is a Lyapunov function for System (2.9).Moreover, the only invariant subset of the set { .L= 0} is {(0, • • • , 0, 0)}.It follows from the Lasalle Invariance Principle [8, p. 200] that, all paths in D approach the origin.Then the disease free equilibrium is globally asymptotically stable in D when R 1 ≤ 1.
Remark 2.12.Biologically, Theorem 2.11 means that when R 1 ≤ 1, the fraction of infected individuals vanishes, but when R 1 > 1 it will remain positive.
When R 1 > 1, the disease free equilibrium of System (2.9) is unstable.We conjecture that in this case System (2.9) admits an endemic equilibrium that is globally asymptotically stable in the interior of D. This conjecture is confirmed by simulations (Figure 4 (b) and (c)).

The asymptotic behaviour of the compartments sizes
For models with varying population size, the knowledge of the dynamic of the fractions is not enough.In fact, if a fraction go to 0 or to a positive number, there are three possible cases for the size.It may go to ∞, go to a positive number, or go to 0. Then it is important to study also the dynamics of the compartments sizes.We consider first the situation where the fractions disease free equilibrium is globally asymptotically stable in its feasible region, that is when R 1 < 1.In this case the fraction of infected individuals goes to zero, while that of susceptible individuals goes to 1. Therefore dN/dt −→ (b − µ)N, and then N −→ ∞ when t −→ ∞.Thus, Due to the results for the model with a unique latent class [12] and the simulations results (figures 5, 6, 7), we made the following conjecture.Conjecture 2.13.Let (S(t), E 1 (t), • • • , E n (t), I(t)) be a solution of System (2.1). (1 The proof of Conjecture 2.13 for a unique latent stage (n = 1) is done in [12].Now we consider the case where the fractions admits an endemic equilibrium that is globally asymptotically stable in the interior of its feasible region.
Theorem 2.14.Let's assume that R 1 > 1 and that System (2.8) admits an endemic equilibrium which is globally asymptotically stable in the interior of D and set Proof.Let's assume that R 1 > 1 and System (2.9) admits and endemic equilibrium We have the sign relation sign As the fractions approach an endemic equilibrium that is in the interior of D, the results follow.
Remark 2.15.Biologically, R 2 is the asymptotic reproduction number of the population, since the birth rate is b and the asymptotic death rate is µ + di * .
We have derived three threshold parameters R 0 , R 1 and R 2 that determine the dynamic of the fractions and that of the compartments sizes.The summary of the results is given in Table 2.

The model with treatment
Now we consider the model with treatment.To the n + 2 compartments of the model above, we add the compartment T of the individual under treatment.We assume that the individuals in T are not infectious and have not an additive death rate as those in I.The model diagram is given in Figure 2.

The model
The model with treatment is given by the following ODE system.

The initial reproduction number
Proof.R T is the product of the contact rate c, the probability to transmit the disease during a Remark 3.2.One gets R T from the formula that gives R 0 (Equation (2.2)) by substituting µ + Thus, we have R T < R 0 , the treatment reduces the reproduction number.
Let τ 1 , • • • , τ n , τ i be the respective fractions of the treated individuals in the compartments Therefore R T can be written as follow R T is the sum of n terms.(1 − τ i ) and (1 − τ 1 ) are factors of all the n terms.While For instance 1 − τ n is a factor of the last term only.
The biological meaning of this is that the treatment of an infectious individual or an individual in the latent class E 1 has an impact on all the infected compartments.While the treatment of an individual in the latent class E m has a direct impact on E m and the infected classes that come after E m only.Therefore, for an optimal strategy of treatment, the priority is to treat the infectious individuals and the people in the earliest stage of latency.
Let's define the treatment force γ by setting γ The critical treatment force to eradicate the epidemic is γ c := 1 − 1/R 0 .If γ > γ c , then the epidemic will dies out.But if γ < γ c , then the epidemic will go on in spite of the treatment.
Proof.The proof of Theorem 3.3 is similar to that of Theorem 2.3.
The feasible region is It is obvious that (1, 0, • • • , 0, 0) is the unique disease free equilibrium of System (3.8).The dynamic of System (3.8) depends on the threshold parameter R 1T given by (3.9) Theorem 3.8.The disease free equilibrium (1, 0, Proof.The proof of Theorem 3.8 is similar to that of Theorem 2.11 Biologically Theorem 3.9 means that if the relative treatment reproduction number R 1T is below one then the fraction of infected individuals will be negligible, while it will persist if R 1T is larger than one.Let's define the relative treatment force γ 1 by setting The critical relative treatment force is If γ 1 ≥ γ 1c then the fraction of infected individuals in the population vanishes, while it persists if γ 1 < γ 1c . When R 1T > 1 the disease free equilibrium of System (3.8) is unstable.The simulations that we made show that in this case there is one endemic equilibrium that is globally asymptotically stable in the interior of ∆ (Figure 12 (c) and (d)).
Conjecture 3.9.If R 1T > 1, then System (3.8) admits one and only one endemic equilibrium that is globally asymptotically stable in the interior of ∆.

