Seven challenges in modeling vaccine preventable diseases

Vaccination has been one of the most successful public health measures since the introduction of basic sanitation. Substantial mortality and morbidity reductions have been achieved via vaccination against many infections, and the list of diseases that are potentially controllable by vaccines is growing steadily. We introduce key challenges for modeling in shaping our understanding and guiding policy decisions related to vaccine preventable diseases.

Dynamics of influenza A drift: the linear three-strain model

We analyze an epidemiological model consisting of a linear chain of three cocirculating influenza A strains that provide hosts exposed to a given strain with partial immune cross-protection against other strains. In the extreme case where infection with the middle strain prevents further infections from the other two strains, we reduce the model to a six-dimensional kernel capable of showing self-sustaining oscillations at relatively high levels of cross-protection. Dimensional reduction has been accomplished by a transformation of variables that preserves the eigenvalue responsible for the transition from damped oscillations to limit cycle solutions.

The dynamics of cocirculating influenza strains conferring partial cross-immunity

We develop a model that describes the dynamics of a finite number of strains that confer partial cross-protection among strains. The immunity structure of the host population is captured by an index-set notation where the index specifies the set of strains to which the host has been exposed. This notation allows us to derive threshold conditions for the invasion of a new strain and to show the existence of an endemic multi-strain equilibrium in a special case. The dynamics of systems consisting of more than two strains can exhibit sustained oscillations caused by an overshoot in the immunity to a specific strain of cross-protection is sufficiently strong.

Pathogen coexistence induced by density-dependent host mortality

A model of two competing infectious diseases with complete cross-protection sharing one host allows for coexistence, provided that the host population is subject to strong density regulation. The phenomenon is caused by the different ways in which host density affects transmission rate and transmission period. The analysis suggest that disease coexistence is most likely when the two diseases differ significantly in virulence and transmission and that the evolutionary stability of the two-disease association depends critically on details in the functional relationship between virulence and transmission.

Slow coevolution of a viral pathogen and its diploid host

We study a population exposed to a lethal infectious disease. Host response is carried at one locus with two alleles while the pathogen occurs in two variants. Based on an SI-type epidemic model we derive explicit equations for the dynamics of each genotype. By assuming small variations in both host and disease, we obtain a separation in time scales between epidemic and evolutionary processes. This allows us to describe explicitly the changes in host and disease gene frequencies. The resulting model has a rich behaviour including multiple stable states and oscillations. However, in the oscillatory situation the model is degenerate excluding the possibility of limit cycles. We show that the degeneracy can only be removed by frequency dependent selection in the pathogen, for example by including direct interaction of virus in a free-living stage. The qualitative conclusions extend to an SIR-type epidemic model, where recovery with immunity from the disease is possible.

Disease-induced natural selection in a diploid host

Using for each genotype an SIR-type model of disease transmission dynamics, we describe natural selection in a continuously breeding diploid host whose disease susceptibility and resistance are carried at one locus with two alleles. The system is transformed into variables that for each disease class describe the number of individuals, the gene frequency, and the deviation from Hardy-Weinberg proportions as measured by Wright's fixation index. An assumption of small variation in disease response among genotypes (slow selection) separates the system to first-order into three blocks. One block describes the population-wide disease dynamics, the second considers the fixation index in each class, and the third block provides the change in gene frequencies. The first two blocks settle to equilibrium at a rate determined by the population turnover time while the last block after a while is dominated by a slowly changing variable, the average gene frequency. The dynamics of the gene frequency take the usual form for a continuous time slow selection model, and this provides explicit, epidemiologically justified expressions for the genotypic fitnesses. We apply the method to other disease transmission patterns (SEI and SIS) and discuss how suitable time averages extend our results to diseases with temporally varying incidence.

The effect of age-dependent host mortality on the dynamics of an endemic disease

Using asymptotic expansions in the ratio between the duration of infection and host lifetime, equilibrium conditions are analyzed for an SIR-type epidemic model with age-dependent mortality and age-independent disease transmission. Disease incidence at equilibrium depends on the distribution of lifetimes. Incidence is maximal if host life span is fixed and, for vanishing higher moments, it decreases with increasing variance of the distribution. The spectrum of the linearization about the endemic equilibrium has two dominant components, one near 0 and one with a large imaginary part. All roots of the characteristic equation have a negative real part so the model is always stable. The roots with a large imaginary part dominate in most cases, indicating that the approach to equilibrium will be through slowly damped oscillations.

Disease regulation of age-structured host populations

A lethal, contagious disease can generate a density-dependent regulation of its host, provided the hosts' contact rate grows with population size. The condition for disease-induced population control is that the expected number of offspring of an infected newborn be less than one. In vertebrates that acquired immunity if they survive infection, the disease changes the age structure of its host population. The steady-state age structure of a disease-regulated host with age-dependent fecundity is computed. Local stability analysis indicates that the equilibrium age structure is always stable. However, when the usual exponentially distributed duration of the disease is replaced by a constant duration, the population can exhibit oscillations with a long period.

Persistence of an infectious disease in a subdivided population

The transmission dynamics of a communicable disease in a subdivided population where the spread among groups follows the proportionate mixing model while the within-group transmission can correspond to preferred mixing, proportionate mixing among subgroups, or mixing between social and nonsocial subgroups, is analyzed. It is shown that the threshold condition for the disease to persist is that either (i) the disease can persist within at least one group through intragroup contacts, or--if (i) does not hold--(ii) the intergroup transmission is sufficiently high. The among-group transmission is computed as an average where each subgroup's reproductive number is weighted according to its intragroup activity level squared and the total number of cases that one infectious individual will cause through intragroup contacts. The model thus allows for a study of the relative importance of communitywide disease transmission and of disease transmission within geographically or socially separate groups.

Epidemiological models with age structure, proportionate mixing, and cross-immunity

Infection by one strain of influenza type A provides some protection (cross-immunity) against infection by a related strain. It is important to determine how this influences the observed co-circulation of comparatively minor variants of the H1N1 and H3N2 subtypes. To this end, we formulate discrete and continuous time models with two viral strains, cross-immunity, age structure, and infectious disease dynamics. Simulation and analysis of models with cross-immunity indicate that sustained oscillations cannot be maintained by age-specific infection activity level rates when the mortality rate is constant; but are possible if mortalities are age-specific, even if activity levels are independent of age. Sustained oscillations do not seem possible for a single-strain model, even in the presence of age-specific mortalities; and thus it is suggested that the interplay between cross-immunity and age-specific mortalities may underlie observed oscillations.