Simple Infection Model Huaizhi Chen

Simple Model The simple model of infection process separates the population into three classes determined by the following functions of age and time: X(a,t) = susceptible population Y(a,t) = infected population Z(a,t) = immune population

Governing Equations The dynamics of the simple model is governed by the following partial differential equations:

Parameters Where

Boundary Conditions We have the following initial conditions: At a = 0 – Y(t)=Z(t)=0, X(t) = B(t), where B(t) is net birth rate at t At t = 0 – X(a), Y(a), Z(a) must be defined.

Additional Classes Other Classes Can be Incorporate into the model – Take Latent Period: We can divide the Y period to H and Y’ Where H is the latent period and Y’ is the infectious period

Latent Period Where we have a new parameter representing the per capita transfer from latent infected to infectious infected.

Maternal Antibodies Temporary immunity can be granted to newly born infants from an immune mother. This can be incorporated into the simple model.

Verticle Transmission Some infections can be passed directly to the new-born offspring of an infected parent. This phenomenon can be represented by tweaking the boundary condition at a = 0 and have X(0,t) = B 1 (t), Y(0,t) = B 2 (t), and Z(0,t) = 0.

Separate Treatments of Male/Female Often, for sexually transmitted diseases, it would be helpful to stratify the variables by sex. Like - X m, X f, Y m, Y f, Z m, Z f and govern those classes with separate dynamics.

Recovery v(t) can be modeled in various manners. Type A (constant) and Type B (step) Proportion Infected

Loss of Immunity We can also modify the simple model to incorporate loss of immunity by incorporate the parameter gamma.

Natural Mortality Mortality can be modeled similarly to recovery. Type I and Type II

Disease-Induced Mortality Generally a constant is used in place of the alpha function; however depending on the disease, it may be advantageous to model alpha by age. For example, malaria and measles exhibit greater mortality in infants.

Transmission The transmission parameter can be modeled by: Where beta is the probability that an infected individual of age a would infect a susceptible of age a’. A specific case with constant probability is

Other Concerns Seasonality – The beta parameter in the transmissions equation can be very seasonal. Nutritional State – Nutritional state of the population can exert changes in the mortality, transmission, and other rate parameters. Homogenous Mixing – Examples have shown important effects of heterogeneity in most real populations.