The asymptotic behaviour of the sizes
For the asymptotic behaviour of the compartments sizes we have similar results as that of the model without treatment.The treatment compartment size T have the same behaviour as that of the infected compartments sizes.

Simulations
In this section we make numerical simulations to validate the theoretical results.We set n = 3, µ = 1, that is we assume that there are 3 latent classes and the life expectancy is the time unit.The other parameters and the initial values are chosen arbitrary to cover all the different scenarios that we have in the previous section.We use the software R and particularly the package deSolve [13] to integrate the ODE systems.

Simulations of the model without treatment
Let's consider first the case with constant solution for System (2.1).For Figure 3  We simulate now cases with varying population sizes.We start by integrating the fractions system.In Figure 4 we have in each case 10 solutions paths of System (2.8) starting at different initial values.In (a) and (b) where R 1 ≈ 0.58 and R 1 ≈ 0.98 respectively, all the 10 solutions paths approach the disease free equilibrium.This validates that when R 1 ≤ 1, the fractions disease free equilibrium is globally asymptotically stable in its feasible region.For (d) and (c), where R 1 ≈ 2.88 and R 1 ≈ 4.32, respectively, all the 10 solutions paths approach an endemic equilibrium.These results confirm that when R 1 > 1, the disease free equilibrium is unstable and that there is an endemic equilibrium that is globally asymptotically stable in the interior of the feasible region.The endemic equilibrium in (c) is different of that in (d).Thus, the endemic equilibrium depends on the parameters values.Now we integrate System (2.1) to validate the results on the asymptotic behaviour of the compartments sizes.We set (S(0), E 1 (0), E 2 (0), E 3 (0), I(0)) = (1000, 200, 100, 200, 100) for the initial values.
In Figure 5 we have the best scenario, the epidemic dies out, while the population goes on growing exponentially.The parameters values satisfy R 0 ≈ 0.82.It confirm that the epidemic cannot invade the population when the basic reproduction number is below one (R 0 < 1).It validates also Conjecture 2.13 (1).
In Figure 6 we have the case with R 0 = 1.We get R 0 = 1 by deducing cβ from the other parameters and using equation (2.2).The population grow exponentially, while the infected compartments sizes go to positive numbers.This simulation confirms Conjecture 2.13 (2).
In Figure 7 all the compartments grow exponentially.But the population growth rate is larger.Therefore, the fractions approach the disease free equilibrium.we have R 0 ≈ 1.24 and R 1 ≈ 0.84.Thus this result confirms Conjecture 2.13 (3).
In Figure 8 all the compartments grow exponentially, while the fractions approach an endemic equilibrium.We have R 0 ≈ 3.55 and R 1 ≈ 2.35.The asymptotic reproduction number of the population is R 2 ≈ 1.50.Thus this result complies with Theorem 2.14 (1).
In Figure 9 we have a case with R 0 ≈ 6.18 and R 1 ≈ 3.84.The fraction approach an endemic equilibrium such that the asymptotic growth rate of the population R 2 = 1.To get this, we have chosen the parameters values such that they satisfy Equation (2.6).All the compartments sizes stabilize.The epidemic has stopped the growth of the population.Thus Theorem 2.14 (2) is confirmed.
In Figure 10 we have the worse scenario, the population vanishes.The fraction approach an endemic equilibrium such that the asymptotic growth rate of the population is below one (1, 0, 0, 0, 0).For (c) cβ = 100, that gives R 1 ≈ 2.88; all the solutions approach the same endemic equilibrium (s            We simulate the fractions system (3.7) to validate the results on the dynamic of the fractions.
In Figure 12 we have in each case 10 solutions paths of System (3.7) starting at different initial values.In (a) and in (b) where we have respectively R 1T ≈ 0.45 and R 1T ≈ 0.99, all the 10 solutions approach the disease free equilibrium.these simulations confirm that if R 1T < 1 or equivalently if γ 1 ≥ γ 1c then the fractions disease free equilibrium is globally asymptotically stable in ∆.In (c) and (d) where R 1T > 1 the 10 solutions approach an endemic equilibrium.
Thus it confirm that if R 1T > 1 or equivalently if γ 1 < γ 1c then the fractions disease free equilibrium is unstable and there is an endemic equilibrium which is globally asymptotically stable in the interior of the feasible set ∆.These simulations validate Theorem 3.8 and Conjecture 3.9.
In Figure 13 we have the best scenario, the epidemic dies out, while the population goes on growing exponentially.The parameters values satisfy R 0 ≈ 3.55 and R T ≈ 0.92.It confirm that the epidemic cannot invade the population when R T < 1.It validates Conjecture 3.10 (1).
In Figure 14 all the compartments grow exponentially.But the population growth rate is larger.Therefore, the fractions approach the disease equilibrium.We have R 0 ≈ 4.25 and In Figure 15 all the compartments grow exponentially, while the fractions approach an endemic equilibrium.We have R 0 ≈ 7.09, R 1 ≈ 4.69, R T ≈ 1.85 and R 1T ≈ 1.52.The asymptotic reproduction number of the population is R 2T ≈ 2.37.Thus this result complies with Theorem 3.11 (3).
In Figure 16 the fractions approach an endemic equilibrium such that R 2T = 1.The sizes of all the compartments stabilize.In spite of the treatment, the epidemic has stopped the growth of the population.Thus this result complies with Theorem 3.11 (1).
In Figure 17 the fractions approach an endemic equilibrium such that R 2T = 0.97.The population vanishes.In spite of the treatment, the epidemic has turned the exponential growth of the population to an exponential decay.Thus this result complies with Theorem 3.11 (3).Now we simulate two epidemics with two different recovery rate θ for the treated individuals in order to check its impact on the epidemic.In Figure 18 we have two cases where all the parameters have the same values except that we set respectively θ = 15 and 150 in (a) and in    The simulations of the treatment model agree with the theoretical results found in Section 3.
In this section we have integrated numerically the different system studied in the previous section.The simulations results validate the theoretical results.Beyond the validations of the theorems, the conjectures are confirmed.

