1. In biology – Dynamics of Malaria Spread Background –Malaria is a tropical infections disease which menaces more people in the world than any other disease –WHO reported in 1978, in Africa alone, one million die annually from malaria before they reach the age of 5!! –It is conservatively estimated that malaria victims number around 100— 200 million annually. –Various schemes have been advocated for control of malaria but success has been limited. –For the control strategies, epidemiological knowledge is required, including the population dynamics leading to the spread of malaria.

2. Dynamics of Malaria Spread Mathematical models play two important roles –They help in clarifying the interaction between different variables of importance and lead to practical insights into the mechanisms underlying the observed phenomena –They are useful in making a quantitative evaluation of the impact of different control strategies, so helping in the selection of optimal strategies for controlling the disease. Aims: –To build models to understand the process –To develop a mathematical theory for the spread of malaria

3. Dynamics of Malaria Spread Biology of malaria –Malaria is an infectious parasitic disease which is transmitted at some stage in the life cycle of a relevant organism called parasite –The parasite are from the species of protozoa in the genus Plasmodium which are transmitted to human by female mosquitoes. –Thus the parasite has two hosts – humans and mosquitoes –The Plasmodium life cycle

4. Dynamics of Malaria Spread –Sporozoite stage: They first reside in the salivary glands of infected female mosquitoes. When an infected female mosquito bite an uninfected human, Plasmodium cells are injected into the human blood. They are carried to the liver where they complete their growth and multiply asexually to produce merozoites. Then they enter the red blood cells, reproduce further asexually and attack healthy red blood cells. When the concentration level reaches a certain high level, the symptoms, such as chill, shivering, various aches and pains, of malaria may appear in the host!! –Incubation period – time interval between the first bite and the appearance of clinical symptoms

5. Dynamics of Malaria Spread –Spread stage: When an uninfected female mosquito bites an infected human, the female mosquito is infected and it can spread malaria to another human. System characterization –Variables and interactions Human population –Two groups – infected & uninfected –Three groups – infected, susceptible & immune –Two categories with seven groups

6. Dynamics of Malaria Spread

7. Mosquito population Climate and geographic factors Mosquito/human interaction Immunity –Deterministic vs stochastic characterization –Time scales Time for sporozoite to enter the liver about 30 minutes Incubation period about 10—30 days Cycles of attack about 1---3 days

8. Dynamics of Malaria Spread Extrinsic incubation period –About 10—11 days at 25 degree –About 15—17 days at 23 degree –About 22 days at 20 degree –Below 19 degree or about 32 degree, no survival Life cycle of mosquito about 14 days Life cycle of human about 70—80 years Duration of immunity –Let T be the time interval of study, It should be be 1—10 years –Discrete vs continuous time characterization

9. Mathematical modeling The modeling of spread of malaria is difficult –The mechanism for the spread of malaria is very complicated; hence great care needs to be taken to ensure a proper trade- off between the reality and the complexity of the model –The model parameters depend on geophysical location. Hence a model with a specified parameter set value for a particular region might not be valid for another region. –The data available for parameter estimation, as well as for validation, are often limited and inadequate –The problem of validation is difficult and requires combining forecasting abilities of the model with biological and epidemiological considerations.

10. Mathematical modeling A variety of models have been built to model the spread of malaria through the population First model in 1916 by Sir Ross Up to now, there 5 typical models –Model I, ODE model by Ross in 1916 –Model II, modified ODE model by Mcdonald in 1950 –Model III, ODEs model by Dietz in 1970 –Model IV ODEs model for seven groups by Dietz in 1974 –Model V, probabilistic model by Bekessy in 1976

11. Model I (Ross, 1916) Assumptions –The changes to the human population are small over the time interval of study, we ignore the variations and treat the total population as being constant –The number of mosquitoes over a season is being treated as constant. –The total population is divided into two groups – uninfected and infected –We don’t differentiate between infectious and non-infectious in the group of affected people, nor between susceptible and immunes in the unaffected group –The proportion of affected people who recover in a unit time is constant –The proportion of unaffected people receiving infectious bites per unit time is constant

12. Model I Variables –t: time –x(t): proportion of human population affected by malaria at time t, and the proportion unaffected is 1-x(t) Parameters –r: the recovery rate, i.e. proportion of affected people who recover in a unit time –h: affected rate, i.e. proportion of unaffected people receiving infectious bites per unit time

13. Model I Changes to x(t) over time interval Change rate Rate of change Initial data, i.e. the proportion of infected at time t=0

14. Model I Analytical solution Limiting affected rate, proportion of infected population under what is called endemic condition Special cases –The whole population is infected –The infection disappears –There is no change

15. Model I Application to newborn children, i.e. x(t) represents the proportion of newborn population affected by malaria at time t & Let p(t) represent the population of infected people detected at time t using some testing procedure. –If the testing procedure is perfect, then p(t)=x(t) –In general, if an infected person is detected with probability k (0<k<1), then we have

16. Model I Parameter estimation (for r, h & k) –By observing the spread of malaria in a cohort of newborn children, we can use the model to obtain estimates for the parameters of the model. –Let n i represent the number of children observed, a i the number found to be infected at time t i for –Estimates the parameters by minimizing

17. Model I

18. Model Parameters estimate Since r is very small, we can treat it as being zero for young children and regard k and h as the two unknown parameters, new estimates

19. Model I

20. Model II (Mcdonald 1950) This model was built to overcome the deficiency of Model I, which yielded very low values for r when fitted to real data. This model is built using the concept of superinfection –When h<r, the amount of new infection per time is fixed –When h>r, one infected the individual never recover

21. Model III (Dietz 1970) Use the concept of superinfection The characterization is done through the total population being divided into different groups based on the number of broods present in an individual. Variables –t: time –x(t): proportion affected by malaria at time t –f j (t): proportion of the total population with exactly j broods.

22. Model III In a small time interval, a change in f j (t) can occur when one or more persons in group (j-1) is bitten by an infections mosquito resulting in an increase in the number of broods by one in the bitten individual, or when one or more persons in group (j+1) recovers partially so that the number of broods in the infected person is reduced by one. Let r define the recovery rate for individual brood and h be the rate of new infections being introduced.

23. Model III Model --- ODE system Initial data Solution for a cohort of newborn babies via discrete Laplace transform

24. Other models Model IV (Dietz etc. 1974) –Seven groups model Model V (Bekessy etc, 1976) –Probabilistic model