# Dr. Muhammad Altaf Khan (City University of Science & Information Technology Peshawar, Pakistan)

Dr. Muhammad Altaf Khan (City University of Science & Information Technology Peshawar, Pakistan) is co-author of a recent paper which presents a deterministic model for dengue virus transmission and aims to investigate the optimal strategy for curtailing the spread of dengue. The model is parameterized using data from the 2017 dengue outbreak in Pakistan, uses optimal control theory and includes two time-dependent control variables determined from sensitivity analysis, insecticide use and vaccination. The results show that the two controls avert the same number of infections in the district considered regardless of the weights on the costs of insecticide use and vaccination, due to the reciprocal relationship between the cost of insecticide use and vaccination.

How was the optimal control theory applied in your model to investigate the best strategy for curtailing the spread of dengue?

The Optimal Control Theory is a powerful technique and it has uses for control systems not only in mathematical models of infectious diseases but also in engineering problems. In this case, we first formulated the model with vaccination and then without vaccination. Then, we evaluated the most appropriate controls for the model by checking the sensitivity of parameters. The sensitivity analysis of the model parameters and its effect on the basic reproduction number shows the possible strategy for dengue elimination.

The results of the sensitivity analysis of the model suggest that a control strategy will

adequately reduce the spread of dengue in the community if it reduces the following components: the transmission probability per contact of infectious mosquitoes with susceptible humans, the mosquito recruitment rate, the mosquito biting rate, the transmission probability per contact of susceptible mosquitoes with infectious humans, disease progression rate in mosquitoes, and vaccine waning rate.

Furthermore, a strategy that increases the human recruitment rate, disease progression rate in humans, human recovery rate mosquito natural death rate, and vaccine efficacy will be effective in curtailing the spread of dengue in the community.

Therefore, we introduced into the transmission model a time-dependent control variable the vaccination rate, which was previously taken as a constant. Although this parameter does not significantly impact the reproduction number, we nevertheless consider this parameter as a time-dependent variable since the vaccine efficacy has a significant impact on the basic reproduction number.

We also considered another control variable for the mosquito population since the mosquito death rate showed a strong significant impact on the basic reproduction number. The mosquito biting rate and transmission probabilities such as the transmission probability from exposed mosquitoes to susceptible humans, transmission probability from infected mosquitoes to susceptible humans, transmission probability from exposed humans to susceptible mosquitoes, transmission probability from infected human to susceptible mosquitoes are another set of candidate parameters to use as a time dependent controls. Based on these results we modelled the controls for dengue elimination.

Could you elaborate on the modelling you used?

Basically models of infectious diseases are usually developed on the basis of real life phenomenon or occurring epidemics. However, this model is formulated on the basis of the dengue literature and dengue epidemiology. Humans are split into categories: susceptible, exposed, vaccinated, infected and the recovered humans while the vector of dengue is split into susceptible, exposed and infected populations. The exposed humans as well as the exposed vectors also take part in the spread of dengue, as well as infected humans and infected mosquitoes. Based on these assumptions the basic model is formulated.

You used two time-dependent control variables determined from sensitivity analysis. These control variables are insecticide use and vaccination. What relationships between these two variables did you find?

We have studied using Optimal Control Theory the use of insecticide and vaccination as effective control measures against epidemics using data obtained from a hospital from the Peshawar District in Pakistan. The application of two time-dependent controls, insecticide use and vaccination, obtained from the sensitivity analysis can reduce the total number of infected humans and mosquitoes in the district; we observe the following results:

(a) A reciprocal relation exists between the cost of insecticide use and vaccination; as the cost of insecticide increases the use of vaccination increases. Only a slight increase in insecticide use is observed when vaccination level decreased due to increase in cost.

(b) The two controls averts about the same number of infections in the district regardless of the weights on the costs; this is due to the reciprocal relation between the cost of insecticide use and vaccination.

(c) Lower weights on the control are more cost-effective than higher weights with insecticide leading to cheaper costs than vaccination. Insecticide led to lower cost compare to vaccination at higher weights; therefore, insecticide is more cost-effect that vaccination at lower weight.

You have found a strong reciprocal relationship between the use of insecticide and vaccination: as the cost of insecticide increases the use of vaccination increases, and that the use of insecticide slightly increases when vaccination levels decrease. What implications do your findings have for the insecticide industry and dengue vaccination programmes?

The results obtained from the numerical solution of the model we did suggest that dengue can be eliminated from the population by following proper vaccination programmes, the mosquitoes elimination by insecticide and, at the same time, the personal protection for the human is required. The health authorities and the public health department could implement strategies discussed in our paper for proper elimination of the diseases and also the insecticide industry and dengue vaccine programmers. For the insecticide industry and dengue vaccine programmers, we advise them to collaborate by increasing coverage of the newly developed vaccine as well as developing new insecticide materials.

Can these two sectors work together and if so what is the rationale for doing so?

Our model simulations show that it is the use of insecticide and the dengue vaccine which could be most useful and effective for dengue elimination; their costs and reciprocal relationship should minimized the disease burden in the population. Vaccination and insecticide programmes can work together. For instance, the dengue vector bites during the day (morning and evening) and insecticides can be effective to decrease the number of mosquitoes where they present. In addition to this, personal prevention and protection, including vaccination as well as monitoring Aedes breeding environments are an essential part of controling dengue.

How would a World Dengue Day assist in combating this disease?

The current increasing number of dengue cases in the developing and developed countries is a big problem for the health authorities. Therefore a united, global response will be required to bring it under control. For this reason we, as dengue researchers (through mathematical modeling), physicians, and public health representatives from around the world propose the establishment of a World Dengue Day, to underscore the effect of dengue worldwide and to encourage a much-needed global response. We are inspired by the World Rabies Day and World TB Day which have raised awareness of these diseases among peoples and led to renewed momentum and greater intensity in the efforts towards their elimination. A World Dengue Day will acknowledge the burden borne by dengue-endemic countries and drive efforts to meet the challenge of dengue. Greater awareness of the disease, together with the development of new diagnostics, drugs and vaccines, will help to make dengue a preventable disease.

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Dr. Muhammad Altaf Khan is an Assistant Professor of mathematics, working in the department of mathematics, City University of Science and Information Technology Peshawar, Pakistan. In 2015, he obtained his PhD in Applied Mathematics from Abdul Wali Khan University, Mardan, Pakistan. His research interests includes infectious diseases modeling and their optimal control.