Discussions and conclusion
We have studied an SEIS model with n serial latent classes with a standard incidence in a population that grows exponentially before the introduction of the disease.The disease induces an additive death rate d for the infectious, affecting hence the dynamic of the population.We have defined the treatment force γ and the relative force γ 1 .Hence we deduced the critical treatment force γ c and the relative critical treatment force γ 1c .If γ > γ c the epidemic dies out.But if γ < γ c then the epidemic will continue in spite of the epidemic.An optimal treatment strategy is to treat in priority the infectious individuals and the individuals in the earliest stage of latency.
All the theoretical results are validated numerical simulations.
The recovery rate θ of the individuals in the compartment T does not intervene in the threshold parameters R T and R 1T .One may think that a high recovery rate (or equivalently a short period of treatment) is better for tackling the epidemic.But in fact when people recover from disease, they join the susceptible compartment increasing hence the number of new infectious contacts.The simulations we made confirm that in the case of endemicity, the fraction of infectious individuals increases with θ .But a long period of treatment means also more expenses, more places, and more personal in hospitals.Some authors assume that the disease induces additive death rates for latent individuals and Similarly to the recovery rate of the treated individuals, the additive death rate d T induced by the disease to the treated individuals does not intervene in these formula.
Our model ignore the limitations of the means of treatment.We have studied the epidemic using only a deterministic model.But a deterministic model fits only when we have a large number of individuals.Therefore, the situation where the population vanishes must be taken with caution.Because when the number of individuals in the population become small, the deterministic setting does not fit anymore.What happens then is very stochastic.The most realistic scenario is that the epidemic dies out first, and the population regrows thereafter.
people in the m th latent class, m = 1, ..., n I Infectious T People under treatment S, E 1 , • • • , E n , I, T Compartments sizes s, e m , i Fractions of the population in the classes S,E m ,I β Probability of transmitting the disease during a contact Transfer rate from E m into I ν m Transfer rate from E m into E m+1 , m = 1, • • • , n − 1 r m Treatment rate of a latent in E m r Treatment rate of an infectious δ Natural recovering rate θ Recovering rate of a treated individual R 0 Basic reproduction number R T Initial reproduction number (with treatment)

