Epidemic models with differential susceptibility and staged progression and their dynamics.

We formulate and study epidemic models with differential susceptibilities and staged-progressions, based on systems of ordinary differential equations, for disease transmission where the susceptibility of susceptible individuals vary and the infective individuals progress the disease gradually through stages with different infectiousness in each stage. We consider the contact rates to be proportional to the total population or constant such that the infection rates have a bilinear or standard form, respectively. We derive explicit formulas for the reproductive number R0, and show that the infection-free equilibrium is globally asymptotically stable if R0>1 when the infection rate has a bilinear form. We investigate existence of the endemic equilibrium for the two cases and show that there exists a unique endemic equilibrium for the bilinear incidence, and at least one endemic equilibrium for the standard incidence when R0>1.


1.
Introduction. Variation of susceptible individuals, possibly caused by genetic factors, age, health, vaccination, or past exposure to the disease, many lead to their differentiation of susceptibility to infection. For example, the efficacy of available vaccinations for many infectious diseases is not perfect. Vaccinated individuals may still contract the disease and the susceptibility varies from individual to individual. Differential susceptibility of infection can occur after vaccination is administered for infectious diseases. Rubeola, more commonly known as the "red measles," is a highly contagious exanthematous viral illness. Prevention of disease is the most effective method of handling rubeola. Despite widespread vaccination programs, however, many women remain susceptible [16].
Implementation of the WHO guidelines for vaccination is universally recognized as one of the most efficient ways of preventing hepatitis B (HB) on a global scale. Vaccinated individuals impose life-threatening conditions on the virus. The induced anti-HBs is generally able to clear an invasion quickly and efficiently. However, if the virus produces mutants (vaccine escape mutants) that are not recognized by these antibodies and prevent them from eliminating the invaders, the vaccine is only partially effective. As a result, vaccinated individuals may still be differentially susceptible to the infection [6].
There is substantial biological evidence demonstrating that the presence of other sexually transmitted diseases (STDs) increases the likelihood of both transmitting and acquiring HIV. Individuals who are infected with other STDs are at least two to five times more likely than uninfected individuals to acquire HIV if they are exposed to the virus through sexual contact. In addition, if an HIV-infected individual is also infected with another STD, that person is more likely to transmit HIV through sexual contact than other HIV-infected persons [1,19,5].
Compartmental differential susceptibility (DS) susceptible-infective-removed (SIR) models were studied in [8] to gain insight into the transmission dynamics of diseases with differential susceptibility whereas it was assumed that the infectives are homogeneous such that there is one group of infectives. While this is true for some diseases, great variability in the infectiousness among infected individuals has also been shown in many studies of infectious diseases [10]. To include both variabilities of susceptibility and infectiousness, we formulate compartmental SIR models with differential susceptibility and staged progression (DSSP) in Section 2. We derive explicit formulas for the reproductive number R 0 using the method of next generation operator in Section 3. As the reproductive number R 0 < 1, we show that the infection-free equilibrium is not only locally, but also globally asymptotically stable for models with bilinear incidence by using a Liapunov function in Section 4. We also investigate existence of the endemic equilibrium whose components are all positive in Section 5. We discuss how the DSSP models can be applied to various situations in Section 6. 2. The model formulation. Suppose that an infectious disease spreads in a population consisting of susceptible, infective (exposed), and removed or recovered individuals. The susceptibles are divided into n groups based on their susceptibilities. A constant influx S 0 enters these susceptible groups and is distributed, based on the inherent susceptibilities of the individuals, in such a way that the input flow into group S i is p i S 0 , with n i=1 p i = 1. We assume that the infectives from susceptible group S i progress through m infection staged-subgroups, I i1 , I i2 , · · · , I im , i = 1, · · · , n, with different infection rates, such that the infected susceptible individuals enter the first subgroup I i1 and then gradually progress from subgroup I i1 finally to subgroup I im . We formulate our models such that if exposed individuals need to be considered in the model, the first subgroup I i1 stands for them with zero infectivity. The model, shown in Figure 1, can be described by the system of differential equations dI ij dt = γ i,j−1 I i,j−1 − (µ ij + γ ij )I ij , i = 1, · · · , n, j = 2, · · · , m,  where µ i is the natural death rate for susceptible individuals in group i, µ ij is the sum of the natural death rate and the disease-induced death rate for infected individuals in group i with infection stage j, λ i is the rate of infection to susceptibles S i , γ ij is the average rate of progress from subgroup I ij to subgroup I i,j+1 , for j = 1, · · · , m− 1, and γ im is the rate at which infectives in subgroup I im removed or recovered. We assume that the removed or recovered individuals, R i , with the death rate δ i , are no longer involved in the disease transmission or that their transmissions are negligible. We write The rate of infection, λ i , is the rate at which a susceptible individual in group S i gets infected and progresses to stage I i1 . It can be determined as where c(N ) is the average number of contacts per individual per unit of time that can be density-dependent, α i is the susceptibility of the individuals in susceptible group S i , β kj is the infectivity of infectives in group I kj , I kj /N is the proportion of infectives in group I kj , N k = S k + If the contact rate is proportional to the total active population N , such that c(N ) := cN , the rate of infection has the bilinear form The rate of change of all the infectives in group i is and the rate of change of the total population in group i satisfies where µ i = min m j=1 µ ij , i = 1, · · · , n. Hence the set Ω defined by Ω := 0 < N i < S 0 pi µi , i = 1, · · · , n is a global attractor for N i > 0, i = 1, · · · , n, and positively invariant for system (2.1) such that, if N i initially satisfies 0 < N i < S 0 pi µi , then it remains in this range. We restrict our investigation of the dynamics of system (2.1) in set Ω hereafter.
3. The reproductive number. System (2.1) has an infection-free equilibrium of which infective components are all zero and susceptible components are positive. Denote this infection-free equilibrium by E 0 . The local stability of E 0 determines epidemic threshold conditions under which the number of infectives will either increase or decrease to zero as a small number of infectives introduced into a fully susceptible population. These threshold conditions are characterized by the reproductive number, denoted by R 0 . The formula of R 0 can be determined by either investigating the eigenvalues of the linearized system about the infection-free equilibrium [7,12,17] or the next-generation operator method [3,2,18]. We derive an explicit formula for R 0 for the models with the standard incidence form as follows.
The Jacobian matrix at the infection-free equilibrium Here we writeD := n l=1 d l .
The local stability of the infection-free equilibrium is determined by matrix J. Using the next generation operator method, we define matrices F and V by Then F is a nonnegative matrix and V is an M -matrix. The reproductive number R 0 equals the spectral radius of the next generation operator F V −1 [18]: The inverse matrix V −1 , as shown in [9], is the lower triangular matrix given by where V ii = −J −1 ii , i = 1, · · · , m, and V ij are defined recursively by