Figure 1 .
Figure 1.The transfer diagram of the SE n IS model with the susceptible class S, the n exposed classes E 1 ,...,E n and the infectious class I

Remark 2 . 1 .
By setting ν n = 0 in System (2.1), the formula dE m /dt = ν m−1 E m−1 − (k m + ν m + µ)E m applies also for m = n.Thus to simplify the formulas we set ν n = 0 in the following.From System (2.1) we have dN/dt = (b − µ)N − dI = (b − µ − di)N, where i = I/N is the fraction of the infectious.Then the population will go on growing whenever the fraction i of infectious individuals is less than (b − µ)/d.But the population would decrease if i grows beyond (b − µ)/d.
Let's consider the fractions s = S/N, e m = E m /N, m = 1, • • • , n and i = I/N.By System (2 i * ) which is globally asymptotically stable in the interior of D. Then we have dN/dt −→ (b − µ − di * )N.The asymptotic growth rate of the population is α = b − µ − di * .

Figure 2 .
Figure 2. The transfer diagram of the SE n ITS model with the susceptible class S, the n exposed classes E 1 , E 2 ,...,E n , the infectious class I and the treatment class T

Theorem 3 . 1 .
The initial reproduction number R T of the epidemic defined by System (3.1) is contact β , the average infectious time 1/(r + µ + d + δ ) and the probability for the exposed individual to become infectious ∑ m=n m=1 k m r m +µ+ν m +k m ∏ l=m−1 l=1 ν l r l +µ+ν l +k l .The latter is the sum of the probabilities to become infectious from the different latent compartments.More precisely k m r m +µ+ν m +k m ∏ l=m−1 l=1 ν l r l +µ+ν l +k l is the probability for the individual to become infectious while being in the compartment E m .

Remark 3 . 4 .Theorem 3 . 5 .
The RHS of Equation(3.3)  is the basic reproduction number of the SEIS model with one latent stage and treatment where k and r e denote respectively the activation rate and the treatment rate of the latent individuals[12].Let k e be the "effective" activation rate given by Equation (2.5).We define the effective "treatment" rate for latent individuals as the treatment rate r e for the treatment model with one latent stage with activation rate k e that has the same treatment initial reproduction number as that of the treatment model with n latent stages.By equations (3.2) and (3.3), one gets the following result.The effective treatment rate of the latent individuals is (3.4) r e = k e − (µ + k e ) m=n ∑ m=1 k m µ + ν m + k m + r m l=m−1

Remark 3 . 7 .
and S, one gets that X(t) is constant if and only if the initial values satisfy System (3.6) and the parameters satisfy Equation (3.5).When the parameters satisfy Equation (3.5), System (3.1) admits infinite equilibria and there is no stability.

3. 4 .
The dynamic of the fractions Now we consider the fractions of the populations in the different compartments, s = S/N, e 1 = E 1 /N, • • • , e n = E n /N, i = I/N and τ = T /N.By System (3.1) one gets (3.7) , the parameters values satisfy equation (2.6), and the initial values satisfy System (2.7).In fact for the initial values we have deduced S(0), E 1 (0), E 2 (0) and I(0) from E 3 (0), by System (2.7).In (a) as in (b) the sizes of all the compartments are constant.This confirms Theorem 2.6.(a)Dynamics of compartments sizes (b) Dynamics of compartments sizes
(a) Dynamics of compartments sizes (b) Dynamics of the compartments sizes
(a) Dynamics of compartments sizes (b) Dynamics of the fractions
(a) Dynamics of compartments sizes (b) Dynamics of the fractions
(a) Dynamics of compartments sizes (b) Dynamics of compartments sizes
Bowong et al. studied a Tuberculosis model with two differential infectivity and n latent classes.Moualeu et al. studied a Tuberculosis Model with n latent classes in , • • • , E 1 , I, S, one gets that E n , • • • , E 1 ,