JAMES M. HYMAN AND JIA LI
It follows from (3.4), iteratively, that where D 1 := diag (α 1 d 1 , · · · , α n d n ) and D 2j := diag β 1j b 1j σ 1j , · · · , β nj b nj σ nj . Then the rank of m j=1 P 1j V j1 is one, and hence the spectral radius of m j=1 P 1j V j1 equals its trace. Therefore the reproductive number for system (2.1) has the explicit formula Theorem 3.1. Let the reproductive number R 0 be defined as in (3.6). Then if R 0 < 1 the infection-free equilibrium E 0 is locally asymptotically stable, and if R 0 > 1 the infection-free equilibrium E 0 is unstable.

DIFFERENTIAL SUSCEPTIBILITY AND STAGED PROGRESSION MODELS 327
If we define the mean transmissibility for group i as and the reproductive number for group i by R 0i := rβ iτi , i = 1, 2, · · · , n, then the reproductive number for the entire population is a weighted average of the reproductive numbers for these groups such that It is similar to derive a formula of R 0 for the models with bilinear incidence form as
Then the infective components of system (2.1) satisfy the system dI dt = −LI + F = −LI + PUBI.

JAMES M. HYMAN AND JIA LI
Then C > 0 is a positive vector. We consider the Liapunov function which is positive definite for I > 0. The derivative of V along the system (2.1) equals
where lim t→∞ ε 2 (t) = 0. Notice that dV dt = 0 only if I = 0, in set Ω, if R 0 < 1. Therefore the infection-free equilibrium is globally asymptotically stable. 5. Endemic equilibrium. As the infection-free equilibrium becomes unstable and the disease spreads in the population, there possibly exists an endemic equilibrium with all components positive. We explore its existence for model (2.1) as follows.
Define function Then, if R 0 < 1 there exists no endemic equilibrium, and if R 0 > 1 there exists a unique endemic equilibrium for model (2.1).
For models with the standard incidence, we let W := r n i=1 m j=1 β ij I ij /N without confusion. Then λ i = W α i . Solutions for S i and I ij have the same formulas as in (5.2) and (5.5). Then substituting them into W , we obtain Define function We have Then if R 0 > 1 there exists at least one positive W such that Q(W ) = 0. That is, there exists at least one endemic equilibrium for model (2.1).
Moreover, it follows from , for all i = 1, · · · , m, then the endemic equilibrium is unique.
In summary, we have Theorem 5.1. If the infection rate has the bilinear form for model (2.1), then there exists no endemic equilibrium provided R 0 < 1, and a unique endemic equilibrium provided R 0 > 1. If the infection rate has the standard form for model (2.1), then there exists at least one endemic equilibrium provided R 0 > 1.
6. Concluding remarks. We have formulated the compartmental differential susceptibility and staged-progression DSSP epidemic models with either the standard or bilinear incidence. The models are a combination of the DS and the SP models, which have been studied intensively but separately. We derived explicit formulas for the reproductive number R 0 for the models with either the standard or the bilinear incidence, using the next generation operator method. The explicit formulas of R 0 for the models well fit in the calculations of R 0 for a variety of epidemiological models in the literature [7,2,13,4]. That is, the reproductive number for each subgroup, R 0i , is defined as a product of the mean number of contacts, the mean infectivity, and the mean duration of infection, and then the reproductive number for the entire population, R 0 , is defined as a weighted average of those R 0i .
For the models with the bilinear incidence, we showed that the infection-free equilibrium is not only locally but also globally asymptotically stable when R 0 < 1 by using a Liapunov function. We also showed that for these models, if R 0 < 1 there exists no endemic equilibrium, and if R 0 > 1 there exists a unique endemic equilibrium. Therefore, we exclude the possibility of the backward bifurcation. For the models with the standard incidence, we showed that if R 0 > 1 there exists at least one endemic equilibrium.
The DSSP model formulated in this paper can be applied to modeling of various disease transmissions. For example, suppose that HIV/AIDS spreads in a population with two groups, one of which consists of individuals who are infected with other STDs and one of which consists of individuals who are free of STDs before first engaging in sexual activities. As shown in [1,19,5], the presence of other STDs increases the likelihood of both transmitting and acquiring HIV. Individuals who are infected with other STDs are at least two to five times more likely than uninfected individuals to acquire HIV, and if an HIV-infected individual is also infected with another STD, that person is more likely to transmit HIV through sexual contact than other HIV-infected persons. We denote the two groups as group 1 and 2, respectively, and assume that all infectives go through 4 infection stages: an early, highly infectious pre-antibody phase, two chronic stages at low infectivities, and a final stage at higher infectiousness [10,14,11,15].
Let α 1 > α 2 and β 1j > β 2j , j = 1, · · · , 4, be the corresponding susceptibilities and infectivities. Then the reproductive number of infection is given by are the reproductive numbers for the two groups, and p is the fraction of the input flow of susceptibles into the group of individuals with infection of other STDs. Based on the formula of the reproductive number in (6.1), it is clear that a strategy to reduce the infection is to decrease the reproductive number for each group, in particular, for the group with the cofactor of other STDs. Meanwhile, it can help bring the transmission under control if the distribution of the input flow into group 1 is reduced which implies that more routinely screening for STDs should be advised for preventing the transmission of HIV/AIDS.
Due to the fact that the efficacy of available vaccinations for many infectious diseases is not perfect. Vaccinated individuals may still contract the disease and their susceptibility varies from individual to individual. To model such a situation, we can divide a population into two groups, one of which has all vaccinated individuals and one of which has all unvaccinated individuals. The infectives may progress the disease through several infection stages. Model (2.1) can be applied and the explicit formulas for the reproductive number can provide helpful guidance for the strategies of control and prevention for the disease.